APPENDIX DATA TABLES FROM
EXPERIMENTAL TECHNIQUES FOR LOW-TEMPERATURE MEASUREMENTS:
Cryostat Design, Material Properties, and Superconductor Critical-Current Testing
JACK W. EKIN
National Institute of Standards and Technology
Boulder, Colorado
Published by Oxford University Press
First Printing 2006, Second Printing 2007, Third Printing 2007, Fourth Printing 2011
Each section is keyed to the chapter of the corresponding number.
Use “Control F” to rapidly find material
1 General information and cryogen properties
A1.1 Term–abbreviation–acronym decoder
A1.2 Fundamental constants
A1.3 SI conversion factors
A1.4 Magnetic units: equivalency table
A1.5 Properties of common cryogenic fluids
A1.6a Cooling power data for 4He, H2, and N2
A1.6b Cooling power data: Amount of cryogenic fluid needed to cool common metals
A1.7 Suppliers of specialty parts and materials
2 Heat transfer
A2.1 Thermal conductivity integrals for technical cryostat materials
A2.2 Emissivity of technical materials at a wavelength of about 10 µm (room temperature)
A2.3 Heat conductance across solid interfaces pressed together
3 Cryostat construction
A3.1 High-thermal-conductivity construction-metal properties: RRR, thermal conductivity,
and electrical resistivity
A3.2 Heat conduction along thin-walled stainless-steel tubing
A3.3 Pipe and tubing sizes
A3.4 Screw and bolt sizes, hexagon socket-head sizes, and load limits
A3.5 Clearances for various types of fits
A3.6 Common braze materials
A3.7 Solder: physical properties
A3.8 Solder fluxes for soft-soldering common metals and alloys
A3.9 Solder: superconducting properties
A3.10 Sticky stuff for cryogenic applications
A3.11 Slippery stuff for cryogenic applications
A3.12 Degassing rates of synthetic materials
A3.13 Vapor pressures of metals
A3.14 Gas permeation constant at room temperature for synthetic materials
4 Cryogenic apparatus wiring
A4.1a Wire gauge size, area, resistivity, heat conduction, and optimum current
A4.1b Wire gauge: Metric and American Wire Gauge (AWG) size comparison
A4.2 Physical properties of common wire materials
A4.3 Residual resistance ratio (RRR) of selected wiring and conductor materials
A4.4 Wire insulation: thermal ratings
A4.5 Thermal anchoring: required wire lengths
A4.6a Thermoelectric voltages of some elements relative to copper
A4.6b Thermoelectric voltages of selected technical materials relative to copper
A4.7 Thermal conductivity of YBCO-coated conductors
5 Temperature measurement tables and controller tuning
A5.1 Vapor pressure vs. temperature (ITS-90) for cryogenic liquids
A5.2 Properties of cryogenic thermometers (~1 – ~300 K)
Data Handbook of Material Properties and Cryostat Design
Appendix Table of Contents
A5.3a Platinum thermometer resistivity vs. temperature above 70 K
A5.3b Platinum thermometer resistivity vs. temperature below 70 K
A5.4 Diode and thermocouple voltage-vs.-temperature tables
A5.5 Magnetic-field correction factors for platinum resistance thermometers
A5.6 Magnetic-field correction factors for zirconium–oxynitride resistance thermometers
A5.7 Temperature-controller tuning with the Ziegler–Nichols method
6 Properties of solids at low temperature
A6.1 Elements: physical properties at room temperature
A6.2 Specific heat vs. temperature for technical materials
A6.3 Debye model values of the molar heat capacity and molar internal
energy as a function of temperature
A6.4 Thermal expansion/contraction of technical materials
A6.5a Ideal electrical resistivity vs. temperature for pure metals
A6.5b Total electrical resistivity vs. temperature for technical alloys and common solders
A6.6 Superconductor properties
A6.7 Thermal conductivity vs. temperature for selected metals, alloys, glasses, and polymers
A6.8a Magnetic mass susceptibility from 1.6 K to 4.2 K of materials commonly used in
cryostat construction
A6.8b Magnetic volume susceptibility at 293 K, 77 K, and 4.2 K of structural materials
commonly used in cryostat construction
A6.8c Ferromagnetic traces at 4.2 K induced by welding and cyclic cooling of
austenitic stainless steels
A6.9 Composition of austenitic stainless steels, nickel steels, and aluminum alloys
A6.10 Mechanical properties of structural materials used in cryogenic systems
7 (i) Specialized resistivity measurement methods
A7.1 Sheet-resistance measurement of unpatterned films
A7.2 van der Pauw method for measuring the resistivity and Hall mobility in
flat isotropic samples of arbitrary shape
A7.3 Montgomery method for measuring the resistivity of anisotropic materials
(ii) Sample-holder material properties
A7.4 Sample-holder materials: thermal contraction on cooling to liquid-helium and
liquid-nitrogen temperatures
A7.5 Superconductor materials: thermal contraction on cooling to liquid-helium
and liquid-nitrogen temperatures
A7.6 Thin-film substrate materials: thermal conductivity and thermal contraction
A7.7 Ultrasonic wire-bond material combinations
8 Sample contacts
A8.1 Overview of contacts for low-Tc and high-Tc superconductors
A8.2 Contact methods for voltage and current connections to bare YBCO superconductors
A8.3 Optimum oxygen-annealing conditions for silver and gold contacts
to Y-, Bi-, and Tl-based high-Tc superconductors
A8.4 Bulk resistivity of common solders, contact pads, and matrix materials
A8.5a Argon ion milling rates of elements
A8.5b Argon ion milling rates of compounds
10 Critical-current analysis parameters
A10.1 Effective critical temperature T*c(B)
A10.2a Scaling parameters for calculating the magnetic-field, strain, and temperature
dependence of the critical current of low-Tc superconductors
A10.2b Summary of scaling relations for utilizing the scaling parameters in Appendix 10.2a
APPENDIXES
DATA HANDBOOK OF MATERIAL PROPERTIES AND CRYOSTAT DESIGN
The following tables provide handbook data for cryostat design and measurements. In many cases the tables serve as a ready collection of general design information for quick reference, including SI conversion factors sorted by function, cooling power data, suppliers of specialty parts, heat conduction down stainless-steel tubing, strengths of bolts, metric equivalents, vacuum-design data, wire properties, magnetic-correction factors for thermometers, and so on. In other cases, the tables provide a convenient aid in material selection, presenting, at a glance, a condensed overview of the temperature dependence of the properties of many materials (sorted by type of material and property). Once a material is selected, more detailed information can then be obtained for that particular material by referring to the extensive references accompanying each table or to the Internet data sources given in Sec. 6.7.2.
The appendix tables are divided into categories corresponding to each chapter, starting with general information and properties of cryogenic fluids, continuing with heat transfer, construction, wiring, thermometers, properties of solids, sample holders, contacts, and ending with information useful for critical-current analysis. The text sections indicated in parentheses with each appendix table contain specific information on the application and interpretation of the data in that table.
A1. General information and cryogen properties (ref. Chapter 1)
A1.1 Term-abbreviation-acronym decoder
When starting in a new field, the jargon can sometimes be daunting. The following is only an brief introductory list, given in rractical terms, but it may be useful to help clarify a few of the more commonly used terms, abbreviations, and acronyms.
An accessible multilingual website containing introductory information on polymers and their designations is http://www.pslc.ws/macrog/index.htm.
Default nomenclature: Alloy compositions in this book are given in weight percent (for example: 2wt%Al or 2%Al) unless specifically indicated as atomic percent (e.g., 2at%Al).
AISI: American Iron and Steel Institute, a designation system for steel alloys (Appendix A6.9)
Alumel: A high electrical-resistivity nickel alloy used for thermocouples consisting of Ni–2%Al–2%Mn–1%Si.
ASTM: Formerly known as the American Society for Testing and Materials, ASTM International provides a global forum for consensus standards for materials, products, systems, and services.
AWG: American Wire Gauge, a designation system for wire size. Appendix A4.1a lists physical information by AWG wire size. Corresponding metric wire sizes are given in Appendix A4.1b.
Bi-2212: A common abbreviation for the high-Tc superconductor material Bi2Sr2CaCu2O8+ (where, typically, << 1). The name is derived from the subscripts of the first four elements in the compound formula. This superconductor is also sometimes referred to as BSCCO (pronounced “bisco”). (Tc ≈ 90 K) Bi-2223: A common abbreviation for the high-Tc superconductor (Bi,Pb)2Sr2Ca2Cu3O10–; also sometimes referred to as BSCCO. (Tc ≈ 110 K) Chromel: A high electrical-resistivity nickel alloy commonly used for resistive wiring and thermocouples, consisting of Ni–10%Ni. Constantan: Another high electrical-resistivity alloy commonly used for resistive wiring and thermocouples, consisting of Cu–45%Ni. Critical current Ic: The maximum amount of current that can be carried by a superconductor before it starts to become resistive. Good commercial superconductors can carry over 1000 A/mm2 in the presence of a 12 T magnetic field applied perpendicular to the wire. Critical magnetic field: There are several definitions of critical magnetic field, described in Sec. 10.3.1. Generally, the so-called upper critical field Hc2 is the practical quantityfor low-Tc superconductors, corresponding to the magnetic field above which all superconductivity is suppressed. Hc2 values at 0 K for low-Tc materials range up to about 30 T, and for high-Tc materials to over 100 T. Typical values are tabulated for practical superconductors in Appendix A6.6 and plotted vs. temperature in Fig. 10.15. The so-called irreversibility or depinning field Hirr is the practical quantity for high-Tc superconductors, plotted vs. temperature in Fig. 10.16. Critical temperature Tc: The temperature below which a superconductor must be cooled before it becomes superconducting. Typical values for practical low-Tc materials range from about 10 K to 40 K; for high-Tc materials, from about 90 K to 130 K. Values are tabulated for the most common superconductors in Appendix A6.6. Cryocooler: A cryogenic refrigerator. Cryogen: Another name for a cryogenic liquid, such as liquid helium (Tboil = 4.2 K), liquid neon (Tboil = 27 K), or liquid nitrogen (Tboil = 77 K). The physical properties of common cryogens are given in Appendix A1.5. CTFE: Polychlorotrifluoroethylene, a type of Teflon. ELI Ti–6Al–4V: Extra Low Interstitial form of Ti–6Al–4V. The mechanical properties of titanium strongly depend on interstitial elements (especially oxygen, nitrogen, and carbon), which affect particularly the fracture toughness. ELI grade is a purer form of titanium with a greater fracture toughness. Ethylene glycol dimethyl terepthalate: Mylar ETP copper: Electrolytic-Tough-Pitch copper [designated by the Unified Numbering System (UNS) as C10300]. This is the copper commonly used to make ordinary copper wire. Eutectic mixture: The alloy composition with the lowest melting-temperature; eutectic compositions are particularly useful as solder materials. FEP: Fluorinated ethylene propylene, a type of Teflon G-10, G-11: Designations for fiberglass-epoxy composites (commonly used as commercial electronic circuit boards) made from layers of fiberglass cloth filled with epoxy. Hastelloy C: A corrosion-resistant nickel alloy consisting of 54%Ni–17%Mo–15%Cr–5%Fe–4%W. HTS: High-Tc (high-critical-temperature) superconductors. Copper-oxide materials having critical temperatures ranging to over 100 K. Also referred to as oxide superconductors or ceramic superconductors. Examples are: YBa2Cu3O7–x (Tc = 92 K), Bi2Sr2CaCu2O8+x (Tc = 85 K), (Bi,Pb)2Sr2Ca2Cu3O10–x (Tc = 110 K), (Tl,Pb)1(Ba,Sr)2Ca2Cu3O10+x (Tc = 115 K), and HgBa2Ca2Cu3O8+x (Tc = 135 K). ISO: International Standards Organization. ITS-90: The International Temperature Scale of 1990. KF flange: Klein Flange, meaning “small flange;” a flexible O-ring vacuum coupling. LTS: Low-Tc (low-critical-temperature) superconductors. These materials (usually with niobium as the core element) have critical temperatures up to about 40 K and are based on a phonon-coupling mechanism between superconducting pairs of electrons. Common examples are: Nb–Ti (Tc = 9.5 K), Nb3Al (Tc = 15 K), Nb–N (Tc = 16 K), Nb3Sn (Tc = 18 K), Nb3Ge (Tc = 23 K), and MgB2 (Tc = 39 K). Lambda point: The temperature (2.177 K) where normal 4He (also designated as He I) transforms into superfluid helium 4He (also designated as He II); see Sec. 1.2.2. Manganin: An alloy commonly used for cryostat wiring and heaters in nonmagnetic applications, consisting of Cu–13%Mn–4%Ni. Martensitic phase transformation: A change in the atomic structure of a metal to a new crystalline phase that is usually harder and more brittle. In stainless steels commonly used in cryogenic apparatus, such as AISI 304, 310, and 316, the martensitic phase transformation is precipitated by cooling to low temperatures or by applied stress (Sec. 6.6.5). The martensitic phase of the metal has a lower fracture toughness and is usually ferromagnetic. Monel: A high-strength, corrosion-resistant, nickel alloy consisting of Ni–30%Cu. n value: An index of the nonlinearity or sharpness of the voltage–current (V–I) curve near the critical current of a superconductor. It is defined by the relation V = c I n (Sec. 10.1.3). Good superconductors have n values above 20 to 30. Nichrome: A highly resistive alloy commonly used for heater wiring, consisting of Ni–20%Cr. OFHC copper: A type of oxygen-free copper [designated by the Unified Numbering System (UNS) as C10200; a higher-purity type is designated as C10100]. PCTFE: Polychlorotrifluoroethylene. PET: Polyethylene terephthalate, Mylar. Phosphor bronze: An alloy commonly used for cryostat and thermometer instrumentation wiring composed of Cu–5%Sn–0.2%P (Grade A). PMMA: Polymethyl methacrylate, Plexiglas. Polyamide: Nylon. Polyimide: Kapton. Phonon: A wave-like displacement of the atoms from their equilibrium positions in a solid, usually thermally generated. Poisson’s ratio: A term used in mechanics (Secs. 3.5.3 and 3.5.4) that is the (negative of the) ratio of the lateral strain to longitudinal strain when a beam is uniformly and elastically stressed along the longitudinal axis. (It simply expresses the fact that the beam becomes narrower as it is stretched, to approximately conserve its volume.) The Poisson’s ratio of metals is typically about 1/3, with values ranging from 0.28 to 0.42 for most materials. PTFE: Polytetrafluoroethylene, a type of Teflon. Quench: A colloquial term for a thermal runaway event; see Thermal runaway. SI: The international system of units (Système International d’Unités). SQUID: Superconducting Quantum Interference Device. A very sensitive magnetometer able to detect magnetic flux as small as a fraction of magnetic flux quantum 2.0678 10–15 Wb). TFE: Tetrafluoroethylene, Teflon. Thermal runaway (quench): A process wherein a small part of a superconductor carrying very high current densities is momentarily heated into the resistive state (by sample movement, friction, or some other disturbance). The resulting electrical (Joule) heating in this portion of the superconductor then heats additional surrounding superconductor material into the resistive state, resulting in a thermal-runaway process with an ever-growing resistive zone and rapidly increasing Joule heating. When measuring the critical current of a superconducting strand, the Joule heating typically locally melts the superconductor unless the current is shut off quickly (in less than a second). More information is given in Sec. 7.5.1. Type I superconductors: Superconducting materials where magnetic field uniformly penetrates the material, suppressing superconductivity at relatively low magnetic fields (typically much less than 1 T). This is the original type of superconductivity discovered in 1911 by Onnes. It was not until nearly fifty years later that practical (high-field) Type II superconductivity was discovered. Type II superconductors: Superconductors wherein magnetic field is localized by circulating supercurrents, confining the field to small regions (vortices) and thereby leaving most of the superconductor free of magnetic field; see Fig. 10.7. This second type of superconductivity, which was discovered half a century after Type I superconductors, allows superconductivity to persist to much higher magnetic fields and comprises the practical superconducting materials from which most of today’s applications are fabricated. Paradoxically, Type II superconductors have a much lower electrical conductivity in the normal (nonsuperconducting) state than that of Type I superconductors (which are typically pure metallic elements). YBCO: The term commonly used for the high-Tc material YBa2Cu3O7– (Tc ≈ 92 K), also sometimes referred to as simply 123 because of the subscripts of the first three elements in the compound. A1.2 Fundamental constants Fundamental Physical Constants a Quantity Symbol Value Avogadro constant NA 6.022 141 99 1023 mol–1 Boltzmann constant kB = R/NA 1.380 650 3 10–23 J∙K–1 electric constant = 1/µc2 8.854 187 817 10–12 F∙m–1 electron volt eV 1.602 176 463 10–19 J elementary charge e 1.602 176 463 10–19 C Lorenz constant (Sec. 6.4.2) LN = (2/3) (kB/e)2 2.443 10–8 V2∙K–2 magnetic flux quantum = h/2e 2.067 833 637 10–15 Wb molar gas constant R 8.314 472 J∙mol–1∙K–1 = 8.314 472 Pa∙m3∙mol–1∙K–1 magnetic constant µ 4π 10–7 = 1.2566 10–6 N∙A–2 = 1.2566 µV∙s∙A–1∙m–1 = 1.2566 µWb∙A–1∙m–1 = 1.2566 µH∙m–1 Newtonian constant of gravitation G 6.673 10–11 m3∙kg–1∙s–2 Planck’s constant h 6.626 068 76 10–34 J∙s speed of light in vacuum c 2.997 924 58 108 m∙s–1 Stefan–Boltzmann constant (Sec. 2.4) = (π2/60)kB4/(h/2π)3c2 5.670 400 10–8 W∙m–2∙K–4 ______________ a From CRC Handbook of Chemistry and Physics (2002), 83rd edition, CRC Press LLC, Boca Raton, Florida. Useful approximate equivalents: Pressure: 1 atm ( 760 torr) ~105 Pa Temperature: 11 000 K ~1 eV Wavelength: 12 000 Ǻ ~1 eV A1.3 SI conversion factors SI: Système International d’Unités (International System of Units) To convert from to multiply by ACCELERATION ft/s2 meter per second2 (m/s2) 3.048 000 E–01 free fall, standard meter per second2 (m/s2) 9.806 650 E+00 in/s2 meter per second2 (m/s2) 2.540 000 E–02 AREA acre meter2 (m2) . 4.046 873 E+03 barn . meter2 (m2) . 1.000 000 E–28 circular mil meter2 (m2) 5.067 075 E–10 ft2 meter2 (m2) 9.290 304 E–02 in2 meter2 (m2) 6.451 600 E–04 mi2 (U.S. statute mile) meter2 (m2) 2.589 988 E+06 section meter2 (m2) 2.589 988 E+06 township meter2 (m2) 9.323 957 E+07 yd2 meter2 (m2) 8.361 274 E–01 BENDING MOMENT OR TORQUE dyne∙centimeter newton∙meter (Nm) 1.000 000 E–07 kgf∙m newton∙meter (Nm) 9.806 650 E+00 ozf∙in newton∙meter (Nm) 7.061 552 E–03 lbf∙in newton∙meter (Nm) 1.129 848 E–01 lbf∙ft newton∙meter (Nm) 1.355 818 E+00 CAPACITY (see VOLUME) DENSITY (see MASS PER UNIT VOLUME) ELECTRICITY AND MAGNETISM ampere hour coulomb (C) 3.600 000 E+01 EMU of capacitance farad (F) 1.000 000 E+09 EMU of current ampere (A) 1.000 000 E+01 EMU of electric potential volt (V) 1.000 000 E–08 EMU of inductance henry (H) 1.000 000 E–09 EMU of resistance ohm () 1.000 000 E–09 ESU of capacitance farad (F) 1.112 650 E–12 ESU of current ampere (A) 3.335 6 E–10 ESU of electric potential volt (V) 2.997 9 E+02 ESU of inductance henry (H) 8.987 554 E+11 ESU of resistance ohm () 8.987 554 E+11 gauss tesla (T) 1.000 000 E–04 gilbert ampere (A) 7.957 747 E–01 maxwell weber (Wb) 1.000 000 E–08 oersted ampere per meter (A/m) 7.957 747 E+01 ENERGY (includes WORK) British thermal unit (thermochemical) joule (J) 1.054 350 E+03 calorie (thermochemical) joule (J) 4.184 000 E+00 electron volt joule (J) 1.602 176 E–19 erg joule (J) 1.000 000 E–07 ft∙lbf joule (J) 1.355 818 E+00 kilocalorie (thermochemical) joule (J) 4.184 000 E+03 kW∙h joule (J) 3.600 000 E+06 W∙h joule (J) 3.600 000 E+03 W∙s joule (J) 1.000 000 E+00 FLOW (see MASS PER UNIT TIME or VOLUME PER UNIT TIME) FORCE dyne newton (N) 1.000 000 E–05 kilogram-force (kgf) newton (N) 9.806 650 E+00 kilopond-force newton (N) 9.806 650 E+00 kip (1000 lbf) newton (N) 4.448 222 E+03 ounce-force (avoirdupois) newton (N) 2.780 139 E–01 pound-force (lbf) newton (N) 4.448 222 E+00 poundal newton (N) 1.382 550 E–01 FORCE PER UNIT AREA (see Pressure) FORCE PER UNIT LENGTH lbf/in newton per meter (N/m) 1.751 268 E+02 lbf/ft newton per meter (N/m) 1.459 390 E+01 HEAT Btu (thermochemical)in/sft2F (k, thermal conductivity) watt per meter kelvin (W/mK) 5.188 732 E+02 Btu (thermochemical)in/hft2F (k, thermal conductivity) watt per meter kelvin (W/mK) 1.441 314 E–01 Btu (thermochemical)/ft2 joule per meter2 (J/m2 ) 1.134 893 E+04 Btu (thermochemical)/h∙ft2∙F (C, thermal conductance) watt per meter2 kelvin (W/m2K) 5.674 466 E+00 Btu (thermochemical)/lb joule per kilogram (J/kg) 2.324 444 E+03 Btu (thermochemical)/lb∙F (c, specific capacity) joule per kilogram kelvin (J/kgK) 4.184 000 E+03 Btu (thermochemical)/s∙ft2∙F watt per meter2 kelvin (W/m2K) 2.042 808 E+04 cal (thermochemical)/cm2 joule per meter2 (J/m2) 4.184 000 E+04 cal (thermochemical)/cm2∙s watt per meter2 (W/m2 ) 4.184 000 E+04 cal (thermochemical)/cm∙s∙C watt per meter kelvin (W/mK) 4.184 000 E+02 cal (thermochemical)/g joule per kilogram (J/kg) 4.184 000 E+03 cal (thermochemical)/gC.…. joule per kilogram kelvin (J/kgK) 4.184 000 E+03 Fhft2/Btu (thermochemical) (R, thermal resistance ) Kelvin meter2 per watt (Km2 /W) 1.761 102 E–01 ft2/h (thermal diffusivity) meter2 per second (m2/s) 2.580 640 E–05 LENGTH angstrom meter (m) 1.000 000 E–10 astronomical unit meter (m) 1.495 98 E+11 fermi (femtometer) meter (m) 1.000 000 E–15 foot (ft) meter (m) 3.048 000 E–01 inch (in) meter (m) 2.540 000 E–02 light year meter (m) 9.460 528 E+15 micron meter (m) 1.000 000 E–06 mil meter (m) 2.540 000 E–05 mile (U.S. statute) meter (m) 1.609 347 E+03 pica (printer’s) meter (m) 4.217 518 E–03 point (printer’s) meter (m) 3.514 598 E–04 rod meter (m) 5.029 210 E+00 yard (yd) meter (m) 9.144 000 E–01 LIGHT footcandle lumen per meter2 (lm/m2) 1.076 391 E+01 footcandle lux (lx) 1.076 391 E+01 MASS grain kilogram (kg) 6.479 891 E–05 gram kilogram (kg) 1.000 000 E–03 hundredweight (long) kilogram (kg) 5.080 235 E+01 hundredweight (short) kilogram (kg) 4.535 924 E+01 ounce (avoirdupois) kilogram (kg) 2.834 952 E–02 pound (lb) (avoirdupois) kilogram (kg) 4.535 924 E–01 slug kilogram (kg) 1.459 390 E+01 ton (assay) kilogram (kg) 2.916 667 E–02 ton (long, 2240 lb) kilogram (kg) 1.016 047 E+03 ton (metric) kilogram (kg) 1.000 000 E+03 ton (short, 2000 lb) kilogram (kg) 9.071 847 E+02 MASS PER UNIT VOLUME (includes DENSITY and MASS CAPACITY) g/cm3 kilogram per meter3 (kg/m3) …… 1.000 000 E+03 oz (avoirdupois)/gal (U.K. liquid) kilogram per meter3 (kg/m3) …… 6.236 027 E+00 oz (avoirdupois)/gal (U.S. liquid) kilogram per meter3 (kg/m3) …… 7.489 152 E+00 oz (avoirdupois)/in3 kilogram per meter3 (kg/m3) …… 1.729 994 E+03 lb/ft3 kilogram per meter3 (kg/m3) …… 1.601 846 E+01 lb/in3 kilogram per meter3 (kg/m3) 2.767 990 E+04 lb/gal (U.K. liquid) kilogram per meter3 (kg/m3) 9.977 644 E+01. lb/gal (U.S. liquid) kilogram per meter3 (kg/m3) 1.198 264 E+02 ton(long, mass)/yd3 kilogram per meter3 (kg/m3) 1.328 939 E+03 POWER Btu (thermochemical)/s watt (W) 1.054 350 E+03 Btu (thermochemical)/min watt (W) 1.757 250 E+01 Btu (thermochemical)/h watt (W) 2.928 751 E–01 cal (thermochemical)/s watt (W) 4.184 000 E+00 cal (thermochemical)/min watt (W) 6.973 333 E–02 erg/s watt (W) 1.000 000 E–07 ft∙lbf/h watt (W) 3.766 161 E–04 ft∙lbf/min watt (W) 2.259 697 E–02 ft∙lbf/s watt (W) 1.355 818 E+00 horsepower (550 ft∙lbf/s) watt (W) 7.456 999 E+02 kilocalorie (thermochemical)/min watt (W) 6.973 333 E+01 kilocalorie (thermochemical)/s watt (W) 4.184 000 E+03 PRESSURE OR STRESS (FORCE PER UNIT AREA) atmosphere (normal = 760 torr) pascal (Pa) 1.013 25 E+05 atmosphere (technical = 1 kgf/cm2 pascal (Pa) 9.806 650 E+04 bar pascal (Pa) 1.000 000 E+05 centimeter of mercury (0 C) pascal (Pa) 1.333 22 E+03 centimeter of water (4 C) pascal (Pa) 9.806 65 E+01 dyne/cm2 pascal (Pa) 1.000 000 E–01 foot of water (39.2 F) pascal (Pa) 2.989 070 E+03 gram-force/cm2 pascal (Pa) 9.806 650 E+01 inch of mercury (32 F) pascal (Pa) 3.386 389 E+03 inch of mercury (60 F) pascal (Pa) 3.376 85 E+03 inch of water (39.2 F) pascal (Pa 2.490 82 E+02 inch of water (60 F) pascal (Pa) 2.488 4 E+02 kgf/cm2 pascal (Pa) 9.806 650 E+04 kgf/m2 pascal (Pa) 9.806 650 E+00 kgf/mm2 . pascal (Pa) 9.806 650 E+06 kip/in2 (ksi) pascal (Pa) 6.894 757 E+06 millibar pascal (Pa) 1.000 000 E+02 millimeter of mercury (0 C) pascal (Pa) 1.333 224 E+02 poundal/foot2 pascal (Pa) 1.488 164 E+00 lbf/ft2 pascal (Pa) 4.788 026 E+01 lbf/in2 (psi) pascal (Pa) 6.894 757 E+03 psi pascal (Pa) 6.894 757 E+03 torr (mm Hg, 0 C) pascal (Pa) 1.333 22 E+02 SPEED (see VELOCITY) STRESS (see PRESSURE) TEMPERATURE degree Celsius (oC) kelvin (K) tK = tC + 273.15 degree Fahrenheit (oF) kelvin (K) tK = (tF + 459.67)/1.8 degree Rankine kelvin (K) tK = tR/1.8 degree Fahrenheit (oF) degree Celsius (C) tC = (tF – 32)/1.8 kelvin (K) degree Celsius (C) tC = tK – 273.15 TIME day (mean solar) second (s) 8.640 000 E+04 day (sidereal) second (s) 8.616 409 E+04 hour (mean solar) second (s) 3.600 000 E+03 minute (mean solar) second (s) 6.000 000 E+01 year (calendar) second (s) 3.153 600 E+07 TORQUE (see BENDING MOMENT) VELOCITY (includes SPEED) ft/h meter per second (m/s) 8.466 667 E–05 ft/min meter per second (m/s) 5.080 000 E–03 ft/s meter per second (m/s) 3.048 000 E–01 in/s meter per second (m/s) 2.540 000 E–02 km/h meter per second (m/s) 2.777 778 E–01 knot (international) meter per second (m/s) 5.144 444 E–01 mi/h (U.S. statute) meter per second (m/s) 4.470 400 E–01 mi/min (U.S. statute) meter per second (m/s) 2.682 240 E+01 mi/s (U.S. statute) meter per second (m/s) 1.609 344 E+03 mi/h (U.S. statute) km/h 1.609 344 E+00 VISCOSITY centipoise pascal-second (Pa∙s) 1.000 000 E–03 centistokes meter2 per second (m2/s) 1.000 000 E–06 ft2/s meter2 per second (m2/s) 9.290 304 E–02 poise pascal-second (Pa∙s) 1.000 000 E–01 poundal∙s/ft2 pascal-second (Pa∙s) 1.488 164 E+00 lb/ft∙s pascal-second (Pa∙s) 1.488 164 E+00 lbf∙s/ft2 pascal-second (Pa∙s) 4.788 026 E+01 slug/ft∙s pascal-second (Pa∙s) 4.788 026 E+01 stoke meter2 per second (m2/s) 1.000 000 E–04 VOLUME (includes CAPACITY) acre-foot meter3 (m3) 1.233 489 E+03 barrel (oil, 42 gal) meter3 (m3) 1.589 873 E–01 board foot meter3 (m3) 2.359 737 E–03 bushel (U.S.) meter3 (m3) 3.523 907 E–02 cup meter3 (m3) 2.365 882 E–04 fluid ounce (U.S.) meter3 (m3) 2.957 353 E–05 foot3 meter3 (m3) 2.831 685 E–02 gallon (Canadian liquid) meter3 (m3) 4.546 090 E–03 gallon (U.K. liquid) meter3 (m3) 4.546 092 E–03 gallon (U.S. dry) meter3 (m3) 4.404 884 E–03 gallon (U.S. liquid) meter3 (m3) 3.785 412 E–03 inch3 meter3 (m3) 1.638 706 E–05 liter meter3 (m3) 1.000 000 E–03 ounce (U.K. fluid) meter3 (m3) 2.841 307 E–05 ounce (U.S. fluid) meter3 (m3) 2.957 353 E–05 peck (U.S.) meter3 (m3) 8.809 768 E–03 pint (U.S. liquid) meter3 (m3) 4.731 765 E–05 quart (U.S. liquid) meter3 (m3) 9.463 529 E–04 tablespoon meter3 (m3) 1.479 000 E–05 teaspoon meter3 (m3) 4.929 000 E–06 ton (register) meter3 (m3) 2.831 685 E+00 yard3 meter3 (m3) 7.645 549 E–01 VOLUME PER UNIT TIME (includes FLOW) ft3/min meter3 per second (m3/s) 4.719 474 E–04 ft3/s meter3 per second (m3/s) 2.831 685 E–02 in3/min meter3 per second (m3/s) 2.731 177 E–07 yd3/min meter3 per second (m3/s) 1.274 258 E–02 gal (U.S. liquid)/day meter3 per second (m3/s) 4.381 264 E–08 gal (U.S. liquid)/min meter3 per second (m3/s) 6.309 020 E–05 WORK (see ENERGY) _____________ Source: Selected excerpts from Metric Practice Guide, Designation: E 380 – 74 (1974), American Society for Testing and Materials, 100 Barr Harbor Drive, West Conshocken, PA 19428-2959; updated with data from Sl10-02 IEEE/ASTM SI 10 American National Standard for Use of the International System of Units (SI): The Modern Metric System (2002), SI10-02 IEEE/ASTM SI 10, 100 Barr Harbor Drive, West Conshocken, PA 19428-2959. A1.4 Magnetic units: Equivalency table a Symbol Quantity Conversion from Gaussian and cgs emu to SI b magnetic flux 1 Mx = 1 G·cm2 108 Wb = 108 V·s B magnetic flux density, magnetic induction 1 G 104 T = 104 Wb/m2 H magnetic field strength 1 Oe 103/(4) A/m m magnetic moment 1 erg/G = 1 emu 103 A·m2 = 103 J/T M magnetization 1 erg/(G·cm3) = 1 emu/cm3 103 A/m 4M magnetization 1 G 103/(4) A/m mass magnetization, specific magnetization 1 erg/(G·g) = 1 emu/g 1 A·m2/kg j magnetic dipole moment 1 erg/G = 1 emu 4 1010 Wb·m J magnetic polarization 1 erg/(G·cm3) = 1 emu/cm3 4 104 T , volume susceptibility c 1 4 , / mass susceptibility d 1 cm3/g 4 103 m3/kg permeability 1 4 107 H/m = 4 107 Wb/(A·m) r relative permeability r w, W energy density 1 erg/cm3 101 J/m3 N, D demagnetizing factor 1 1/(4) a Table based on R. B. Goldfarb and F. R. Fickett (1985), NBS STP 696, National Bureau of Standards. U.S. Government Printing Office, Washington, D.C. b Gaussian units are the same as cgs emu for magnetostatics; Mx = maxwell, G = gauss, Oe = oersted; Wb = weber, V = volt, s = second, T = tesla, m = meter, A = ampere, J = joule, kg = kilogram, H = henry. c Volume susceptibility is dimensionless but is sometimes expressed in cgs units as emu/cm3 or emu/(cm3·Oe). d Mass susceptibility is sometimes expressed in cgs units as emu/g or emu/(g·Oe). A1.5 Properties of common cryogenic fluids. (Sec. 1.2) Additional data on the vapor-pressure vs. temperature dependence of these cryogenic fluids are given in Appendix A5.1. Fluid: Property: 3He 4He H2 * (Para) H2 * (Normal) Ne N2 Ar O2 CH4 (Methane) Molecular Weight 3.0160 4.0026 2.0159 2.0159 20.179 28.013 39.948 31.999 16.043 Critical Temp. [K] 3.324 5.195 32.93 33.18 44.49 126.2 150.7 154.6 190.6 Critical Pressure [atm] 1.145 2.245 12.67 12.98 26.44 33.51 47.99 49.77 45.39 Boiling Point [K] 3.191 4.230 20.27 20.27 27.10 77.35 87.30 90.20 111.7 Melting Point [K] 4.2 (at 140 atm) 13.80 13.95 24.56 63.15 83.81 54.36 90.72 Liquid Density at B.P. [g/mL] 0.05722 0.1247 0.07080 0.07080 1.207 0.8061 1.395 1.141 0.4224 Gas Density at 0C and 1 atm [g/L] 0.1345 0.1785 0.08988 0.08988 0.8998 1.250 1.784 1.429 0.7175 Vapor Density at B.P. [g/L] 24.51 16.76 1.339 1.339 9.577 4.612 5.774 4.467 1.816 Liquid Thermal Conductivity at B.P. [mW/(m•K)] — 18.66 103.4 103.4 155.0 145.8 125.6 151.6 183.9 Liquid Isobaric Specific Heat at B.P. [J/(g•K)] 24.80 5.299 9.659 9.667 1.862 2.041 1.117 1.699 3.481 Latent Heat of Vaporization at B. P. 7.976 J/g (0.4564 J/mL) 20.75 (2.589) 445.4 (31.54) 445.4 (31.54) 85.75 (103.5) 199.2 (160.6) 161.1 (224.9) 213.1 (243.1) 510.8 (215.8) Latent heat of Fusion at M.P. [J/g] — 30.5 — 58.2 16.6 25.5 27.8 13.8 58.7 Vap. Pres. of Solid at M. P. [kPa] — — 7.04 7.20 43.46 12.52 68.89 0.146 11.5 Magnetic Susceptibil- ity [10–6 cm3/mol] a (+ paramagnetic) –2.02 (gas) –5.44 (liq., 20.3K) –3.99 (gas, >~ 293K) –6.96
(gas) –12.0
(gas) –19.32
(gas) +3 449 (gas)
+7 699 (liq., 90K)
+10 200 (sol., 54K) –17.4
B.P. ≡ boiling point; M.P. ≡ melting point.
Principal source of data: E.W. Lemmon, NIST, evaluated from equations of state referenced in Appendix A5.1.
Data on solids:
V. Johnson (1960), NBS, Wright Air Development Div. (WADD) Technical Report 60-56, Part II. U.S. Government Printing Office, Washington, D.C.
D. H. J. Goodall (1970), A.P.T. Division, Culham, Culham Science Center, Abingdon, Oxfordshire, UK.
K. Timmerhaus and T. Flynn (1989), Cryogenic Process Engineering, Plenum Press, New York.
* Hydrogen can exist in two different molecular forms: higher-energy orthohydrogen (nuclear spins aligned) and lower-energy parahydrogen (nuclear spins opposed). The equilibrium ratio is determined by temperature: at room temperature and above, hydrogen consists of about 25 % para and 75 % ortho (so-called normal hydrogen), but at the atmospheric boiling temperature of liquid hydrogen (20.27 K) and below, the equilibrium shifts almost completely to parahydrogen (99.79 % para and 0.21 % ortho at 20.27 K).
a CRC Handbook of Chemistry and Physics (2002), 83rd edition, CRC Press, Boca Raton, Florida.
A1.6a Cooling power data for 4He, H2, and N2 (Sec. 1.2)
Tabulated values are consumption rates resulting from 1 W dissipated directly in the indicated cryogenic liquid at atmospheric pressure.
Cryogenic liquid Volume of liquid boiled off
from 1 W
[L/h] Flow of gas at 0oC, 1 atm
from 1 W
[L/min] Enthalpy change at 1 atm pressure
[J/g]
4He
1.377
16.05
87 (4.2 K–20 K)
384 (4.2 K–77 K)
1542 (4.2 K–300 K)
H2
0.1145 1.505 590 (20 K–77 K)
3490 (20 K–300 K)
N2
0.0225 0.243 233.5 (77 K–300 K)
Data compiled from:
V. Johnson (1960), NBS, Wright Air Development Div. (WADD) Technical Report 60-56, Part II, U.S. Government Printing Office, Washington, D.C.
D. H. J. Goodall (1970), A.P.T. Division, Culham Science Center, Abingdon, Oxfordshire, UK.
A1.6b Cooling power data: Amount of cryogenic fluid needed to cool common metals a,b (Sec. 1.2)
Cryogenic Fluid: 4He
(Tb = 4.2 K) H2
(Tb = 20.3 K) N2
(Tb = 77.3 K)
Initial Temp. of Metal: 300 K
[L/kg] 77 K
[L/kg] 300 K
[L/kg] 77 K
[L/kg] 300 K
[L/kg]
Using the latent heat of vaporization only Aluminum 58 2.6 5.4 0.25 1.01
Copper 27 1.8 2.4 0.17 0.46
Stainless Steel 30 1.2 2.8 0.12 0.54
Using both the latent heat and the enthalpy of the gas Aluminum 1.60 0.22 1.03 0.14 0.64
Copper 0.80 0.15 0.51 0.092 0.29
Stainless Steel 0.80 0.10 0.52 0.064 0.34
Tb is the boiling temperature at atmospheric pressure.
a Determined from data by J. B. Jacobs (1962), Adv. Cryog. Eng. 8, 529.
b For temperature combinations other than those given in this table, see Jacobs (1962, reference above).
A1.7 Suppliers of specialty parts and materials
The following is a list of suppliers of specialty parts and materials for constructing measurement cryostats. It is provided as a convenience to save time locating less-common items. These are not complete listings of suppliers and information can change over time, but at least they are a place to start. They may also serve as points of reference if contact information has changed.
Updated supplier information is listed for cryogenic instrumentation annually each December in the Cold Facts Buyer’s Guide, Cryogenic Society of America, http://www.cryogenicsociety.org/. Suppliers for general physics instrumentation are updated each August in the Physics Today Buyers Guide, American Institute of Physics, http://www.physicstoday.org/guide/.
Trade names, products, and companies cited here do not constitute or imply endorsement by NIST or by the U. S. government, and do not imply that they are the best available for the purpose.
Adhesives (see Appendix A3.10)
Coaxial cables for cryogenic applications (Secs. 4.7.1, 4.8)
Solid dielectric coaxial cables for lower frequency applications (< 1 GHz) where dimensional stability of the terminations on thermal cycling is not needed (see Sec. 4.8): Axon Cable Inc., 390 E. Higgins Rd., Suite 101, Elk Grove Village, IL 60007, Tel. 708-806-6629, Fax. 708-806-6639, http://www.axon-cable.com/. Supplier of miniature coaxial cable; stock number SM50 comes standard with Teflon™ dielectric and jacket; PXC47K08 can also be supplied with a Teflon™ jacket. Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/. Micro-Coax, 206 Jones Blvd., Pottstown, PA 19464-3465, Tel. 610-495-0110, 800-223-2629, Fax. 610-495-6656, http://www.micro-coax.com/. Oxford Instruments–Cryospares, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/cryospares/. Precision Tube, Coaxitube Div., 620 Naylor Mill Road, Salisbury, MD 21801, Tel. 410-546-3911, Fax. 410-546-3913, http://www.precisiontube.com/. Catalog contains helpful information on the electrical selection of coaxial cables. RS, United Kingdom, Tel. +44-1536-201201, Fax. +44-1536-201-501, http://www.rs-components.com. Supplier of miniature coaxial cable with Teflon™ dielectric and jacket; “RF cable MCX” stock numbers: 388-530 (50 Ω), 388-546 (75 Ω).; (for low frequencies, where impedance matching is not a concern, the 75 Ω might be better since the capacitance is a bit lower). Storm Products Co., Microwave Sales Office, 10221 Werch Drive, Woodridge, IL 60517, Tel. 630-754-3300, 888-347-8676, Fax. 630–754-3500, http://www.stormproducts.com/. Expanded dielectric coaxial cables for higher frequency applications (> 1 GHz) where dimensional stability of the terminations on thermal cycling is needed (see Sec. 4.8):
Storm Products Co., Microwave Sales Office, 10221 Werch Drive, Woodridge, IL 60517, Tel. 630-754-3300, 888-347-8676, Fax. 630–754-3500, http://www.stormproducts.com/; expanded dielectric coaxial cables, for example cable #421-193.
Connectors (Secs. 4.1, 4.6, 4.7, 4.8)
Alligator clips: smooth, flat jaws; 7/32” jaw opening, #20 wire or smaller; crimp connection:
Mueller Electric Co., part number (PN) BU-34C, http://www.muellerelectric.com/ (distributed by Allied Electronics, Inc., PN 860-4340, Tel. 800-433-5700, http://www.alliedelec.com/ or Newark Electronics, PN 28F497, Tel. 800-263-9275, http://www.newark.com/.
Rf connectors:
Fischer Connectors, Tel. 1-800-551-0121, http://www.fischerconnectors.com/
Lemo Connectors, http://www.lemousa.com/.
Vacuum lead-throughs (room temperature):
Cerama-Seal, 1033 State Route 20, New Lebanon, NY 12125, Tel. 10518-794-7800, Fax 518-794-8080, http://www.ceramaseal.com/.
Detoronics Corp., 10660 East Rush St., So. El Monte, CA 91733-3432, Tel. 818-579-7130, Fax 818-579-1936, http://www.detoronics.com/.
Contacts (springy devices) (Sec. 7.4.3)
Beryllium–copper clad circuit board for making microsprings:
Specialty order from Q-Flex, 1220 S. Lyon St., Santa Ana, CA 92705, Tel. 714-835-2868, Fax. 714-835-4772, http://www.q-flex.com/.
Fuzz Buttons:
Techknit, Cranford, NJ, http://www.fuzzbuttons.com/.
Pogo Pins:
Emulation Technology, Inc., Santa Clara, CA, http://www.emulation.com/pogo/.
Cryogenic accessories and consumables
Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/.
Oxford Instruments–Cryospares, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/cryospares/.
Cryogenic measurement systems – Complete (Sec. 1.4)
Cryo Industries; 11124 S. Willow St., Manchester, NH 03103; Tel. 603-621-9957; cryo@cryoindustries.com; http://www.cryoindustries.com/.
Janis Research Co.; 2 Jewel Dr. P. O. Box 696, Wilmington, MA 01887-0696; Tel. 978-657-8750, http://www.janis.com/.
Oxford Instruments, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/.
Precision Cryogenic Systems, Inc.; 1171 West Rockville Rd., Indianapolis, Indiana 46234; Tel. 317-272-0880, http://www.precisioncryo.com/.
Quantum Design; 6325 Lusk Glvd., San Diego, CA 92121–3733; Tel. 858-481-4400, Fax. 858-481-7410, http://www.qduse.com/.
Current leads (Secs. 4.9, 4.10)
Flexible superconducting braid:
Supercon Inc., 830 Boston Turnpike, Shrewsbury, MA 01545, http://www.supercon-wire.com/ (by special order).
Low-Tc and high-Tc superconductors – see Superconducting wire
Vapor-cooled leads:
American Magnetics Inc., P.O. Box 2509, 112 Flint Road, Oak Ridge, TN 37831-2509, USA, http://www.americanmagnetics.com/.
Cryomagnetics Inc., 1006 Alvin Weinberg Drive, Oak Ridge, TN 37830, USA, http://www.cryomagnetics.com/.
Current power supplies; low-ripple, series-transistor regulated (Sec. 9.2)
Alpha Scientific Electronics, Hayward, CA, 510-782-4747, http://www.alphascientific.com/.
Inverpower Controls Ltd., Burlington, Ontario, Canada, 905-639-4692, http://www.inverpower.com/.
Walker LDJ Scientific Inc., Worcester, MA 01606, 508-852-3674, http://www.walkerscientific.com/ (current ≤ 500 A).
Dewars for measurement systems—metal and fiberglass–epoxy
American Magnetics Inc., P.O. Box 2509, 112 Flint Road, Oak Ridge, TN 37831-2509, USA, http://www.americanmagnetics.com/.
Cryomagnetics Inc., 1006 Alvin Weinberg Drive, Oak Ridge, TN 37830, USA, http://www.cryomagnetics.com/.
International Cryogenics, 4040 Championship Drive, Indianapolis, IN 46268, Tel. 317-297-4777, Fax. 317-297-7988, http://www.intlcryo.com/.
Janis Research Co.; 2 Jewel Dr. P. O. Box 696, Wilmington, MA 01887-0696; Tel. 978-657-8750, http://www.janis.com/.
Oxford Instruments, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/.
Precision Cryogenic Systems, Inc., 7804 Rockville Road, Indianapolis, Indiana 46214, Tel. 317–273-2800, Fax. 317-273-2802, prcry@iquest.net, http://www.precisioncryo.com/.
Tristan Technologies, Inc., 6185 Cornerstone Court East, Suite 106, San Diego, CA 92121, Tel. 877-436-1389, http://www.tristantech.com/.
Epoxies and pastes — Conductive (Secs. 7.4.1 and 8.3.2)
Silver-based epoxy:
Ted Pella, Inc., P.O. Box 492477, Redding, CA 96049-2477, Tel. 800-237-3526; Fax. 530-243-3761, http://www.TedPella.com/.
Silver paste:
Ted Pella, Inc., P.O. Box 492477, Redding, CA 96049-2477, Tel. 800-237-3526; Fax. 530-243-3761, http://www.TedPella.com/.
Heaters, thin film (Secs. 1.4, 5.4, 7.3.1, and 7.4.1)
Minco Products, Inc., 7300 Commerce Lane, Minneapolis, MN 55432-3177, Tel. 763-571-3121, Fax. 763-571-0927, Info@minco.com, http://www.minco.com/ .
Liquid-level monitors (Sec. 1.6.2)
Janis Research Co.; 2 Jewel Dr. P. O. Box 696, Wilmington, MA 01887-0696; Tel. 978-657-8750, http://www.janis.com/.
Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/.
Oxford Instruments, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/.
Lubricants (see Appendix A3.11)
Magnets, superconducting (Secs. 1.4, 1.5, 9.1.4, 9.2.1)
American Magnetics Inc., P.O. Box 2509, 112 Flint Road, Oak Ridge, TN 37831-2509, USA, http://www.americanmagnetics.com/.
American Superconductor Corp., Two Technology Dr., Westborough, MA 01581, Tel. 508-836-4200, Fax. 508-836-4248, http://www.amsuper.com/ (high-Tc magnets).
Cryomagnetics Inc., 1006 Alvin Weinberg Drive, Oak Ridge, TN 37830, http://www.cryomagnetics.com/.
Oxford Instruments, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/.
SuperPower, Inc., 450 Duane Ave., Schenectady, NY 12304, Tel. 518-346-1414, Fax. 518-346-6080, http://www.igc.com/superpower/ (high-Tc magnets).
Materials, less common and specialty sizes (Secs. 3.2, 3.4, 6.5.2, 7.3, 7.4)
Metals – general supplier of high purity metals and metallic compounds:
ESPI, 1050 Benson Way, Ashland, OR 97520, Tel. 800-638-2581, Fax. 800-488-0060, http://www.espimetals.com/.
Aluminum – high conductivity wires:
Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA 15069, http://www.alcoa.com/
Sumitomo Chemical, Japan, http://www.sumitomo-chem.co.jp/english/
Swiss Federal Institute of Technology, Zurich, Switzerland, Tel. +41 44 632 1111, Fax. +41 44 632 1010, http://www.ethz.ch
Copper – high conductivity, oxygen free; (see Appendix A3.1 for a listing of the various types):
Copper & Brass Sales, Tel. 800-926-2600, Fax. 888-926-2600, http://www.copperandbrass.com/ (OFHC ™ copper tubes).
Farmer’s Copper & Industrial Supply 800-231-9450, Fax. 409-765-7115, http://www.farmerscopper.com/ (OFHC ™ copper tubes).
McMaster–Carr, http://www.mcmaster.com/.
Fiberglass–epoxy composite tubes; custom sizes (made from G-10, G-11, G-13):
A & M Composites, P.O. Box 3281, Big Spring, TX 79721, Tel. 432-267-6525, Fax. 432-267-6599, http://www.amcctx.com/.
Microwave circuit board (TMM™) (with a thermal-expansion coefficient less than that of G-10 circuit board, so as to give better dimensional stability):
Rogers Corp., One Technology Dr., P.O. Box 188, Rogers, CT 06263-0188, Tel. 860-774-9605, Fax. 860-779-5509, http://www.rogers-corp.com/ .
Titanium tubes – less common sizes:
Titanium Sports Technologies (TST), 1426 E. Third Ave., Kennewick, WA 99336, Tel. 509-586-6117, http://www.titaniumsports.com/.
Mechanical actuators and linear motors (Sec. 3.6)
Energen, Inc., 650 Suffolk St., Lowell, MA 01854, http://www.energeninc.com/index.htm.
Soldering materials (Secs. 3.3.4, 4.5, 4.6, 8.3.2, 8.3.3)
Indium-alloy solders:
Indium Corp. of America, Indalloy® solders, Tel. 315-853-4900 or 800-4-INDIUM, askus@indium.com, http://www.indium.com/.
Lake Shore Cryotronics, Ostalloy® solders, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/ .
Umicore Indium Products, Ostalloy® solders, http://www.thinfilmproducts.umicore.com/.
Solder flux:
• Combined solder and flux paste:
Fusion Automation, Inc., http://www.fusion-inc.com/ Model SSX-430-830.
Multicore Kester 135, http://www.kester.com/.
• Mild flux:
Alpha HF260, http://www.alphametals.com/distributors/pdfs/2001134214.pdf.
Litton ESF33, http://www.amsuper.com/products/library/003-TechNote_Soldering.pdf.
• Unactivated rosin flux:
Kester, Tel. 800-253-7837, Fax. 847-390-9338, technicalservice@kester.com, http://www.kester.com/ designated “Plastic core” RNA (rosin non-activated).
Solder with antimony to minimize embrittlement and cracking at cryogenic temperatures:
Kester, Tel. 800-253-7837, Fax. 847-390-9338, technicalservice@kester.com, http://www.kester.com/.
Strain gauges, accessories, and gauge adhesives for cryogenic service (Sec. 9.4.4)
Vishay Intertechnology, Inc., Vishay Micro-Measurements Division, http://www.vishay.com/.
Sticky stuff: (see Appendix A3.10)
Superconducting wire (Secs. 4.9, 4.10, Chapters 9 and 10)
Updated links to superconductor suppliers are available at http://superconductors.org/Links.htm.
Low-Tc (Nb–Ti and Nb3Sn):
Alstom Magnets & Superconductors, 90018 Belfort Cedex, France, Tel. +33 (0)3 84 55 32 26, Fax. +33 (0)3 84 55 70 93, http://www.powerconv.alstom.com/.
Bochvar, 5 ulitsa Rogova, Moscow 123060, Tel. (095) 190-49-93[1], 190-82-97[2], Fax. (095) 196-41-68, e-mail: post@bochvar.ru, http://www.bochvar.ru.
European Advanced Superconductor (EAS), Ehrichstraße 10, 63450 Hanau, Germany, Tel. (+49) (6181) 43 84-41 00, Fax. (+49) (6181) 43 84-44 00, http://www.advancedsupercon.com/.
Furukawa Electric, 6-1, Marunouchi 2-chome, Chiyoda-ku, Tokyo 100, Japan, Tel. 81-3-3286-3001, Fax. 81-3-3286-3747,3748, http://www.furukawa.co.jp/english.
Kobe Steel, Ltd., Shinko Building, 10-26, Wakinohamacho, 2-chome, Chuo-ku, Kobe, Hyogo 651-8585, Japan, Tel. 81-78-261-511, Fax. 81-78-261-4123, http://www.kobelco.co.jp/english.
Outokumpu, http://www.outokumpu.com/.
Oxford Superconducting Technology, 600 Milik St., P.O. Box 429, Carteret, NJ 07008-0429, Tel. 732 541 1300, Fax. 732 541 7769, http://www.oxford-instruments.com/.
Shape Metal Innovations (SMI); Nb3Sn powder-in-tube (PIT) process, Tel. +31 53 4340704, JLSMI@worldonline.nl.
Sumitomo, One North Lexington Ave., White Plains, NY 10601, Tel. 914-467-6001, Fax. 914-467-6081, http://www.sumitomoelectricusa.com/.
Supercon Inc., 830 Boston Turnpike, Shrewsbury, MA 01545, http://www.supercon-wire.com/.
Western Superconducting Material Technology Corp., P.O. Box 51 Xi’an Shaanxi, 710016 P.R. China.
Low-Tc (MgB2):
Columbus Superconductor S.R.L., Corso F. Perrone 24, 16152 Genova, Italy, Tel. +39 (0)10 65 98 784, Fax. +39 (0)10 65 98 732.
Diboride Conductors, http://www.diboride.biz/.
Hyper Tech Research, Inc., 110 E. Canal St., Troy, OH 45373-3581, Tel. 937-332-0348, http://www.hypertechresearch.com/.
High-Tc (Bi-2212):
Oxford Superconducting Technology, 600 Milik St., P.O. Box 429, Carteret, NJ 07008-0429, Tel. 732 541 1300, Fax. 732 541 7769, http://www.oxford-instruments.com/.
Showa Electric Wire and Cable Co., Ltd., http://www.swcc.co.jp/eng/index.htm.
High-Tc (Bi-2223):
American Superconductor Corp., Two Technology Drive, Westborough, MA 01581, Tel. 508.836.4200, Fax. 508.836.4248, http://www.amsuper.com/.
European Advanced Superconductor (EAS), Ehrichstraße 10, 63450 Hanau, Germany, Tel. (+49) (6181) 43 84-41 00, Fax. (+49) (6181) 43 84-44 00, http://www.advancedsupercon.com/.
Innova Superconductor Technology Co. Ltd, 7 Rongchang Dongjie, Longsheng Industrial Park, Beijing 100176, People’s Republic of China.
Sumitomo, One North Lexington Ave., White Plains, NY 10601, Tel. 914-467-6001, Fax. 914-467-6081, http://www.sumitomoelectricusa.com/.
Trithor GmbH, Heisenbergstrasse 16, D-53359 Rheinbach, Germany, Tel.: +49 (0) 2226 – 90 60 – 0, Fax. +49 (0) 2226 – 90 60 – 900, http://www.trithor.com/.
High-Tc (YBCO):
American Superconductor Corp., Two Technology Drive, Westborough, MA 01581, Tel. 508.836.4200, Fax 508.836.4248, http://www.amsuper.com/.
Fujikura, http://www.fujikura.co.jp/ie_e.html.
SuperPower, 450 Duane Avenue, Schenectady, NY 12304, Tel.: 518/346-1414, Fax. 518/346-6080, http://www.igc.com/superpower/.
Theva GmbH, Rote-Kreuz-Str. 8, D-85737 Ismaning Germany, Tel. +49 89 923346-0, Fax. +49 89 923346-10, info@theva.com, http://www.theva.com/.
Thermometers and accessories (Chapter 5)
Beryllium-oxide high-thermal-conductivity chips:
Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/.
Capacitance bridges:
Automatic bridges—Andeen–Hagerling Inc., Cleveland, OH, Tel. 440-349-0370, Fax. 440-349-0359, http://www.andeen-hagerling.com/.
Capacitance controller card—Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/.
General Radio capacitance bridges (5 digit) available from IET Labs Inc., Westbury, NY, Tel. 800-899-8438, Fax. 516-334-5988, http://www.ietlabs.com/ or Tucker Electronics, Dallas TX, Tel. 800-527-4642, Fax. 214-348-0367, http://www.tucker.com/.
Grease – thermally conducting:
Apiezon N grease – Apiezon Products, M & I Materials Ltd., Manchester, UK, Tel. +44 (0)161 864 5419, Fax. +44 (0)161 864 5444, http://www.apiezon.com/.
Cry-Con grease – available, for example, from Janis Research Co., Accessories and Ancillary Equipment, http://www.janis.com/.
Thermometers for cryogenic temperatures and calibration services:
Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/.
Oxford Instruments–Cryospares, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/cryospares/.
Scientific Instruments, Inc., West Palm Beach FL 33407, Tel. 561-881-8500, Fax. 561-881-8556, http://www.scientificinstruments.com/.
Tinsley Manufacturing, supplier of rhodium–iron resistance thermometers in wire form.
Temperature controllers:
Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/.
Oxford Instruments, Witney, Oxfordshire, UK OX294TL, Tel. +44(0)1865 881437, Fax. +44(0)1865 884045, http://www.oxinst.com/.
Thermocouple wire (Secs. 5.1.1, 5.1.2, 5.1.4, 5.1.6, and 5.5.9)
Omega Engineering, P.O. Box 4047, Stamford, Connecticut 06907-0047, 800-848-4286 or 203-359-1660, Fax. 203-359-7700, http://www.omega.com/.
River Bend Technology Centre, Northbank, Irlam, Manchester M44 5BD, United Kingdom, http://www.omega.co.uk/.
Vacuum accessories (Secs. 3.3.1, 3.7)
C-ring metal seals:
American Seal & Engineering Co., P.O. Box 1038, Orange, CT 06477, 800-878-2442, http://www.ameriseal.com.
Garlock–Helicoflex, P.O. Box 9889, Columbia, SC 20290, Tel. 800-713-1880, http://www.helicoflex.com.
Hydrodyne, 325 Damon Way, Burbank, CA 91505, Tel. 818-841-9667, http://www.hydrodyne.com.
Nicholsons Sealing Technologies Ltd., Hamsterley, Newcastle upon Tyne, UK, NE17 7 SX, Tel. +44 (0)1207 560505, http://www.nicholsons.com.
Dynamic seals: O-rings, spring-loaded PTFE:
Bal Seal Engineering Co., Inc., 620 West Ave., Santa Ana, CA 92707-3398, Tel. 714-557-5192.
Vacuum flanges and fixtures: Ladish Tri-Clover, and ISO KF; available from general vacuum-equipment suppliers such as:
Duniway Stockroom Corp., Tel. 800-446-8811, http://www.duniway.com/.
Kurt J. Lesker Co., Tel. 800-245-1656, http://www.lesker.com/.
O-rings, indium wire:
Indium Corp. of America, 1676 Lincoln Ave., Utica, NY. 13503.
O-rings, metal:
Perkin Elmer, Beltsville, MD, Tel. 301-937-4010.
Screws (silver plated to prevent galling, precleaned, and optionally vented for vacuum systems):
McMaster–Carr, http://www.mcmaster.com/.
U-C Components, Morgan Hill, CA, http://www.uc-components.com/.
Wire (Sec. 4.1, 4.2, and 4.3)
Phosphor-bronze twisted-wire pairs for thermometer leads:
Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/ Quad-Twist™ cryogenic wire.
Pure indium wire for indium O-rings:
Indium Corp. of America, Tel. 315-853-4900 or 800-4-INDIUM, askus@indium.com, http://www.indium.com/.
Stripper (chemical) for polyimide (KaptonÔ) wire insulation:
Miller–Stephenson chemical, George Washington Hwy., Danbury, CT 06810, Tel. 203-743-4447, Fax. 203-791-8702, support@miller-stephenson.com, MS-111 stripping agent.
A2. Heat-transfer (ref. Chapter 2)
A2.1 Thermal conductivity integrals for technical cryostat materials a (see also Fig. 2.1 in Sec. 2.2)
The thermal conductivity integrals tabulated below are referenced to 4 K. Steady-state heat conduction q.cond through a solid member of uniform cross section A and length L may be determined between two arbitrary temperatures T1 and T2 by taking the difference between the two corresponding 4 K integral values:
q.cond ≡ A/L T1T2 (T) dT = A/L { 4KT2 (T) dT – 4KT1 (T) dT },
where (T) is the temperature-dependent thermal conductivity.
Data for materials other than those tabulated may be estimated well enough for cryostat-design purposes by using data for similar materials, especially if they have a low thermal conductivity and do not contribute much to the total heat influx. For example, most commercial glasses, as well as many plastics and disordered polymers can be represented (within a factor of about two) by the integral values given for Pyrex. Values for Manganin can be approximated by those given for Constantan, and values for Inconel and Monel alloys are between those of stainless steel and Constantan.
Greater care must be given to the highly conducting materials. Phosphorus deoxidized copper is the type of copper used most often in pipe, rods, and bars. Electrolytic tough pitch copper is the material from which copper electrical wires are usually made.
The temperature dependence of the thermal conductivity of additional cryostat construction materials is given in Appendix A6.7.
Thermal Conductivity Integrals
4KT dT [kW/m] [W/m]
COPPER COPPER ALLOYS
ALUMINUM
STAINLESS STEEL CONST-ANTAN GLASS
POLYMERS
T(K) Elect. Tough Pitch b
Phos. Deox. Be/Cu 98 Cu
2 Be German Silver
60 Cu 25 Zn 15 Ni Com-mon Pure 99 Al b Mn/Al
98.5 Al
1.2 Mn plus traces Mg/Al
96 Al
3.5 Mg plus traces Average Types 303,304, 316, 347 Average Pyrex Quartz Boro-Silicate Teflon Perspex Nylon
6 0.80 0.0176 0.0047 0.00196 0.138 0.0275 0.0103 0.00063 0.0024 0.211 0.113 0.118 0.0321
8 1.91 0.0437 0.0113 0.00524 0.342 0.0670 0.025 0.00159 0.0066 0.443 0.262 0.238 0.0807
10 3.32 0.0785 0.0189 0.010 0.607 0.117 0.0443 0.00293 0.0128 0.681 0.44 0.359 0.148
15 8.02 0.208 0.0499 0.030 1.52 0.290 0.112 0.00816 0.0375 1.31 0.985 0.669 0.410
20 14.0 0.395 0.0954 0.0613 2.76 0.534 0.210 0.0163 0.0753 2.00 1.64 1.01 0.823
25 20.8 0.635 0.155 0.102 4.24 0.850 0.338 0.0277 0.124 2.79 2.39 1.44 1.39
30 27.8 0.925 0.229 0.153 5.92 1.23 0.490 0.0424 0.181 3.68 3.23 1.96 2.08
35 34.5 1.26 0.316 0.211 7.73 1.67 0.668 0.0607 0.244 4.71 4.13 2.59 2.90
40 40.6 1.64 0.415 0.275 9.62 2.17 0.770 0.0824 0.312 5.86 5.08 3.30 3.85
50 50.8 2.53 0.650 0.415 13.4 3.30 1.24 0.135 0.457 8.46 7.16 4.95 6.04
60 58.7 3.55 0.930 0.568 17.0 4.55 1.79 0.198 0.612 11.5 9.36 6.83 8.59
70 65.1 4.68 1.25 0.728 20.2 5.89 2.42 0.270 0.775 15.1 11.6 8.85 11.3
76 68.6 5.39 1.46 0.826 22.0 6.72 2.82 0.317 0.875 17.5 13.0 10.1 13.1
80 70.7 5.89 1.60 0.893 23.2 7.28 3.09 0.349 0.943 19.4 13.9 11.0 14.2
90 75.6 7.20 1.99 1.060 25.8 8.71 3.82 0.436 1.11 24.0 16.3 13.2 17.3
100 80.2 8.58 2.40 1.23 28.4 10.2 4.59 0.528 1.28 29.2 18.7 15.5 20.4
120 89.1 11.5 3.30 1.57 33.0 13.2 6.27 0.726 1.62 40.8 23.7 20.0 26.9
140 97.6 14.6 4.32 1.92 37.6 16.2 8.11 0.939 1.97 54.2 28.7 24.7 33.6
160 106 18.0 5.44 2.29 42.0 19.4 10.1 1.17 2.32 69.4 33.8 29.4 40.5
180 114 21.5 6.64 2.66 46.4 22.5 12.2 1.41 2.69 85.8 39.0 34.2 47.5
200 122 25.3 7.91 3.06 50.8 25.7 14.4 1.66 3.06 103.0 44.2 39.0 54.5
250 142 35.3 11.3 4.15 61.8 33.7 20.5 2.34 4.06 150.0 57.2 51.0 72.0
300 162 46.1 15.0 5.32 72.8 41.7 27.1 3.06 5.16 199.0 70.2 63.0 89.5
a Data from:
V. Johnson (1960), NBS, Wright Air Development Div. (WADD) Technical Report 60-56, Part II. U.S. Government Printing Office, Washington, D.C.
D. H. J. Goodall (1970), A.P.T. Division, Culham Science Center, Abingdon, Oxfordshire, UK.
b The high thermal conductivity of nearly pure metals is variable and strongly depends on their impurity content; see Sec. 6.4.2.
A2.2 Emissivity of technical materials at a wavelength of about 10 m (room temperature) (Sec. 2.4)
Material Emissivity
polished highly
oxidized common condition
Metallic:
Ag 0.01
Cu 0.02 0.6
Au 0.02
Al 0.03 0.3
Brass 0.03 0.6
Soft-solder 0.03
Nb, crystalline, bulk 0.04
Lead 0.05
Ta 0.06
Ni 0.06
Cr 0.07
Stainless Steel 0.07
Ti 0.09
Tin (gray), single crystal 0.6
Nonmetallic:
IMI 7031 varnish 0.9
Phenolic lacquer 0.9
Plastic tape 0.9
Glass 0.9
Compiled from:
American Institute of Physics Handbook (1972), 3rd edition, Chapter 6, McGraw–Hill, New York.
M. M. Fulk, M. M. Reynolds, and O. E. Park (1955), Proc 1954 Cryogenic Eng. Conf., Nat. Bur. Stands. (U.S.) Report No. 3517, p. 151. U.S. Government Printing Office, Washington, D.C.
W. H. McAdams (1954), Heat Transmission, 3rd edition, McGraw–Hill, New York.
W. T. Ziegler and H. Cheung (1957), Proc 1956 Cryogenic Engineering Conference, National Bureau of Standards, p. 100. U.S. Government Printing Office, Washington, D.C.
Emissivities of additional materials at room temperature are available in the technical reference section of The Temperature Handbook (2002), p. Z-171. Omega Engineering Inc., Stamford, Connecticut (http://www.omega.com/).
A2.3 Heat conductance across solid interfaces pressed together with 445 N force (45 kgf or 100 lbf) (Sec. 2.6)
Heat conductance at a force level F other than 445 N can be determined by multiplying these data by the ratio F / 445 N. In addition to these data, see Fig. 2.7 for heat conductance values covering a wide range of temperatures (0.1 K to 300 K) for pressed contacts of gold/gold, indium/copper, copper/copper, and stainless/stainless. Data are also given in Fig. 2.7 for solder, grease, and varnish joints.
Interface Materials 4.2 K 77 K y *
Gold/Gold 2 10–1 W/K a 1.3 a
Copper/Copper 1 10–2 W/K b 3 10–1 W/K b 1.3 a
Steel/Steel 5 10–3 W/K b 3 10–1 W/K b
Sapphire/Sapphire 7 10–4 W/K a 3 a
* Values of y are for calculating the heat conductance at temperatures below 4.2 K by using Eq. (2.14) in Sec. 2.6.
a R. Berman and C. F. Mate (1958), Nature 182, 1661.
b R. Berman (1956), J. Appl. Phys. 27, 318.
A3. Cryostat construction (ref. Chapter 3)
A3.1 High-thermal-conductivity construction-metal properties: RRR, thermal conductivity, and electrical resistivity (Sec. 3.2.2)
RRR 293K/4K, the residual resistivity ratio; a thermal conductivity; electrical resistivity
The thermal conductivities of additional construction materials are shown in Fig. 2.1 and tabulated in Appendix A6.7.
Material RRR a,f
(293K/4K) 293 K g
[W/(m∙K)] 4.2 K f,h
[W/(m∙K)] 293 K f,g
[∙cm] 77 K f
[∙cm] Use Comments
Copper
High purity
(99.999 % pure) c,d ~2000 394 ~11300 1.68 0.19 Very high thermal-cond. parts. Thermal conductivity can be increased by annealing; see footnotes c and d.
Oxygen-free c,d,e
Grade C10100 b,c,d
(99.99 % pure)
Electronic grade C10200 b,c,d
(99.95 % pure)
~150
~100
394
390
~850
~560
1.72
1.72
0.19
0.19 High thermal-cond. foil, rods, plates, and tubes. Thermal conductivity can be increased by annealing; see footnotes c and d.
ETP
Grade C11000 b,c
~100 390 ~560 1.71 — High thermal-cond. rods, plates, wire, and wire braid. ETP electrolytic-tough-pitch copper
Contains about 0.3 % oxygen—cannot be used for hydrogen brazing
Thermal conductivity of cold-worked ETP copper can be increased by annealing; see footnote c.
Phosphorus deoxidized
Grade C12200 3 to 5 339 ~14 to 24 2.03 — Tubes
Brass
Free cutting brass
Grade C36000 ~2.5 125 ~4.5 7.2 4.7
Beryllium copper, annealed
Grade C17000–C17300 1.5 to 2.5 ~84
depends on processing ~1.8 to 3.0 6.4 to 10.7
depends on processing 4.2 to 8.5
depends on processing
Aluminum
99.999 % high purity
~1000 235 ~3400 2.76 0.23
Grade 1100
~14 222 ~45 —
Grade 6063
~7 218 ~22 —
Grade 5052
~1.4 138 ~2.8 4.93 —
a The listed RRR values are nominal and can vary by about 50 % from sample to sample for the purer grades, depending on the amount and type of impurities as well as cold-work condition.
b Unified Numbering System (UNS) grade numbers for metals and alloys.
c The thermal and electrical conductivity of deformed and coldworked high-purity, oxygen-free, and ETP copper can be increased (depending on the amount of cold work) by annealing. Heat in vacuum (<~ 10–4 torr) or argon at about 500 oC for about an hour. If vacuum or argon are not readily available, copper can be heated in air, but a surface scale forms, which can be removed afterward with dilute nitric acid.
d Although this is not commonly done, further increase in the thermal and electrical conductivity can be obtained by oxidizing the magnetic iron impurities in high-purity and oxygen-free copper (but not in ETP copper, which contains too many impurities other than iron). The RRR of oxygen-free copper is typically increased from ~100 as received, to ~800 after oxidation; the RRR of high purity (99.99 %) copper is typically increased from ~1500 as received, to more than 10 000 after oxidation. Heat the copper part at about 1000 oC in oxygen at about 0.13 Pa to 1.3 Pa (10–3 torr to 10–2 torr) pressure. About a day of annealing is required for small parts, up to a month for large copper billets [ref. F. R. Fickett (1974), Mater. Sci. Eng. 14, 199–210].
e Sources of oxygen-free copper are not as plentiful as ETP copper, especially in tube form. However, if high-thermal-conductivity tubes are needed or if hydrogen brazing is to be done, oxygen-free copper is required. Suppliers of oxygen-free copper are listed in Appendix A1.7 under Material, copper.
f From C. A. Thompson, W. M. Manganaro, and F. R. Fickett (1990), Cryogenic Properties of Copper, Wall Chart, NIST, and the references cited therein. U.S. Government Printing Office, Washington D.C.
g Metals Handbook (1961), Vol. 1, Properties and Selection of Materials, 8th edition, ASM International, Materials Park, Ohio.
h Calculated from the Wiedemann–Franz–Lorenz law, Eq. (2.4): = LN T / , where LN is the Lorenz constant; this results in
(4.2 K) = (293 K) (293K/4K) (4.2 K/293 K).
A3.2 Heat conduction along thin-walled stainless-steel tubing a (Sec. 3.2.2)
The heat conduction values tabulated in this table may be simply scaled to lengths other than 10 cm (inversely proportional) and wall thicknesses other than those listed in column 2 (directly proportional).
The tabulated values of conducted heat assume no gas cooling of the tubing. If the gas boiled off by the conducted heat were to cool the tubing with 100 % efficiency, the resultant heat flow would be 1/10 of the values given for T = 77 K and l/32 of those for T = 300 K.
Tube O.D.
[inches] Wall
Thickness
[inches (mm)] Cross Sectional
Area
[cm2] Heat conducted [milliwatts]
along 10 cm of tubing with one end at 4 K and the other at:
T=77 K T=300 K
1/8 0.004” (0.10 mm) 0.0098 3.1 mW 30 mW
3/16 0.004” (0.10 mm) 0.0149 4.7 45
1/4 0.004” (0.10 mm) 0.020 6.3 61
3/8 0.006” (0.15 mm) 0.045 14 137
1/2 0.006” (0.15 mm) 0.060 19 184
5/8 0.006” (0.15 mm) 0.075 24 230
3/4 0.006” (0.15 mm) 0.091 29 277
1 0.006” (0.15 mm) 0.121 38 370
1 1/4 0.010” (0.25 mm) 0.251 80 770
1 1/2 0.010” (0.25 mm) 0.302 96 924
2 0.015” (0.38 mm) 0.604 191 1847
All dimensions are in inches.
a From:
V. Johnson (1960), NBS, Wright Air Development Div. (WADD) Technical Report 60-56, Part II. U.S. Government Printing Office, Washington, D.C.
D. H. J. Goodall (1970), A.P.T. Division, Culham Science Center, Abingdon, Oxfordshire, UK.
A3.3 Pipe and tubing sizes a,b (Sec. 3.5)
Type K Copper Tubing
Brass Pipe
Steel and PVC Pipe, Schedule 40
Soft Copper Refrigeration Tubing
Nominal Size
[inches] Internal Diameter External Diameter Internal Diameter External Diameter Internal Diameter External Diameter Internal Diameter External Diameter
1/8 NA c NA NA NA 0.269 0.405 0.065 0.125
1/4 0.30 0.375 0.410 0.540 0.364 0.540 0.190 0.250
3/8 0.40 0.500 0.545 0.675 0.493 0.675 0.311 0.375
1/2 0.53 0.625 0.710 0.840 0.622 0.840 0.436 0.500
5/8 0.65 0.750 NA NA NA NA 0.555 0.625
3/4 0.75 0.875 0.920 1.050 0.824 1.050 0.680 0.750
1 1.00 1.125 1.185 1.315 1.049 1.315
1-1/4 1.25 1.375 1.530 1.660 1.380 1.660
1-1/2 1.48 1.625 1.770 1.900 1.610 1.900
2 1.96 2.125 2.245 2.375 2.067 2.375
2-1/2 2.44 2.625 2.745 2.875 2.469 2.875
3 2.91 3.125 3.334 3.500 3.068 3.500
3-1/2 3.39 3.625 3.810 4.000 3.548 4.000
4 3.86 4.125 4.296 4.500 4.026 4.500
5 4.81 5.125 5.298 5.562 5.047 5.562
6 5.74 6.125 6.309 6.625 6.065 6.625
a From B. Brandt (2002), National High-Field Magnet Laboratory, Florida State University, personal communication.
b All dimensions are in inches.
c NA ≡ Not Available.
A3.4 Screw and bolt sizes, hexagon socket-head sizes, and load limits (Sec. 3.3.1)
Maximum load and minimum engaged thread length are determined for stainless-steel (SS) bolts assuming a yield strength of 414 MPa (60 ksi). Hexagon socket-head diameters and heights are given to facilitate laying out bolt circles on vacuum flanges.
Screw a
Size –
Number of threads per inch
Major Diam.
[inches (mm)]
Nearest Standard Metric Size
Maximum b
load
(SS bolts)
[lbf (kN)]
Engaged length c
(SS into SS)
[inches (mm)]
Number c engaged threads
(SS into SS)
Engaged length
(SS into Al)
[inches (mm)]
Number engaged threads
(SS into Al)
Socket head diameter d
[inches]
Max Min
Socket head height d
[inches]
Max Min Tap drill size
(inch, number, & letter drills)
Clearance drill size
(number & inch drills)
0-80 0.0600 (1.524) M1.6 0.35 108 (0.48) 0.0328 (0.833) 2.6 0.0654 (1.66) 5.2 0.096 0.091 0.060 0.057 3/64 51
1-64 0.0730 (1.854) M2 0.4 157 (0.70) 0.0396 (1.01) 2.5 0.0786 (2.00) 5.0 0.118 0.112 0.073 0.070 53 47
1-72 “ “ 167 (0.74) 0.0407 (1.03) 2.9 0.0831 (2.11) 6.0 “ “ “ “ 53 47
2-56 0.0860 (2.184) “ 222 (0.99) 0.0471 (1.20) 2.6 0.0938 (2.38) 5.3 0.140 0.134 0.086 0.083 50 42
2-64 “ “ 236 (1.05) 0.0482 (1.22) 3.1 0.0100 (0.25) 6.4 “ “ “ “ 50 42
3-48 0.0990 (2.515) M2.5 0.45 292 (1.30) 0.0539 (1.37) 2.6 0.0107 (0.27) 5.2 0.161 0.154 0.099 0.095 47 37
3-56 “ “ 314 (1.40) 0.0558 (1.42) 3.1 0.115 (2.93) 6.5 “ “ “ “ 46 37
4-40 0.1120 (2.845) M3 0.5 362 (1.61) 0.0602 (1.53) 2.4 0.118 (2.99) 4.7 0.183 0.176 0.112 0.108 43 31
4-48 “ “ 396 (1.76) 0.0625 (1.59) 3.0 0.129 (3.27) 6.2 “ “ “ “ 3/32 31
5-40 0.1250 (3.175) “ 477 (2.12) 0.0688 (1.75) 2.8 0.139 (3.53) 5.6 0.205 0.198 0.125 0.121 38 29
5-44 “ “ 499 (2.22) 0.0703 (1.79) 3.1 0.145 (3.68) 6.4 “ “ “ “ 37 29
6-32 0.1380 (3.505) M4 0.7 545 (2.42) 0.0741 (1.88) 2.4 0.144 (3.65) 4.6 0.226 0.218 0.138 0.134 36 27
6-40 “ “ 609 (2.71) 0.0775 (1.97) 3.1 0.161 (4.08) 6.4 “ “ “ “ 33 27
8-32 0.1640 (4.166) “ 841 (3.74) 0.0914 (2.32) 2.9 0.186 (4.73) 6.0 0.270 0.262 0.164 0.159 29 18
8-36 “ “ 884 (3.93) 0.0932 (2.37) 3.4 0.196 (4.98) 7.1 “ “ “ “ 29 18
10-24 0.1900 (4.826) M5 0.8 1 050 (4.68) 0.103 (2.61) 2.5 0.201 (5.11) 4.8 0.312 0.303 0.190 0.185 26 9
10-32 “ “ 1 200 (5.34) 0.109 (2.76) 3.5 0.230 (5.83) 7.3 “ “ “ “ 21 9
12-24 0.2160 (5.486) “ 1 450 (6.45) 0.120 (3.05) 2.9 0.244 (6.20) 5.9 — — — — 16 2
12-28 “ “ 1 550 (6.88) 0.123 (3.13) 3.5 0.261 (6.62) 7.3 — — — — 15 2
1/4-20 0.2500 (6.350) M6 1.0 1 910 (8.49) 0.138 (3.51) 2.8 0.278 (7.05) 5.6 0.375 0.365 0.250 0.244 7 17/64
1/4-28 “ “ 2 180 (9.71) 0.146 (3.71) 4.1 0.318 (8.07) 8.9 “ “ “ “ 3 17/64
5/16-18 0.3125 (7.938) M8 1.25 3 150 (14.0) 0.177 (4.48) 3.2 0.366 (9.30) 6.6 0.469 0.457 0.312 0.306 F 21/64
5/16-24 “ M8 1.0 3 480 (15.5) 0.185 (4.69) 4.4 0.405 (10.3) 9.7 “ “ “ “ I 21/64
3/8-16 0.3750 (9.525) M10 1.5 4 650 (20.7) 0.214 (5.43) 3.4 0.451 (11.5) 7.2 0.562 0.550 0.375 0.368 5/16 25/64
3/8-24 “ M10 1.0 5 270 (23.4) 0.226 (5.75) 5.4 0.511 (12.3) 12.3 “ “ “ “ Q 25/64
7/16-14 0.4375 (11.112) M12 1.75 6 380 (28.4) 0.251 (6.37) 3.5 0.530 (13.5) 7.4 0.656 0.642 0.438 0.430 U 29/64
7/16-20 “ M12 1.25 7 120 (31.7) 0.263 (6.68) 5.3 0.592 (15.0) 11.8 “ “ “ “ 25/64 29/64
1/2-13 0.5000 (12.700) M12 1.75 8 510 (37.9) 0.289 (7.34) 3.8 0.619 (15.7) 8.1 0.750 0.735 0.500 0.492 27/64 33/63
1/2-20 “ M12 1.25 9 600 (42.7) 0.305 (7.74) 6.1 0.698 (17.7) 14.0 “ “ “ “ 29/64 33/64
9/16-12 0.5625 (14.288) M16 2.0 10 900 (48.6) 0.327 (8.30) 3.9 0.706 (17.9) 8.5 — — — — 31/64 37/64
9/16-18 “ M16 1.5 12 200 (54.2) 0.343 (8.72) 6.2 0.788 (20.0) 14.2 — — — — 33/64 37/64
5/8-11 0.6250 (15.875) M16 2.0 13 600 (60.3) 0.364 (9.25) 4.0 0.790 (20.1) 8.7 0.938 0.921 0.625 0.616 17/32 41/64
5/8-18 “ M16 1.5 15 400 (68.3) 0.385 (9.78) 6.9 0.894 (22.7) 16.1 “ “ “ “ 37/64 41/64
3/4-10 0.7500 (19.050) M20 2.5 20 100 (89.3) 0.442 (11.2) 4.4 0.973 (24.7) 9.7 1.125 1.107 0.750 0.740 21/32 49/64
3/4-16 “ M20 1.5 22 400 (99.5) 0.464 (11.8) 7.4 1.09 (25.6) 17.4 “ “ “ “ 11/16 49/64
Al ≡ aluminum, SS ≡ stainless steel
a ANSI screw thread standard.
b It is good practice to derate these maximum loads by about a factor of 2 safety margin.
c To prevent galling and seizing, especially for stainless steel into stainless steel, use silver-plated stainless-steel bolts or coat them with MoS2. Sources of such screws are listed in Appendix A1.7 under Vacuum accessories, Screws.
d From R. O. Parmley, ed. (1997), Standard Handbook of Fastening and Joining, McGraw–Hill.
A3.5 Clearances for various types of fits
When machining parts that need to slip or slide over each other, the required gap varies with the type of fit desired and the diameter of the part. The following table can be used as a rough guide. For more critical parts, follow the detailed specifications in the Machinery’s Handbook (2000), Industrial Press, Inc., New York.
Be sure to adjust the gap for any difference in thermal contractions between the two materials.
The clearance gaps tabulated below are appropriate only for moving parts that are protected from repeated air exposure. Beware of liquid air films that can freeze movable parts (see the tip in Sec. 1.5.1 for preventing this). Also, the gap between a dip probe and the inner wall of a dewar (or the bore of a magnet) must be much larger than the clearances indicated below. A 1 mm to (preferably) 2 mm gap is needed to accommodate frost that can form on surfaces during repeated insertion and removal of probes from a dewar.
Type of Fit Approximate Gap for a 1/8th inch (3.2 mm) diameter shaft
[10–3 inch] Approximate Gap for a 1 inch (25 mm) diameter shaft
[10–3 inch] Approximate Gap for a 5 inch (127 mm) diameter shaft
[10–3 inch]
Running fit 0.3 1 2
Sliding fit 0.15 0.5 1
Push fit ~ 300 oC) to activate flux
49Bi–18Pb–12Sn–21In 58 °C
136 °F 9.01 2.43 10 23.0 43 Eutectic alloy; expands slightly on solidification and then shrinks slowly over several hours.
50Bi–25Pb–12.5Sn–12.5Cd
(Wood’s metal)
65–70 °C
149–158 °F 9.60 3.1 31 Contains Cd, whose fumes are TOXIC.
Similar to Ostalloy® 158
50Bi–26.7Pb–13.3Sn–10Cd
(Cerrobend)
70 °C
158 °F 9.58 4.0 18 22.0 41 Contains Cd, whose fumes are TOXIC.
66.3In–33.7Bi
72 °C
162 °F
7.99
Eutectic; very low melting temperature solder for thin Ag or Au films and contacting high-Tc superconductors; low strength
Mild ZnCl2 solution
55.5Bi–44.5Pb
(Cerrobase)
124 °C
255 °F 10.44 4.0 4 44 Contracts slightly on solidification
a Indium Corp. of America, http://www.indium.com/
b J. Ross (2002), Canfield Corp., personal communication
A3.8 Solder fluxes for soft-soldering common metals and alloys a (Sec. 3.3.4)
Material
Flux
Mild b Corrosive c Special Flux and/or Solder d
Aluminum •
Aluminum–Bronze •
Beryllium Copper •
Brass • •
Copper • •
Copper–Chromium •
Copper–Nickel •
Copper–Silicon •
Gold •
Inconel •
Lead • •
Magnesium •
Monel •
Nickel •
Nichrome •
Platinum •
Silver • •
Stainless Steel •
Steel •
Tin • •
Tin–Zinc • •
Zinc •
a Information from J. F. Smith and D. M. Borcina, Lead Industries Assoc., Inc., New York, New York.
b Mild fluxes: rosin, rosin in alcohol, paste of petroleum jelly, zinc chloride, or ammonium chloride. After soldering, wash away flux with a solution of soap and water, or isopropanol. Be aware that fluxes other than pure rosin, or rosin dissolved in alcohol, will leave chloride residues trapped in the solder that eventually react with ambient moisture to form hydrochloric acid, which attacks electronic circuits and perforates thin (0.1 mm) stainless-steel tubing. For soldering copper electronic circuitry, use only pure rosin flux, not “activated” rosin flux or pastes. See Sec. 3.3.4 for more information.
c Corrosive flux: zinc-chloride solution (zinc dissolved in hydrochloric acid). After soldering, wash away flux with water or isopropanol; then neutralize the pH by blotting the area with a baking-soda/water solution or ammonia/detergent/water solution.
d Special Flux and/or Solder: Appendix A3.7 has information on highly-corrosive stainless-steel soldering fluxes as well as types of solders and fluxes that work with aluminum. After soldering, wash away corrosive acid fluxes with water or isopropanol; then neutralize the pH by blotting the area with a baking-soda/water solution or ammonia/detergent/water solution.
A3.9 Solder: Superconducting properties a (Sec. 3.3.4)
Tc ≡ superconducting transition temperature of the solder
Hc ≡ superconducting critical field of the solder
Solder
[wt%] Tc
[K] Hc (1.3K)
[T] Melting Temperature
[oC]
60Sn–40Pb 7.05 0.08 182–188
50Sn–50Pb 7.75 0.20 182–216
30Sn–70Pb
7.45 0.15 182–257
95Sn–5Sb
3.75 0.036 232–240
50In–50Sn 7.45 0.64 117–125
50In–50Pb
6.35 0.48 180–209
97.5Pb–1.0Sn–1.5Ag 7.25 0.11 309
a From W. H. Warren and W. G. Bader (1969), Rev. Sci. Instrum. 40, 180–182.
A3.10 Sticky stuff for cryogenic applications (Sec. 3.3.5)
Material
Application and comments
Epoxies
Araldite Type 1 a
Eccobond 2 b Low-viscosity, unfilled epoxy. Robust and good adhesion at cryogenic temperatures.
Scotch-Weld DP-460 c
High performance urethane, two-part epoxy, Duo-Pak™ cartridge.
Silver-based epoxy d Electrically and thermally conductive epoxy.
Stycast 1266 e Low-viscosity, unfilled epoxy. High thermal expansion, but thin films of this epoxy do not crack and provide good adhesion at cryogenic temperatures. Crack resistance can be improved by heating to 90 oC for 4 h after epoxy has hardened.
Stycast 2850 FT e High-viscosity epoxy; filled with silica powder to provide a low thermal expansion matching that of copper.
Tapes
Fiberglass Electrical Tape Tough under cryogenic cycling and withstands cycling to higher-temperatures when soldering.
Kapton Tape A robust tape, well suited for providing tough, durable electrical insulation between cryostat parts.
Masking Tape
All-purpose tape. The adhesion improves with thermal cycling. Tape becomes brittle with age and eventually becomes difficult to remove.
Mylar Electrical Tape (3M #56ƒ, “yellow” tape)f Maintains adhesion better than Kapton tape upon cryogenic cycling, but thinner (10–3 inch) and therefore better suited for applications where strength is not paramount. Commonly used for electrically isolating samples from Cu sample holders. Dielectric strength is 5500 V.
Teflon Pipe-thread Tape
Excellent for wrapping wires to supports structure for mechanical support, or fastening samples to sample holders, especially where you do not want to deal with sticky tape that is hard to remove. For the same reason, this is also the best tape for corralling fine delicate wires. To protect small wires from mechanical damage, place a layer of tape under the wires as well as over them when wrapping them to a support structure.
Varnish and Glues
Bostik Multibond Glue g
All-purpose glue that holds well at cryogenic temperatures. Easier to work with if thinned with acetone or methyl-ethyl-ketone.
Duco Household Cement (model-airplane glue) All-purpose glue that survives thermal cycling well. Can be thinned or removed with acetone. Not good for wires because the acetone dissolves varnish insulation. Good for sticking samples to the sample rod in a vibrating sample magnetometer.
IMI 7031 varnish (formerly GE 7031) varnish h Easier to work with if thinned to the consistency of water with ethanol (acetone also acts as a thinner, but it makes the varnish stringy and eats wire insulation). Baking the varnish under a heat lamp decreases drying time.
Loctite™ i
Low viscosity adhesive used in machine shops as a substitute for lock nuts, interference fits, or silver soldering. Good for securing tight-fitting metal parts. Cures at room temperature, but can be loosened by moderate heating with a torch. Works OK at cryogenic temperatures.
White Shellac
Useful for adhering sapphire to sapphire.
Miscellaneous
Apiezon Black Wax j Meltable adhesive.
Beeswax, and Alox 350 (Tmelt=38 oC to 43 oC), and Alox 2138F (Tmelt=71 oC)k
Low-strength fillers. Although they yield at low stress, they are sometimes useful as magnet-coil filling agents to minimize the probability of thermal-runaway events that can otherwise result from microfracturing of epoxies used to impregnate superconducting coils.
BluTack l
A gummy clay-like adhesive for generally attaching leads to support structures or mechanically holding almost anything in place.
Dental Floss (waxed or no-wax) Excellent for tying things together (like samples to samples holders) and overwrapping fragile instrumentation leads wound onto heat sinks. Waxed floss is a little easier to stick in place during wrapping and tying.
Silver paste d Electrically and thermally conductive weak adhesive.
Suppliers of specialty materials include:
a Ciba Specialty Chemicals Corp, 4917 Dawn Ave., East Lansing, MI 48823, Tel. 517-351-5900, Fax 517-351-9003, http://www.araldite.com/
b Emerson and Cuming Corp., http://www.emersoncuming.com/
c 3M, http://www.3M.com/; distributed by MSC Industrial Supply, PN: 65861684, (Duo-Pak cartridge PN: 65861569), Tel. 800-645-7270, http://www.mscdirect.com/; or McMaster–Carr Supply Co., PN: 7467A26, http://www.mcmaster.com/
d Ted Pella, Inc., P.O. Box 492477, Redding, CA 96049-2477, Tel. 800-237-3526; Fax. 530-243-3761, http://www.TedPella.com/
e Emerson & Cumming, http://www.emersoncumming.com/
f Essex Brownell Inc., 4670 Shelby Drive, Memphis, TN 38118, Tel. 800-805-4636, Fax. 219-461-4165; or from http://www.mpsupplies.com/3mtape56.html
g Bostik Pty. Ltd., 51–71 High Street, Thomastown, Vic., Australia 3074, Tel 3-465-5211
h Insulating Materials Inc., 1 W. Campbell Rd., Schenectady, NY 12306, Tel. 518-395-3200, Fax. 518-395-3300; small quantities available from Lake Shore Cryotronics, Westerville, OH 43081, Tel. 614-891-2244, Fax. 614-818-1600, http://www.lakeshore.com/
i Loctite, a Hendel Company, http://www.loctite.com/
j Apiezon Products, M&I Materials Ltd., P.O. Box 136, Manchester, M601AN, England. Tel. +44 161 875 4442, http://www.apiezon.com/
k Alox Corp., Niagara Falls, NY
l Bostik Findley, http://www/bostikfindley-us.com/
A3.11 Slippery stuff for cryogenic applications
Material
Application and comments
Lubricant coatings
Graphite
Available as dry powder or spray-on coatings
Molybdenum disulfide Spray coatings; good for higher forces
Teflon ™ Spray coatings, low coefficient of friction
Thicker lubricant coatings
Emralon ® a Flurocarbon lubricant in an epoxy mixture for thicker lubricating coatings or for making cast parts with a low coefficient of friction
Bearing materials
Kel-F ™ b Polychlorotrifluoroethylene. Stronger than Teflon ™
Nylon ™ Stronger than Teflon™, but higher coefficient of friction
Teflon ™
Polyamide, low coefficient of friction, but softer than other materials
Teflon™ materials reinforced with Nylon™, fiberglass and other materials
Flurogold ® c
Reinforced Teflon™
Parmax ® d High strength polymer, similar uses as Torlon ™
Rulon ® b Teflon reinforced with Nylon, fiberglass, or other materials; available in various formulations. Type J has the lowest coefficient of friction of the Rulon® series. Applications include retainer rings for cryogenic ball-bearing raceways.
Teflon-coated Kapton ™ e Useful, for example, as a cryogenic gasket material since the Teflon™ coating deforms for good sealing, but the stronger Kapton™ base keeps the gasket from extruding.
Torlon ™ e PolyAmide-Imide (Teflon™-Kapton™ combination) high strength polymer used for wear and friction parts. Capable of performing under continuous stress at temperatures to 260°C. Low coefficient of linear thermal expansion and high creep resistance provide good dimensional stability. Available as sheet, rod, or tube.
a Acheson Colloids Co., http://www.achesonindustries.com/
b San Diego Plastics, Inc., http://www.sdplastics.com/
c Granor Rubber and Engineering, http://www.granor.com.au/, Conroy & Knowlton Inc., http://conroyknowlton.com/materials.htm
d Mississippi Polymer Technologies, http://www.mptpolymers.com/
e Boedeker, http://www.boedeker.com/
A3.12 Degassing rates of synthetic materials a (Sec. 3.8.3)
Degassing rates of metals are given in Fig. 3.18.
Material Degassing rate at room temperature before baking
[Pa∙m3∙s–1∙m–2] Baking temperature
[oC] Degassing rate at room temp. after 24 h bake
[Pa∙m3∙s–1∙m–2]
Araldite ATI epoxy b 3.4 10–4 85 —
Mycalex b 2.7 10–6 300 —
Nylon 31 b 1.1 10–4 120 8.0 10–7
Perspex b 1.3 10–5 85 7.8 10–6
Polythene b 4.0 10–4 80 6.6 10–6
PTFE (Teflon) c 2.0 10–4 — 4.7 10–7 a
Viton A b 1.3 10–4 200 2.7 10–6
Polyimide (Kapton) d — 200* 6.6 10–8
— 300* 4.0 10–8
Kalrez e — 300 4.0 10–8
Viton E60C e — 150 ~1 10–6
— 300 3.0 10–8
* 12 h bake
a Compiled by G. F. Weston (1985), Ultrahigh Vacuum Practice, Butterworth, London.
b R. S. Barton and R. P. Govier (1965), J. Vac. Sci. Tech. 2, 113.
c B. B. Dayton (1959), Trans. 6th Nat. Symp. Vac. Technol., I, p. 101.
d P. W. Hait (1967), Vacuum 17, 547.
e L. DeChernatony (1977), Vacuum 27, 605.
A3.13 Vapor pressures of metals a (Sec. 3.8.3)
Tabulated values in the three right-hand columns are expressed as the temperature required to produce the vapor pressures indicated at the head of each column.
These data are plotted in Figs. 3.19a and 3.19b.
Metal Melting Temperature [K] Temperature [K] giving a vapor pressure P
P = 1.3310-9 Pa P = 1.3310–7 Pa P = 1.3310–5 Pa
Ag Silver 1234 721 800 899
Al Aluminum 932 815 906 1015
Au Gold 1336 915 1020 1150
Ba Barium 983 450 510 583
Be Beryllium 1556 832 925 1035
C Carbon — 1695 1845 2030
Ca Calcium 1123 470 524 590
Cd Cadmium 594 293 328 368
Ce Cerium 1077 1050 1175 1325
Co Cobalt 1768 1020 1130 1265
Cr Chromium 2176 960 1055 1175
Cs Cesium 302 213 241 274
Cu Copper 1357 855 945 1060
Fe Iron 1809 1000 1105 1230
Ge Germanium 1210 940 1030 1150
Hg Mercury 234 170 190 214
In Indium 429 641 716 812
Ir Iridium 2727 1585 1755 1960
K Potassium 336 247 276 315
La Lanthanum 1193 1100 1220 1375
Mg Magnesium 923 388 432 487
Mn Manganese 1517 660 734 827
Mo Molybdenum 2890 1610 1770 1975
Na Sodium 371 294 328 370
Ni Nickel 1725 1040 1145 1270
Pb Lead 601 516 580 656
Pd Palladium 1823 945 1050 1185
Pt Platinum 2043 1335 1480 1655
Re Rhenium 3463 1900 2100 2350
Rh Rhodium 2239 1330 1470 1640
Sb Antimony 903 447 526 582
Se Selenium 490 286 317 356
Sn Tin 505 805 900 1020
Sr Strontium 1043 433 483 546
Ta Tantalum 3270 1930 2120 2370
Th Thorium 1968 1450 1610 1815
Ti Titanium 1940 1140 1265 1410
W Tungsten 3650 2050 2270 2520
Zn Zinc 693 336 374 421
Zr Zirconium 2128 1500 1665 1855
a From G. F. Weston (1985), Ultrahigh Vacuum Practice, Butterworth, London, who extracted the data from compilations by R. E. Honig (1962), RCA Rev. 23, 567; and R. E. Honig (1969), RCA Rev. 30, 285.
A3.14 Gas permeation constant at room temperature for synthetic materials a [for use with Eq. (3.27) of Sec. 3.8.3]
Additional gas permeations rates are given for:
• helium through glass in Fig. 3.20
• helium through ceramics in Fig. 3.21
• hydrogen through metals in Fig. 3.22.
Permeation constant K in [m2 s–1] at 23 oC
Material Nitrogen Oxygen Hydrogen Helium Argon
Polythene b 9.9 10–13 3.0 10–12 8.2 10–12 5.7 10–12 2.7 10–12
PTFE (Teflon) b 2.5 10–12 8.2 10–12 2.0 10–11 5.7 10–10 4.8 10–12
Perspex b — — 2.7 10–12 5.7 10–12 —
Nylon 31 b — — 1.3 10–13 3.0 10–13 —
Polystyrene b — 5.1 10–13 1.3 10–11 1.3 10–11 —
Polystyrene c 6.4 10–12 2.0 10–11 7.4 10–11 – —
Polyethylene c 6–11 10–13 2.5–3.4 10–12 6–12 10–12 4–5.7 10–12 —
Mylar 25-V-200 c — — 4.8 10–13 8.0 10–13 —
CS2368B (Neoprene) b 2.1 10–13 1.5 10–12 8.2 10–12 7.9 10–12 1.3 10–12
Viton A b — — 2.2 10–12 8.2 10–12 —
Polyimide (Kapton) d 3.2 10–14 1.1 10–13 1.2 10–12 2.1 10–12 —
a Compiled by G. F. Weston (1985), Ultrahigh Vacuum Practice, Butterworth, London.
b J. R. Bailey (1964), Handbook of Vacuum Physics, Vol. 3, Part 4, Pergamon Press.
c D. W. Brubaker and K. Kammermeyer, Ind. Eng., Chem. 44, 1465 (1952); 45, 1148 (1953); 46, 733 (1954).
d D. E. George, in an article by W. G. Perkins (1973), J. Vac. Sci. Technol. 10, 543.
A4. Cryogenic apparatus wiring (ref. Chapter 4)
A4.1a Wire gauge size, area, resistivity, heat conduction, and optimum current (Secs. 4.1, 4.2, and 4.9.1)
To obtain the resistance-per-length for wire materials other than copper, multiply the room-temperature values for copper in the fourth column of the table by the ratio 293K/Cu 293K, where 293K and Cu 293K are the resistivity values of the new material and copper, respectively. Ratio values for several common wire materials at room temperature follow the table.
At 77 K and 4.2 K, the resistance-per-length may be similarly calculated by using the low-temperature resistivity data given in Appendix A4.2.
In practice, resistivity values may vary from those tabulated below because of different impurity concentrations, alloy concentrations, and heat treatments.
American Wire Gauge (AWG) and Brown & Sharpe (B&S) are the same gauge.
The nearest common metric wire sizes are given in the next table, A4.1b.
American Wire Gauge (AWG) or
Brown & Sharpe (B&S) Diameter 20C a
[mm] Cross-sectional area at 20C a
[mm2] Resistance of annealed copper wire at 20C a
[/km] Heat conducted along 1 m of copper wire between the indicated temperatures b
[W] Optimum current for 1 m of copper wire with one end at 4 K and the other at temperature Tupper c
[A]
300K–4.2K 300K–76K 76K–4.2K Tupper = 290K Tupper = 77K
0000 11.68 107.2 0.161 17.4 10.0 7.35 536 1072
000 10.40 85.03 0.203 13.8 7.94 5.83 425 850
00 9.266 67.43 0.256 10.9 6.30 4.62 337 674
0 8.252 53.48 0.322 8.66 5.00 3.67 267 535
1 7.348 42.41 0.407 6.87 3.96 2.91 212 424
2 6.543 33.63 0.513 5.45 3.14 2.31 168 336
3 5.827 26 67 0.646 4.32 2.49 1.83 133 267
4 5.189 21.15 0.815 3.43 1.98 1.45 106 212
5 4.621 16.77 1.03 2.72 1.57 1.15 84 168
6 4.115 13.30 1.30 2.15 1.24 0.912 66 133
7 3.665 10.55 1.63 1.71 0.985 0.724 53 106
8 3.264 8.366 2.06 1.36 0.781 0.574 42 84
9 2.906 6.634 2.60 1.08 0.620 0.455 33 66
10 2.588 5.261 3.28 0.852 0.491 0.361 26 53
11 2.305 4.172 4.13 0.676 0.390 0.286 21 42
12 2.053 3.309 5.21 0.536 0.309 0.227 16 33
13 1.828 2.624 6.57 0.425 0.245 0.180 13 26
14 1.628 2.081 8.28 0.337 0.194 0.143 10 21
15 1.450 1.650 10.4 0.267 0.154 0.113 8.2 16
16 1.291 1.309 13.2 0.212 0.122 0.0898 6.5 13
17 1.150 1.038 16.6 0.168 0.0969 0.0712 5.2 10
18 1.024 0.8231 21.0 0.133 0.0769 0.0565 4.1 8.2
19 0.9116 0.6527 26.4 0.106 0.0610 0.0446 3.3 6.5
20 0.8118 0.5176 33.3 0.0838 0.0483 0.0355 2.6 5.2
21 0.7230 0.4105 42.0 0.0665 0.0383 0.0282 2.0 4.1
22 0.6439 0.3255 53.0 0.0527 0.0304 0.0223 1.6 3.2
23 0.5733 0.2582 66.8 0.0418 0.0241 0.0177 1.3 2.6
24 0.5105 0.2047 84.2 0.0332 0.0191 0.0140 1.0 2.0
25 0.4547 0.1624 106 0.0263 0.0152 0.0111 0.81 1.6
26 0.4049 0.1288 134 0.0209 0.0120 0.00884 0.64 1.3
27 0.3606 0.1021 169 0.0165 0.00954 0.00700 0.51 1.0
28 0.3211 0.08098 213 0.0131 0.00756 0.00556 0.40 0.81
29 0.2859 0.06422 268 0.0104 0.00600 0.00440 0.32 0.64
30 0.2548 0.05093 339 0.00825 0.00476 0.00349 0.25 0.51
31 0.2268 0.04039 427 0.00654 0.00377 0.00277 0.20 0.40
32 0.2019 0.03203 538 0.00519 0.00299 0.00220 0.16 0.32
33 0.1798 0.02540 679 0.00411 0.00237 0.00174 0.13 0.25
34 0.1601 0.02014 856 0.00326 0.00188 0.00138 0.10 0.20
35 0.1426 0.01597 1080 0.00259 0.00149 0.00110 0.080 0.16
36 0.1270 0.01267 1360 0.00205 0.00118 0.000869 0.063 0.13
37 0.1131 0.01005 1720 0.00163 0.000939 0.000689 0.050 0.10
38 0.1007 0.007967 2160 0.00129 0.000744 0.000546 0.040 0.080
39 0.08969 0.006318 2730 0.00102 0.000590 0.000433 0.032 0.063
40 d 0.07988 0.005010 3440 0.000812 0.000468 0.000344 0.025 0.050
a Data obtained partially from calculations (see following footnotes) and partially from tabulations in the CRC Handbook of Chemistry and Physics (1987; 2002), CRC Press, Inc., Boca Raton, Florida; and from the Machinery’s Handbook (2000), 26th edition, Industrial Press, New York.
b Heat conduction for a length other than 1 m is obtained by dividing the values in the table by the desired wire length (in meters). In obtaining the values for heat conduction, it was assumed that there was no gas cooling of the wire. If helium gas boil-off were used to cool the wire with maximum efficiency, the resultant heat flow would be 1/12 of the values given for an upper temperature of 300 K, and 1/4 of the values shown for 77 K. Calculations were based on the thermal conductivity integrals of electrolytic-tough-pitch (ETP) copper, Appendix A2.1. From V. Johnson (1960), National Bureau of Standards; Wright Air Development Division (WADD) Technical Report 60-56, Part II; and D. H. J. Goodall (1970), A.P.T. Division, Culham Laboratory.
c Optimum current is for steady-state operation. For wires that carry current with only a low duty cycle, the optimum current should be adjusted to a higher value because in that case the Joule heating is intermittent, whereas the heat flow down the current lead is continuous. Optimum current for a length other than 1 m is obtained by dividing the values in the table by the desired wire length (in meters). Values were calculated from Eqs. (4.1) and (4.2) in Sec. 4.9.1, which were derived by R. McFee (1959), Rev. Sci. Instrum. 30, 98–102.
d For wire sizes smaller than #40 AWG, the diameter can be calculated by using a ratio of 1 : 1.123 for consecutive AWG sizes.
Room-temperature resistivities for several common wire materials relative to copper.
These ratios can be used to obtain the resistance-per-length for wire materials other than copper by multiplying the room-temperature values given for copper in the fourth column of the above table by the ratio 293K/Cu 293K. (Calculated from Appendix A4.2 and the CRC Handbook of Chemistry and Physics 2002.)
Material 293 K/Cu 293 K
Aluminum 1.579
Brass (70%Cu–30%Zn) 3.62
Constantan 29
Manganin 28
Nichrome 64
Phosphor Bronze 7.5
Platinum 6.26
Silver 0.946
Tungsten 3.15
A4.1b Wire gauge: Metric and American Wire Gauge (AWG) size comparison (Secs. 4.1 and 4.2)
American Wire Gauge (AWG) or
Brown & Sharpe (B&S) Nearest common metric gauge wire diameter at 20C
[mm] Metric wire
cross-sectional area at 20C
[mm2] Resistance of annealed copper wire at 20C
[Ω/km]
5 4.750 17.72 1.0
6 4.250 14.19 1.2
7 3.750 11.04 1.5
8 3.350 8.814 1.9
9 2.800 6.158 2.8
10 2.500 4.910 3.5
11 2.240 3.941 4.3
12 2.000 3.142 5.4
13 1.800 2.545 6.7
14 1.600 2.011 8.5
15 1.400 1.539 11.1
16 1.250 1.227 13.9
17 1.120 0.9852 17.4
18 1.000 0.7854 21.8
19 0.900 0.636 26.9
20 0.800 0.503 34.0
21 0.710 0.396 43.2
22 0.630 0.312 54.8
23 0.560 0.246 69.4
24 0.500 0.196 87.1
25 0.450 0.159 108
26 0.400 0.126 136
27 0.355 0.0990 173
28 0.315 0.0779 219
29 0.280 0.0616 278
30 0.250 0.0491 348
31 0.224 0.0394 434
32 0.200 0.0314 544
33 0.180 0.0255 672
34 0.160 0.0201 850
35 0.140 0.0154 1110
36 0.125 0.0123 1390
37 0.112 0.00985 1740
38 0.100 0.00785 2180
39 0.090 0.0064 2700
40 0.080 0.0050 3400
A4.2 Physical properties of common wire materials: Composition, resistivity, melting temperature, thermal expansion, magnetoresistance, and magnetic susceptibility a (Sec. 4.2)
Wire Material Chemical Composition Resistivity
at 293 K
at 77 K
at 4.2 K
[∙cm] Melting Range Coef. Thermal Expansion
[oC–1] Magnetores.
R/R0 @
4.2 K and 10 T (perpendicular to wire) d Volume Susceptibility
[SI]
Copper (ETP)
100 wt% Cu
1.68
0.21
~0.02
1056–1083oC 1.68 10–5
(20–100oC) 188 % f 3.2 10–5 @R.T.b
2.5 10–5 @4.2K b
Constantan 55 wt% Cu
45 wt% Ni 49.9 1300–1340°C c 1.5 10–5
(20–100oC)
–2.56 % Ferromagnetic d
Ferromagnetic d
Manganin 83 wt% Cu
13 wt% Mn
4 wt% Ni
48.2
45.4
42.9
1100–1160°C c
(85wt%Cu–
15wt%Mn) 1.9 10–5
(20–100oC) –2.83 % 0.0027 @R.T.d
0.022 @76K d
0.0125 @4.2K d
Nichrome
80 wt% Ni
20 wt% Cr
109
107
106
1400oC 1.73 10–5
(20–1000oC) 0.69 % 5.2 10–4 @R.T.d
8.3 10–4 @76K d
5.6 10–3 @4.2K d
Phosphor Bronze A 94.8 wt% Cu
5 wt% Sn
0.2 wt% P
12.8
11.0
10.7 950–1050oC 1.78 10–5
(20–300oC) 4.5 % e,g –5.2 10–5 @R.T.d
–4.7 10–5 @76K d
–3.3 10–5 @4.2K d
a Except where otherwise cited, data were compiled from Metals Handbook, (1961), Vol. 1, Properties and Selection of Materials (1995), 8th edition, American Society for Metals, Metals Park, Ohio; Temperature Measurement and Control, Lake Shore Cryotronics, Inc., Westerville, Ohio; and C. A. Thompson, W. M. Manganaro, and F. R. Fickett (1990), Cryogenic Properties of Copper, Wall Chart, NIST, and the references cited therein.
b F. R. Fickett (1992), Adv. Cryog. Eng. (Mater.), 38B, 1191–1197.
c T. B. Massalski, ed. (1990), Binary Alloy Phase Diagrams, ASM International, Materials Park, Ohio
d M. Abrecht, A. Adare, and J. W. Ekin (2007), Rev. Sci. Inst. 78, 046104. Susceptibilities at 4.2 K were determined from magnetization vs. magnetic field data; room-temperature and 76 K susceptibilities were calculated from the magnetization measured at H = 100 Oe, except where noted.
e The magnetoresistance of phosphor bronze varies with (trace) impurities in the wire.
f The magnetoresistance of pure copper is strongly dependent on its purity; it can be determined from a normalized “Kohler” plot, such as that shown in Fig. 5.16 of F. R. Fickett, Chapter 5 in Materials at Low Temperatures, R. P. Reed and A. F. Clark, eds., ASM International, Metals Park, Ohio.
g At 76 K and 10 T, the magnetoresistance of phosphor bronze is much smaller than at 4 K, decreasing to about ΔR/R0 = 0.08% (Abrecht et al. 2006, footnote d).
A4.3 Residual Resistance Ratio (RRR) of selected wiring and conductor materials (Sec. 4.2)
RRR R293 K/R4 K = ρ293 K/ρ4 K
RRR values of additional materials are tabulated in Appendix A3.1.
Material
Resistivity at 293 K
[cm]
Resistivity at 4 K
[cm]
RRR
(293 K/4 K)
Copper
Electrolytic-Tough-Pitch, ETP (common wire, rod, and plate material)
1.68 ~0.015 ~110
Oxygen-free copper a,b
99.95% pure; annealed ~500oC for ~1 h in argon or vacuum (<~ 10–4 torr) 1.68 ~0.010 ~160 Oxygen-free copper a 99.95% pure; unannealed 1.71 ~0.038 ~45 Copper ground strap a (1/4 inch wide flexible braid) 1.74 ~0.070 ~25 Silver foil a (rolled) 1.61 ~0.019 ~85 Aluminum 99.9995 % c (pure rolled foil annealed 350oC for 1 h) 2.65 ~0.0005 ~5000 a Measured by R. McDonough (1995), unpublished data, National Institute of Standards and Technology, Boulder, Colorado. b See annealing information in footnote c of Appendix A3.1. c Measured by P. Kirkpatrick (1997), unpublished data, National Institute of Standards and Technology, Boulder, Colorado. A4.4 Wire insulation: Thermal ratings a (Sec. 4.3) Wire Insulation Thermal Rating Polyvinyl Formal (Formvar) 105 oC 221 oF Tetrafluoroethylene (Teflon) 200 oC 392 oF Polyimide (Kapton) 220 oC 428 oF a Data from Temperature Measurement and Control (2002), Sec. 3, Lake Shore Cryotronics, Inc., Westerville, Ohio. A4.5 Thermal anchoring: Required wire lengths (Sec. 4.4) Tabulated values give the tempering length required to bring the designated wire material to within 1 mK of the heat-sink temperature Ts. T1 is the temperature where the lead was last thermally anchored. a Tempering Length for Various Wire Gauges b [cm] Material T1 [K] Ts [K] 0.005 mm2 (#40 AWG)c (~0.080mm)d [cm] 0.013 mm2 (#36 AWG) (~0.125mm) [cm] 0.032 mm2 (#32 AWG) (~0.200mm) [cm] 0.21 mm2 (#24 AWG) (~0.500mm) [cm] Copper 300 80 1.9 3.3 5.7 16.0 300 4 8.0 13.8 23.3 68.8 Phosphor-Bronze 300 80 0.4 0.6 1.1 3.2 300 4 0.4 0.7 1.3 3.8 Manganin 300 80 0.2 0.4 0.4 2.1 300 4 0.2 0.4 0.7 2.0 Stainless Steel 304 300 80 0.2 0.3 0.6 1.7 300 4 0.2 0.3 0.5 1.4 a From D. S. Holmes and S. S. Courts (1998), Chapter 4 in Handbook of Cryogenic Engineering, ed. J. G. Weisend II, Taylor & Francis, Philadelphia, Pennsylvania; based on an earlier calculation by J. G. Hust (1970), Rev. Sci. Instrum. 41, 622–624. (The difference in values between the earlier and later evaluations for copper stems from the use of mean thermal conductivity values by Hust and thermal-conductivity integrals by Holmes and Courts.) b The calculated results pertain to wires with thin, well bonded insulation such as Formvar or polyimide (not NylonÔ or Teflon sleeve insulation) in a vacuum environment (i.e., not cooled by surrounding gas). The insulation-plus-adhesive layer attaching the wire to the heat sink is assumed to have a thickness about equal to the wire diameter and a thermal conductivity typical of varnish, namely 0.01, 0.02, and 0.05 W/(m•K) at 4 K, 20 K, and 78 K, respectively. The length of untempered conductor between T1 and Ts is assumed to be 25 cm; however, increasing this length by a factor of 10 shortens the required tempering length by a factor of less than two. c American Wire Gauge (Appendix A4.1a). d Nearest metric wire size (Appendix A4.1b). A4.6a Thermoelectric voltages of some elements relative to copper a (Sec. 4.6) Tabulated thermoelectric voltages are relative to copper with the reference junction at 0 oC. A positive sign means that, in a simple thermoelectric circuit, the resultant voltage direction produces a current from the material to the copper at the reference junction (0 oC). Values have been ordered by their absolute magnitude at –100 oC or, when not available, at +100 oC. Thus, the higher a material’s position in the table, the closer its thermoelectric voltage matches that of copper. Element –200 oC [mV] –100 oC [mV] 0 oC [mV] +100 oC [mV] +200 oC [mV] Gold –0.02 –0.02 0 +0.02 +0.01 Silver –0.02 –0.02 0 –0.02 –0.06 Iridium –0.06 +0.02 0 –0.10 –0.34 Rhodium –0.01 +0.03 0 –0.06 –0.22 Carbon — — 0 –0.06 –0.29 Indium — — 0 –0.07 — Zinc +0.12 +0.04 0 0.00 +0.06 Cadmium +0.15 +0.06 0 +0.14 +0.52 Thallium — — 0 –0.18 –0.53 Tungsten +0.62 +0.22 0 +0.36 +0.79 Lead +0.43 +0.23 0 –0.32 –0.74 Cesium +0.41 +0.24 0 — — Tin +0.45 +0.25 0 –0.34 –0.76 Cerium — — 0 +0.38 +0.63 Tantalum +0.40 +0.27 0 –0.43 –0.90 Magnesium +0.56 +0.28 0 –0.32 –0.73 Platinum +0.19 +0.37 0 –0.76 –1.83 Aluminum +0.64 +0.43 0 –0.34 –0.77 Molybdenum — — 0 +0.69 +1.36 Thorium — — 0 –0.89 –2.09 Lithium –0.93 +0.63 0 +0.06 — Sodium +1.19 +0.66 0 — — Rubidium +1.28 +0.83 0 — — Calcium — — 0 –1.27 –2.96 Palladium +1.00 +0.85 0 –1.33 –3.06 Mercury — — 0 –1.36 –3.16 Potassium +1.80 +1.15 0 — — Cobalt — — 0 –2.09 –4.91 Nickel +2.47 +1.59 0 –2.24 –4.93 Antimony — — 0 +4.13 +8.31 Bismuth +12.58 +7.91 0 –8.10 –15.40 Germanium –45.81 –26.25 0 +33.14 +70.57 Silicon +63.32 +37.54 0 –42.32 –82.40 a Calculated from thermal emf data compiled in the American Institute. of Physics Handbook (1972), 3rd edition, Chapter 4, McGraw–Hill. A4.6b Thermoelectric voltages of selected technical materials relative to copper a (Sec. 4.6) Tabulated thermoelectric voltages are relative to copper with the reference junction at 0 oC. A positive sign means that in a simple thermoelectric circuit the resultant voltage direction produces a current from the material to the copper at the reference junction (0 oC). Values have been ordered by their absolute magnitude at +100 oC. The higher a material’s position in the table, the closer its thermoelectric voltage matches that of copper. Technical Material –200 oC [mV] –100 oC [mV] 0 oC [mV] +100 oC [mV] +200 oC [mV] Silver Coin (90 Ag–10 Cu) — — 0 +0.04 +0.07 60 Ni–24 Fe–16 Cr — — 0 –0.09 +0.18 Copper–Beryllium — — 0 –0.09 –0.21 Manganin — — 0 –0.15 –0.28 Yellow Brass — — 0 –0.16 –0.34 Copper Coin (95 Cu–4 Sn–1 Zn) — — 0 –0.16 –0.35 Phosphor Bronze — — 0 –0.21 –0.49 Solder (50 Sn–50 Pb) — — 0 –0.30 — Solder (96.5 Sn–3.5 Ag) — — 0 –0.31 — 18-8 Stainless Steel — — 0 –0.32 –0.79 80 Ni–20 Cr — — 0 +0.38 +0.79 Spring Steel — — 0 +0.56 +0.80 Gold–Chromium — — 0 –0.93 –2.15 Iron –2.73 –1.47 0 +1.13 +1.71 Alumel +2.58 +1.66 0 –2.05 –4.00 Chromel P –3.17 –1.83 0 +2.05 +4.13 Nickel Coin (75 Cu–25 Ni) — — 0 –3.52 –7.84 Constantan +5.54 +3.35 0 –4.27 –9.28 a Calculated from thermal emf data compiled in the American Institute of Physics Handbook (1972), 3rd edition, Chapter 4, McGraw–Hill. A4.7 Thermal conductivity of YBCO coated conductors (Sec. 4.10) Thermal conductivity values are tabulated separately for each of the major component materials of YBCO coated conductors. For any particular YBCO conductor, the total thermal conductivity total of the composite conductor is the sum of the contribution of each component i weighted by its fractional cross-sectional area. Thus, for tape conductors total = Σ (di/D) i , where di is the layer thickness of the ith component and D is the total thickness of the tape. Thermal conductivity [W/(mK)] Material 20 K [W/(m∙K)] 50 K a [W/(m∙K)] 90 K a [W/(m∙K)] 110 K [W/(m∙K)] 295 K [W/(m∙K)] YBCO(a–b) 123 phase b (Calc. from melt-textured data) 14 27 22 21 ~18 YBCO(c) 123 phase b (Calc. from melt-textured data) (a–b/c ≈ 6.3) 3.5 4.4 3.2 3.0 ~2.8 YBCO(a–b) 123 + 40% 211 phase b melt textured 10 19 16 15 ~14 YBCO c sintered 5 8 5 5 5 Ag d depends on Ag purity (Sec. 2.2) 1180 620 560 450 Cu (RRR = 100) e 2430 1220 497 452 397 Inconel 625 f = 24.7992 10–6 T2 + 1.989348 10–2 T + 7.899798 (valid 116 K–1255 K) 7.4 9.8 Hastelloy C-276 f UNS N10276 = 3.565928 10–6 T2 + 1.349819 10–2 T + 5.726708 (valid 105 K–811 K) 7.0 (100 K) 10 Nichrome f UNS N06003 (77.3%Ni, 21%Cr) = 2.099567 10–6 T2 + 1.480732 10–2 T + 8.265973 (valid 273 K–1073 K) 13 a The abrupt rise in thermal conductivity below Tc is due to condensation of electrons into superconducting pairs, eliminating them as scatterers of phonons, and thus enhancing the dominant phonon contribution to the thermal conductivity. b M. Ikebe, H. Fujishiro, T. Naito, K. Noto, S. Kohayashi, and S. Yoshizawa (1994), Cryogenics 34, 57–61. c A. Jezowski, J. Mucha, K. Rafalowicz, J. Stepien–Damm, C. Sulkowski, E. Trojnar, A. J. Zaleski, and J. Klamut (1987), Phys. Lett. A 122, 431–433. d Calculated from the resistivity of silver (Appendix A6.5a) by using the Wiedemann–Franz Law (Sec. 2.2) for electronic thermal conduction ( = LN T/, where LN = 2.44 10–8 V2/K2). e Cryogenic Materials Properties Program CD, Release B-01 (June 2001), Cryogenic Information Center, 5445 Conestoga Ct., Ste. 2C, Boulder, Colorado 80301-2724; Ph. (303) 442-0425, Fax (303) 443-1821. f R. Radebaugh et al. (2003), http://www.cryogenics.nist.gov/ and the references listed therein. A5. Temperature measurement tables and controller tuning (ref. Chapter 5) A5.1 Vapor pressure vs. temperature (ITS-90) for cryogenic liquids a (Sections 5.1.6 and 5.4.1) Values are tabulated from 200 kPa (~2 atmospheres) down to the triple point (i.e., the solidification temperature). A table of triple points follows the vapor-pressure table. Atmospheric pressure is 101.325 kPa, corresponding to 760 mm Hg at 0 oC and standard gravity. Stratification: Because of temperature stratification within the cryogenic liquid, these vapor pressure data are useful when lowering the temperature of a cryogenic liquid (pumping on cryogenic bath), but not when raising the temperature (pressurizing the bath). In the latter case, the bottom of the bath is much colder than the surface unless a resistive heater is used to establish thermal equilibrium throughout the depth of the liquid. This can take a half hour or more depending on the heater power. Further information and methods to ensure accurate temperature measurements in cryogenic liquids are given in Secs. 1.2.1 and 5.4.1. The techniques to minimize temperature stratification in the cryogen liquid are particularly important. Hydrostatic pressure head: Even when pumping on cryogenic fluids, be aware that after reaching a given pressure, the temperature at the sample depth in a static bath can increase slowly from the hydrostatic pressure of liquid above the sample. If there is a lot of turbulent mixing of the cryogenic liquid, such as from bubbles that occur during pumping or from a relatively large steady-state heat leak into the bath, the error is minimal. However, if the liquid is static for a while (tens of minutes or more), the temperature at the sample location beneath the liquid surface can rise. This hydrostatic pressure-head correction can be significant, especially for the case of the more dense cryogens, such as liquid nitrogen. The correction is given by ∆P = h, where ∆P is the hydrostatic pressure increase, is the mass density of the cryogenic liquid, and h is the height of liquid above the sample. From the cryogen mass densities tabulated in Appendix A1.5 and the SI conversion factors in Appendix A1.3, we find that the pressure increase amounts to about 1.22 kPa at a depth of 1 m in liquid helium, and about 7.90 kPa at the same depth in liquid nitrogen. At atmospheric pressure, for example, this corresponds to a temperature correction of +13 mK in liquid helium, and a correction of +536 mK in liquid nitrogen. The temperature correction increases at lower bath pressures. Thus, for approximate temperature measurement, vapor-pressure data are fine, but for accuracies better than those just noted, it’s safest to use a cryogenic thermometer in close thermal proximity to the sample (Sec. 5.3.1). Note that neither of these errors (stratification or pressure head) occurs in superfluid helium (i.e., at temperatures below the dashed line in the 4He column below). Superfluid helium has an extremely high thermal conductivity (see Sec. 1.2.2) and thus, in this case, vapor-pressure data serve to determine sample temperature very accurately. An extensive tabulation of additional physical properties of cryogenic liquids is given in Appendix A1.5. Vapor pressure vs. temperature for cryogenic liquids Pressure [kPa] 3He b [K] 4He c [K] Para* H2 d [K] Ne e [K] N2 f [K] Ar h [K] O2 i [K] CH4 j [K] 200 5.036 22.805 29.558 83.626 94.290 97.245 120.622 190 4.970 22.596 29.357 83.115 93.722 96.672 119.894 180 4.901 22.379 29.148 82.584 93.130 96.077 119.137 170 4.829 22.153 28.931 82.030 92.514 95.454 118.347 160 4.754 21.918 28.703 81.451 91.869 94.805 117.521 150 4.676 21.672 28.466 80.845 91.194 94.123 116.655 140 4.594 21.413 28.216 80.207 90.483 93.406 115.744 130 4.507 21.142 27.952 79.533 89.732 92.649 114.782 120 4.416 20.855 27.672 78.819 88.936 91.845 113.762 110 3.269 4.319 20.550 27.376 78.059 88.087 90.989 112.674 105 3.224 4.269 20.389 27.219 77.659 87.641 90.538 112.102 101.325† 3.191 4.230 20.268 27.100 77.355 87.302 90.196 111.667 100 3.178 4.216 20.224 27.057 77.244 87.178 90.070 111.508 95 3.130 4.162 20.053 26.888 76.812 86.696 89.584 110.890 90 3.080 4.106 19.875 26.713 76.363 86.195 89.077 110.248 85 3.028 4.048 19.689 26.530 75.895 85.672 88.549 109.576 80 2.974 3.988 19.496 26.339 75.405 85.124 87.995 108.874 75 2.918 3.925 19.293 26.138 74.891 84.550 87.414 108.137 70 2.859 3.859 19.081 25.927 74.349 83.945 86.802 107.360 65 2.797 3.790 18.857 25.704 73.777 86.155 106.540 60 2.732 3.717 18.620 25.466 73.170 85.467 105.668 58 2.705 3.687 18.522 25.367 72.916 85.180 105.303 56 2.677 3.656 18.421 25.266 72.655 84.884 104.929 54 2.649 3.624 18.317 25.162 72.387 84.580 104.543 52 2.620 3.591 18.211 25.054 72.111 84.268 104.147 50 2.590 3.558 18.101 24.944 71.826 83.945 103.738 48 2.559 3.524 17.988 24.830 71.533 83.612 103.316 46 2.528 3.489 17.871 24.712 71.230 83.268 102.880 44 2.495 3.452 17.750 24.590 70.916 82.912 102.429 42 2.462 3.415 17.627 70.591 82.543 101.962 40 2.427 3.377 17.498 70.254 82.160 101.476 38 2.392 3.337 17.364 69.903 81.762 100.971 36 2.355 3.295 17.226 69.537 81.346 100.445 34 2.317 3.252 17.081 69.155 80.912 99.895 32 2.277 3.208 16.929 68.755 80.456 99.318 30 2.236 3.161 16.770 68.334 79.978 98.712 29 2.214 3.137 16.689 68.116 79.729 98.397 28 2.193 3.112 16.604 67.891 79.473 98.073 27 2.170 3.087 16.517 67.660 79.210 97.740 26 2.147 3.061 16.428 67.422 78.938 97.396 25 2.124 3.035 16.336 67.177 78.659 97.042 24 2.100 3.008 16.243 66.923 78.370 96.677 23 2.075 2.979 16.145 66.661 78.071 96.299 22 2.049 2.951 16.044 66.390 77.762 95.908 21 2.023 2.921 15.940 66.109 77.441 95.502 20 1.996 2.890 15.832 65.817 77.108 95.080 19 1.968 2.858 15.719 65.513 76.761 94.641 18 1.939 2.825 15.602 65.196 76.399 94.183 17 1.909 2.791 15.481 64.864 76.020 93.704 16 1.878 2.755 15.354 64.516 75.623 93.201 15 1.846 2.718 15.220 64.151 75.205 92.673 14 1.812 2.679 15.080 63.765 74.763 92.115 13 1.776 2.638 14.931 63.356 74.296 91.523 12 1.739 2.594 14.773 73.798 90.894 11 1.699 2.549 14.605 73.265 10 1.658 2.500 14.424 72.690 9 1.613 2.448 14.230 72.067 8 1.565 2.392 14.018 71.383 7 1.513 2.331 70.625 6.5 1.485 2.298 70.213 6 1.455 2.263 69.773 5.5 1.424 2.227 69.301 4 1.318 2.087 67.633 3.5 1.277 2.039 66.960 3.0 1.231 1.986 66.201 2.5 1.181 1.926 65.327 2.0 1.123 1.858 64.290 1.8 1.097 1.827 63.814 1.6 1.068 1.793 63.290 1.4 1.038 1.757 62.707 1.2 1.004 1.716 62.049 1.0 0.966 1.670 61.289 0.9 0.946 1.644 60.859 0.8 0.923 1.616 60.387 0.7 0.898 1.585 59.860 0.6 0.871 1.551 59.266 0.5 0.841 1.512 58.578 0.45 0.824 1.490 58.188 0.4 0.806 1.467 57.760 0.35 0.786 1.441 57.281 0.3 0.763 1.411 56.741 0.25 0.739 1.378 56.115 0.2 0.710 1.339 55.368 0.15 0.675 1.292 54.439 0.1 0.631 1.230 0.09 0.620 1.214 0.08 0.609 1.197 0.07 0.596 1.179 0.06 0.582 1.158 0.05 0.566 1.134 0.04 0.547 1.106 0.03 0.524 1.072 0.02 1.026 0.01 0.956 0.008 0.935 0.006 0.910 0.004 0.875 0.002 0.822 † Atmospheric pressure Reference are listed after next table. Boiling temperature and triple points for cryogenic liquids a Cryogenic Liquid Boiling Temperature (ITS-90) 1 atm (101.325 kPa, 760 mm Hg) [K] Triple Point Temperature (ITS-90) [K] Pressure [kPa] 3He b 3.1905 — 4He c 4.230 2.1768† 4.856 H2 (para) d, * 20.268 13.80 7.04 Neon e 27.100 24.557 43.46 Nitrogen f 77.355 63.151 12.52 Liquid Air g 78.903 59.75‡ 5.26 Argon h 87.302 83.806 68.89 Oxygen i 90.196 54.359 0.146 Methane j 111.67 90.694 11.70 Footnotes and references for both of the above tables: * Hydrogen can exist in two molecular forms: higher-energy orthohydrogen (nuclear spins aligned) and lower-energy parahydrogen (nuclear spins opposed). The equilibrium ratio is determined by temperature: at room temperature and above, hydrogen consists of about 25 % para and 75 % ortho (so-called “normal” hydrogen), but at the atmospheric boiling temperature of liquid hydrogen (20.27 K) and below, the equilibrium shifts almost completely to parahydrogen (99.79 % para and 0.21 % ortho at 20.27 K). The boiling temperatures of parahydrogen and normal hydrogen are nearly equal. † Superfluid point ‡ Solidification point a Data were evaluated by E. W. Lemmon, (2003) from equations of state given in references c through j; National Institute of Standards and Technology, Boulder, Colorado, personal communication. b E. W. Lemmon (2002), National Institute of Standards and Technology, Boulder, Colorado, personal communication. c R. D. McCarty and V. D. Arp (1990), Adv. Cryog. Eng. 35, 1465–1475. d B. A. Younglove (1982), J. Phys. Chem. Ref. Data 11, Suppl. 1, 1–11. e R. S. Katti, R. T. Jacobsen, R. B. Stewart, and M. Jahangiri (1986), Adv. Cryog. Eng. (Mater.) 31, 1189–1197. f R. Span, E. W. Lemmon, R. T. Jacobsen, W. Wagner, and A. Yokozeki (2000), J. Phys. Chem. Ref. Data 29(6), 1361–1433. g E. W. Lemmon, R. T. Jacobsen, S. G. Penoncello, and D. G. Friend (2000), J. Phys. Chem. Ref. Data 29(3), 1–54. h C. Tegeler, R. Span, and W. Wagner (1999), J. Phys. Chem. Ref. Data 28(3), 779–850. i R. Schmidt and W. Wagner (1985), Fluid Phase Equilibria 19,175–200. j U. Setzmann and W. Wagner (1991), J. Phys. Chem. Ref. Data 20(6), 1061–1151. A5.2 Properties of cryogenic thermometers (~1 K to ~300 K) a (Sections 5.1.2, 5.1.3, and 5.5) This table is designed for use in conjunction with the reference compendium, Sec. 5.5, where comments on the properties and practical use of each type of thermometer are given (in corresponding order). Sensor Type Temp. Range Accuracy* (± value) Reproducibility† (± value) Long-term Calibration Drift Inter-change-ability‡ Magnetic Field Use Best Use Cost Metallic Resistance Sensors (positive temperature coefficient): Platinum 77 K to 800 K With impurity correction: 20 K to 77 K (Appendix A5.3b) Without indiv. calib: 0.6 K at 70 K 0.2 K at 300 K With indiv. calibration: 20 mK at 77 K 35 mK at 300 K 55 mK > 330 K
200 mK > 480 K 10 mK from 77 K to 305 K 10 mK/yr from 77 K to 273 K Yes Recommended above 70 K; error < 0.1 % with standard correction factors given in Appendix A5.5 Measurements above 77 K
Excellent reproducibility interchange-ability, low mag. field error
Many shapes & sizes Low without calibration
High with individual. calibration
Rh–Fe 0.5 K to 900 K With indiv. calibration:
10 mK 4.2 K
25 mK 100 K
35 mK 300K 10 mK from 1.4 K to 325 K
High purity, strain-free: 0.1 mK at 4.2 K 20 mK/yr from 1.4 K to 325 K No Not recommended below ~77 K Secondary standard thermometer
Measurements over a wide temp. range down to 0.5 K High with indiv. calib.
Semiconductor-like Resistance Sensors (negative temperature coefficient):
Germanium 0.05 K to 100 K Must be indiv. calibrated.
With indiv. calib:
5 mK at < 10 K
15 mK at < 20 K
35 mK at < 50 K 0.5 mK at 4.2 K
1 mK/yr at 4.2 K
10 mK/yr at 77 K No Not recommended Secondary standard thermometer
Excellent reproducibility High with indiv. calib.
Zirconium oxynitride (Cernox) 0.3 K to 420 K Must be indiv. calibrated.
With indiv. calib:
5 mK at 4.2 K
20 mK at 20 K
50 mK at 100 K
140 mK at 300 K
230 mK at 400 K 3 mK at 4.2 K 25 mK/yr over the range 1 K to 100 K
0.05% of reading 100 K to 300 K No Recommended
Lowest error
Correction factors given in Appendix A5.6 One of the best sensors for use in mag. fields
Good sensitivity over wide temp. range
Fast response time as chip High with indiv. calib.
Carbon glass 1 K to ~325 K Must be indiv. calibrated.
With indiv. calib:
5 mK at < 10 K 20 mK at 20 K 55 mK at 50 K 0.75 mK at 4.2 K –5 mK/yr at 4.2 K –30 mK/yr at 15 K –100 mK/yr at 77 K –600 mK/yr at 300 K No Recommended Correction factors given by Sample et al. (1982) One of the best sensors for use in mag. fields High sensitivity at 4.2 K, low sens. >100K
Fragile; calib. easily invalidated High with indiv. calib.
Bi ruthenate/ ruthenium oxide 0.05 K to
40 K With indiv. calib:
5 mK at 0.05 K
7 mK at 1.4 K
11 mK at 4.2 K
77 mK at 20 K
470 mK at 77 K
1.7 K at 200 K
7 K at 300 K 10 mK at 4.2 K 15 mK/yr at 4.2 K Yes, but only within each lot Recommended Most useful below 20 K
Calibration interchangeability (20-40 mK) for sensors of the same lot High with indiv. calib.
Diode Voltage Sensors:
Silicon diode 1.4 K to 475 K Without calib:
1 K < 100 K
1% at 100 K–
475 K
With indiv. calib: 20 mK 1.4-10 K 55 mK 10-475 K 5 mK at 4.2 K
20 mK at 77 K
15 mK at 300 K 10 mK/yr at 4.2 K
40 mK/yr at 77 K
25 mK/yr at 300 K Yes Not recommended below ~60 K Relatively inexpensive, easily measured, interchangeable thermometer
Small size Medium without calib.
High with indiv. calib.
GaAlAs diode 1.4 K to 325 K Must be indiv. calibrated.
With indiv. calib:
15 mK < 20 K 50 mK at 50 K 110 mK at 300 K 5 mK at 4.2 K 15 mK/yr at 4.2 K 50 mK/yr over the range 77 K to 330 K No Acceptable error (~10 times less than Si diode, but >10 times greater error than Cernox When diode sensor is required in mag. field High with indiv. calib.
Special Purpose Sensors:
Thermocouples
Chromel–AuFe(0.07%)
Type E (Chromel–CuNi)
1.4 K to 325 K
3 K to 1274 K
1.7 K from 73 K to 273 K 20 mK at 77 K
— Yes
Not recommended
Difficult to use in magnetic field Low mass sensor
– Chromel-AuFe for lower temp. range
– Type E for higher temp. range Requires low dc voltage measurement
Capacitance Sensor 1 K to 290 K
Used as transfer control element only, not absolute measurement > 500 mK
10 mK after cooling and stabilizing 1 K/yr No Recommended for temperature control Excellent magnetic field stability for temp. control Sensor: medium
Requires capacitance measurement
Definitions:
* Accuracy: The difference between the measured and true temperature value.
† Reproducibility: The change in apparent temperature when the sensor is subjected to repeated thermal cycling from room temperature.
‡ Interchangeability: The ability to substitute one sensor for another with little change in calibration.
a Information compiled from Temperature Measurement and Control (2000, 2002), Lake Shore Cryotronics, Westerville, Ohio; L. M. Besley (1993), National Measurements Laboratory, Sydney, Australia; and L. G. Rubin (2002), Francis Bitter National Magnet Laboratory, Cambridge, Massachusetts.
References:
Carbon resistors: H. H. Sample, L. J. Neuringer, and L. G. Rubin (1974), Rev. Sci. Instrum., 45, 64–73.
Carbon-glass thermometers: H. H. Sample, B. L. Brandt, and L. G. Rubin (1982), Rev. Sci. Instrum. 53, 1129–1136.
Cernox: B. L. Brandt, D. W. Liu, and L. G. Rubin (1999), Rev. Sci. Instrum. 70, 104–110.
Platinum thermometers: B. L. Brandt, L. G. Rubin, and H. H. Sample (1988), Rev. Sci. Instrum. 59, 642–645.
Thermocouples: H. H. Sample and L. G. Rubin (1977), Cryogenics 17, 597–606.
A5.3a Platinum-resistance-thermometer resistivity vs. temperature) above 70 K DIN EN 60751 (Sections 5.1.3, 5.1.6, and 5.5.1)
This table gives a standard calibration of the temperature dependence of platinum resistance thermometers (PRTs) for use above liquid-nitrogen temperature. This calibration is not recommended at lower temperatures, however, because the role of impurity resistivity increases in the low-temperature range. A standard calibration below 70 K that compensates for varying impurity resistivities is given in the next table, Appendix A5.3b.
For platinum thermometers having an ice-point resistance R273K other than 100 Ω, multiply the resistance values in this table by the ratio of the ice-point resistance to 100 Ω. For example, for a 500 Ω PRT, multiply all the values in the table by 5 to obtain the calibration for such a thermometer.
Calculation of temperature values (ITS-90): Temperatures other than those listed in the table can be calculated according to DIN EN 60751 for Class A and Class B platinum resistance thermometers {alpha [R100C – R0C]/[100 R0C] = 0.00385}, where the temperature T is in kelvins:
For T < 273.15 K (0 C): R(T) = R0 [1 + A (T – 273.15) + B (T – 273.15)2 + C (T – 373.15) (T – 273.15)3] For T ≥ 273.15 K (0 C):
R(T) = R0 [1 + A (T – 273.15) + B (T – 273.15)2]
where the constants in these two equations have the values
A = 3.9083 10–3 C–1
B = –5.775 10–7 C–2
C = –4.183 10–12 C–4
R0 = 100 Ω
Interchangeability tolerance:
Class A: ∆T (K) = ± (0.15 + 0.002T – 273.15)
Class B: ∆T (K) = ± (0.3 + 0.005T – 273.15)
DIN EN 60751 resistance vs. temperature (ITS-90) for platinum-resistance thermometers. (Thermometer resistance at 273.15 K is 100 , alpha a = 0.00385.)
T
[K] R
[Ω] Interchange-ability Tolerance
[K]
Class A, B T
[K] R
[Ω] Interchange-ability Tolerance
[K]
Class A, B T
[K] R
[Ω] Interchange-ability Tolerance
[K]
Class A, B
60 12.80 140 46.71 225 81.04
65 14.98 145 48.77 230 83.02
70 17.16 ±0.6, ±1.3 150 50.82 ±0.4, ±0.9 235 85.00
75 19.32 155 52.87 240 86.98
80 21.47 160 54.91 245 88.95
85 23.62 165 56.95 250 90.92 ±0.2, ±0.4
90 25.75 170 58.98 255 92.89
95 27.88 175 61.01 260 94.85
100 30.00 ±0.5, ±1.2 180 63.03 265 96.81
105 32.12 185 65.05 270 98.77
110 34.22 190 67.06 273.15 100.00
115 36.32 195 69.07 275 100.72
120 38.41 200 71.07 ±0.3, ±0.7 280 102.67
125 40.49 205 73.07 285 104.62
130 42.57 210 75.07 290 106.57
135 44.64 215 77.06 295 108.51
220 79.05 300 110.45 ±0.2, ±0.4
a Alpha [R100C – R0C]/100 R0C]
A5.3b Platinum-resistance-thermometer resistivity vs. temperature below 70 K (Sections 5.1.3, 5.1.6, and 5.5.1)
This table provides a calibration for platinum resistance thermometers (PRTs) if they must be used for thermometry below ~70 K (such as a when a platinum film-type sensor is used for a rapid time response). In this low-temperature range, differences in impurity levels in the thermometers lead to significant errors in the standard calibration table (Appendix A5.3a). Consequently, the temperature should be determined by calculating the Z ratio (defined below).
With this scheme, the impurity resistivity of an individual PRT can be compensated. The PRT’s resistance is first measured at or near liquid-helium temperature in order to determine its impurity resistivity R4.2K. This is then used with the measured resistance of the sensor RT to determine the ratio:
Z ≡ (RT – R4.2K)/(R273K – R4.2K),
where R273K is the sensor’s ice-point resistivity. The value of this Z-ratio can then be used to determine temperature from the following table.
Values of ∆Z/∆T are also provided to facilitate interpolation between tabulated temperature values.
This Z-ratio procedure has a typical error of about 25 mK down to 30 K, below which temperature the error increases to about 120 mK at 14 K (Besley and Kemp 1978). Use below 14 K is not recommended.
Z-ratio for platinum resistance thermometers a
T
[K] 106 Z 106 ∆Z/∆T
[K–1]
14.0 908.7 256
14.5 1043.5 284
15.0 1192.9 314
15.5 1358.2 347
16.0 1540.4 382
16.5 1740.3 418
17.0 1958.9 457
17.5 2197.2 497
18.0 2456.2 539
18.5 2736.7 583
19.0 3039.7 629
19.5 3366.1 677
20.0 3716.7 726
21.0 4493.6 829
22.0 5376.0 937
23.0 6368.3 1049
24.0 7474.2 1164
25.0 8696.8 1282
26.0 10038 1401
27.0 11500 1522
28.0 13083 1644
29.0 14788 1766
30.0 16615 1887
31.0 18562 2007
32.0 20628 2126
33.0 22812 2242
34.0 25111 2356
35.0 27523 2468
36.0 30045 2576
37 32674 2681
38 35406 2783
39 38238 2880
40 41166 2974
42 47293 3151
44 53758 3311
46 60528 3456
48 67572 3586
50 74862 3701
52 82368 3803
54 90065 3892
56 97929 3970
58 105937 4037
60 114071 4095
65 134839 4205
70 156050 4274
75 177535 4316
80 199174 4337
85 220881 4344
90 242600 4342
95 264293 4334
100 285935 4322
a Table values were calculated by L. M. Besley and R. C. Kemp (1978), Cryogenics 18, 497–500, from data on 50 high-quality platinum thermometers. Data were compiled by C. G. Kirby and R. E. Bedford, and J. Kathnelson (1975), Metrologia 11, 117–124, and J. P. Compton and S. D. Ward (1975), Temperature Measurement, p. 91, Institute of Physics London Conference Series No. 26. See also the discussion and summary given in G. K. White (1989), Experimental Techniques in Low-Temperature Physics, pp. 100–104, Oxford University Press.
A5.4 Diode and thermocouple voltage-vs.-temperature tables a (Sections 5.1.3, 5.1.6, 5.5.7, and 5.5.9)
Silicon Diode b
DT-470
Curve 10
[K] [V]
Thermocouple c
Chromel vs.
Au–0.07at%Fe
[K] [mV] Type E d Thermocouple
Chromel vs. Constantan
[K] [mV] Type K e Thermocouple
Chromel vs.
Alumel
[K] [mV] Type T f
Thermocouple
Copper vs.
Constantan
[K] [mV]
1.4 1.6981 1.4 –5.298 3.0 –9.836 3.0 –6.458 3.0 –6.258
2.0 1.6879 3.0 –5.281 5.6 –9.830 6.0 –6.455 6.5 –6.252
3.8 1.6390 4.8 –5.259 9.0 –9.818 10.0 –6.449 11.0 –6.240
9.0 1.4505 7.0 –5.223 13.5 –9.796 14.5 –6.438 16.5 –6.218
12.0 1.3681 10.5 –5.174 19.0 –9.757 19.5 –6.421 22.0 –6.189
15.5 1.2946 19.0 –5.032 25.0 –9.701 25.0 –6.395 29.0 –6.140
20.0 1.2144 26.0 –4.193 32.0 –9.620 32.0 –6.353 38.0 –6.062
24.0 1.1360 48.0 –4.549 40.0 –9.507 40.0 –6.291 48.0 –5.954
25.0 1.1246 58.0 –4.381 50.0 –9.337 48.0 –6.215 60.0 –5.800
26.0 1.1190 70.0 –4.173 60.0 –9.135 58.0 –6.102 75.0 –5.575
27.0 1.1152 80.0 –3.995 70.0 –8.903 65.0 –6.010 90.0 –5.320
28.0 1.1121 90.0 –3.813 80.0 –8.648 75.0 –5.863 105 –5.034
32.0 1.1026 100 –3.627 90.0 –9.367 85.0 –5.699 120 –4.719
36.0 1.0949 110 –3.437 105 –7.906 95.0 –5.516 135 –4.377
44.0 1.0809 120 –3.244 120 –7.394 105 –5.317 155 –3.878
60.0 1.0527 135 –2.948 135 –6.839 120 –4.988 175 –3.328
77.35 1.0203 150 –2.645 150 –6.240 135 –4.624 195 –2.734
100 0.9755 165 –2.337 170 –5.383 150 –4.227 220 –1.930
120 0.9338 180 –2.042 190 –4.456 165 –3.799 245 –1.059
140 0.8907 200 –1.600 210 –3.470 185 –3.187 270 –0.125
170 0.8240 220 –1.169 235 –2.161 205 –2.526 300 +1.062
200 0.7555 245 –0.623 260 –0.767 230 –1.646 330 +2.325
230 0.6856 270 –0.071 290 +0.995 260 –0.519 360 +3.664
273.15 0.5833 300 +0.599 320 +2.843 295 +0.869 395 +5.310
320 0.4707 305 +0.716 350 +4.770 350 +3.130 430 +7.042
360 0.3734 310 +0.843 385 +7.115 395 +5.000 470 +9.111
400 0.2746 315 +0.994 420 +9.557 460 +7.616 510 +11.276
440 0.1746 320 +1.194 460 +12.443 510 +9.613 555 +13.805
475 0.0906 325 +1.484 475 +13.557 575 +12.279 575 +14.968
a Data from Lake Shore Cryotronics, Inc. (2002), Temperature Measurement and Control , Westerville, Ohio.
b Accuracy: 1 K at < 100 K, 1% at 100 K–475 K. Reproducibility: ±5 mK at 4.2 K, ±20 mK at 77 K, ±15 mK at 300 K. See Appendix A5.2 for more information. c For accuracy information, refer to L. L. Sparks and R. L. Powell (1973), J. Res. Nat. Bur. Std. 76A, 263–283. Thermocouple voltages are referenced to zero at 273 K. d Accuracy: 1.7 K from 73 K to 273 K. See Appendix A5.2 for more information. Thermocouple voltages are referenced to zero at 273 K. e Accuracy: 2.2 K from 73 K to 273 K. Thermocouple voltages are referenced to zero at 273 K. f Accuracy: 1.0 K from 73 K to 273 K. Thermocouple voltages are referenced to zero at 273 K. A5.5 Magnetic-field correction factors for platinum resistance thermometers (Sections 5.1.6, 5.2 and 5.5.1) These magnetic-field correction factors were calculated from magnetoresistance data obtained by Brandt et al. (1988) and are tabulated here as relative temperature errors (Tapparent – Tactual)/Tactual [%]. The corrections were nearly the same for measurements on thirteen platinum resistance thermometers (PRTs) of varying purity (alpha = 3.85 to 3.925 10–3 C–1), as well as varying construction types (wire-wound and thick-film types), manufacturers, and ice-point resistances (100 Ω to 500 Ω). For all the sensors, the standard deviation of the correction is simply about 10 % of the correction value itself, irrespective of temperature or magnetic field. Example: Suppose we wish to correct the reading a platinum resistance thermometer that indicates an apparent temperature of 100 K in a magnetic field of 10 T. From the table below we see that, at 100 K and 10 T, this temperature reading would actually be too high by 0.40 %. Thus, the actual temperature would be 100 K / (1 + 0.004) = 99.6 K. The standard deviation for this correction would be about 10 % of the correction (0.1 0.40 %), or only 0.04 % (i.e., 0.04 K). The orientation of the magnetic field has a negligible effect on these correction factors for film-type PRTs, but the effect is significant for wire-wound PRTs. The data below correspond to wire-wound PRTs oriented with the applied magnetic field parallel to the long axis of their package (see Fig. 5.17). Thus, to use the table, it is recommended that wire-wound PRTs be installed with this orientation. Alternatively, when the sensor must be used in varying field orientations, a thin-film PRT is preferred. Magnetic-Field Correction Factors for Platinum Resistance Thermometers a Tabulated values are (Tapparent – Tactual)/Tactual (in percent) at magnetic field B. The standard deviation of these corrections is about 10 % of the tabulated values. T [K] B = 5 T [%] B = 10 T [%] B = 15 T [%] B = 19 T [%] 40 1.5 4.1 6.7 8.9 50 1.05 2.9 5.0 6.7 60 0.59 1.8 3.2 4.5 70 0.27 0.97 1.9 2.8 80 0.18 0.67 1.4 2.1 90 0.13 0.50 1.0 1.6 100 0.11 0.40 0.85 1.4 120 0.068 0.26 0.57 0.91 150 0.038 0.16 0.35 0.56 200 0.019 0.085 0.19 0.31 220 0.017 0.074 0.17 0.28 250 0.015 0.058 0.14 0.22 300 0.010 0.030 0.080 0.13 a Correction factors calculated from magnetoresistance data from B. L. Brandt, L. G. Rubin, and H. H. Sample (1988), Rev. Sci. Instrum. 59, 642–645. A5.6 Magnetic-field correction factors for zirconium-oxynitride resistance thermometers (Sections 5.1.6, 5.2 and 5.5.4) These magnetoresistance correction factors were interpolated to even temperature values by using a cubic-spline fit to magnetoresistance data measured by Brandt et al. (1999). The corrections are applicable to a wide range of zirconium-oxynitride (Cernox) resistance thermometers having 4.2 K resistances from about 300 to 8000 , and 4.2 K dimensionless sensitivities (dR/R)/(dT/T) in the range –0.74 to –1.9 (with –1.2 to –1.9 recommended). The data are presented as the percentage change in resistance rather than temperature (unlike platinum in Appendix A5.5), because Brandt et al. found that, for zirconium-oxynitride sensors, the standard deviations in resistance were much smaller than the standard deviations in temperature. [This results from the wide range of dimensionless sensitivities S (dR/R)/(dT/T) of the individual sensors and the differing temperature dependences of S.] The standard deviation of the magnetoresistance correction is given as a quantity below each magnetoresistance correction. The standard deviations show that this correction procedure is most beneficial at temperatures below about 3 K and from about 6 K to 15 K. Example: Let us assume that a particular zirconium-oxynitride thermometer indicates an apparent temperature of about 10 K in a magnetic field of 16 T. We wish to correct for the magnetoresistance error of this particular sensor. From the table below, we see that, at 10 K and 16 T, the sensor’s resistance would exceed its zero-field resistance by 1.40 %. We use the dimensionless sensitivity S (dR/R)/(dT/T) supplied with the calibration data for our particular sensor to calculate the equivalent shift in its apparent temperature. For our sensor, suppose that S = –1.16 at 10 K. Then, we would calculate (Tapparent – Tactual)/Tactual = (1.40 %)/(–1.16) = –1.21 % and accordingly adjust the apparent temperature reading by this percentage to obtain the actual temperature. Since Tapparent – Tactual is negative at 10 K and 16 T, the apparent temperature is lower than the actual temperature, and it would need to be increased by 1.21 % to give the actual temperature. From the standard deviation of ±0.30 for the resistivity correction at 10 K and 16 T, we find the standard deviation of the temperature correction is ±0.26 %. At liquid-nitrogen temperatures, the effect of magnetic-field orientation on these correction factors was observed to be insignificant. However, at 4.2 K the situation was more complex, giving rise to a positive orientation effect in some sensors, negative in others (for example, apparent temperature shifts of –0.2 % to +0.4 % were observed at 16 T; Brandt et al. 1999). Since the tabulated corrections were determined for field perpendicular to the film surface (canister aligned parallel to the field), it is best to orient these sensors accordingly so they can be used with the correction data. Magnetic Field Correction for Zirconium-Oxynitride (Cernox) Resistance Thermometers a Tabulated values are [R(B) – R(0)]/R(0) (in percent) at magnetic field B. The standard deviation of the correction is shown as a factor (also in percent) below each correction. T [K] B=2 T [%] B=4 T [%] B=6 T [%] B=8 T [%] B=10 T [%] B=12 T [%] B=14 T [%] B=16 T [%] B=18 T [%] B=20 T [%] B=23 T [%] B=26 T [%] B=29 T [%] B=32 T [%] 2 –2.20 –4.47 –5.52 –6.10 –6.61 –7.19 –7.85 –8.59 –10.12 –10.39 –10.96 –12.87 –14.87 –16.92 ±0.39 ±1.08 ±1.58 ±1.83 ±1.90 ±1.79 ±1.59 ±1.35 ±1.09 ±0.85 ±1.00 ±1.28 ±1.80 ±2.35 2.5 –1.21 –2.55 –3.09 –3.25 –3.35 –3.55 –3.88 –4.14 –5.07 –5.65 –6.31 –8.06 –9.87 –11.74 ±0.25 ±0.77 ±1.20 ±1.46 ±1.55 ±1.51 ±1.35 ±1.13 ±0.86 ±0.59 ±1.40 ±1.80 ±2.25 ±2.76 3 –0.65 –1.42 –1.65 –1.57 –1.43 –1.37 –1.47 –1.69 –2.17 –2.73 –3.42 –5.00 –6.73 –8.55 ±0.16 ±0.52 ±0.88 ±1.12 ±1.24 ±1.24 ±1.15 ±0.98 ±0.76 ±0.57 ±1.73 ±2.16 ±2.62 ±3.12 3.5 –0.38 –0.76 –0.80 –0.57 –0.28 –0.06 0.02 –0.11 –0.34 –0.75 –1.57 –3.02 –4.63 –6.34 ±0.11 ±0.35 ±0.64 ±0.87 ±1.00 ±1.04 ±1.00 ±0.92 ±0.79 ±0.71 ±1.92 ±2.37 ±2.84 ±3.30 4.2 –0.17 –0.30 –0.20 0.11 0.50 0.82 1.07 1.06 1.02 0.79 0.03 –1.20 –2.63 –4.22 ±0.03 ±0.20 ±0.42 ±0.61 ±0.73 ±0.81 ±0.81 ±0.85 ±0.74 ±0.72 ±2.03 ±2.50 ±2.97 ±3.43 5 –0.09 –0.07 0.10 0.42 0.82 1.20 1.47 1.62 1.22 1.55 1.00 0.06 –1.23 –2.65 ±0.02 ±0.15 ±0.31 ±0.44 ±0.55 ±0.59 ±0.64 ±0.67 ±0.65 ±0.68 ±1.89 ±2.37 ±2.84 ±3.30 6 –0.08 0.02 0.21 0.51 0.90 1.29 1.61 1.85 2.01 1.96 1.69 0.91 –0.12 –1.26 ±0.02 ±0.07 ±0.19 ±0.31 ±0.40 ±0.45 ±0.51 ±0.55 ±0.58 ±0.68 ±1.87 ±2.31 ±2.78 ±3.25 7 –0.06 0.03 0.23 0.52 0.89 1.26 1.60 1.88 2.26 2.08 1.88 1.29 0.48 –0.48 ±0.01 ±0.03 ±0.11 ±0.21 ±0.27 ±0.34 ±0.39 ±0.45 ±0.53 ±0.66 ±1.75 ±2.17 ±2.62 ±3.08 8 –0.05 0.03 0.24 0.52 0.85 1.19 1.52 1.80 1.97 2.03 1.78 1.38 0.73 –0.11 ±0.02 ±0.03 ±0.07 ±0.13 ±0.18 ±0.24 ±0.30 ±0.37 ±0.48 ±0.62 ±1.53 ±1.95 ±2.38 ±2.81 9 –0.04 0.02 0.21 0.46 0.76 1.07 1.37 1.63 1.71 1.86 1.62 1.32 0.78 0.08 ±0.03 ±0.03 ±0.06 ±0.10 ±0.14 ±0.18 ±0.24 ±0.32 ±0.44 ±0.57 ±1.36 ±1.76 ±2.16 ±2.56 10 –0.05 0.01 0.16 0.38 0.64 0.92 1.18 1.40 1.54 1.65 1.43 1.17 0.72 0.16 ±0.03 ±0.04 ±0.06 ±0.09 ±0.12 ±0.15 ±0.21 ±0.30 ±0.40 ±0.52 ±1.23 ±1.59 ±1.96 ±2.34 12 –0.05 –0.01 0.08 0.23 0.43 0.66 0.85 1.01 1.21 1.24 1.04 0.81 0.48 0.09 ±0.02 ±0.04 ±0.05 ±0.08 ±0.11 ±0.13 ±0.18 ±0.26 ±0.34 ±0.43 ±1.02 ±1.31 ±1.62 ±1.95 15 –0.03 –0.03 0.02 0.12 0.24 0.37 0.50 0.61 0.70 0.76 0.48 0.32 0.07 –0.27 ±0.02 ±0.03 ±0.05 ±0.06 ±0.08 ±0.11 ±0.15 ±0.20 ±0.26 ±0.33 ±0.77 ±0.99 ±1.23 ±1.48 20 –0.03 –0.04 –0.04 –0.01 0.03 0.09 0.13 0.17 0.20 0.22 –0.03 –0.17 –0.37 –0.65 ±0.01 ±0.02 ±0.03 ±0.05 ±0.06 ±0.08 ±0.11 ±0.14 ±0.18 ±0.23 ±0.51 ±0.67 ±0.83 ±1.00 30 –0.01 –0.02 –0.03 –0.04 –0.04 –0.05 –0.05 –0.07 –0.09 –0.12 –0.35 –0.49 –0.64 –0.83 ±0.00 ±0.01 ±0.01 ±0.02 ±0.03 ±0.05 ±0.07 ±0.09 ±0.11 ±0.13 ±0.27 ±0.34 ±0.42 ±0.51 40 0.00 –0.01 –0.01 –0.02 –0.04 –0.06 –0.08 –0.10 –0.14 –0.17 –0.38 –0.50 –0.61 –0.76 ±0.00 ±0.00 ±0.01 ±0.01 ±0.02 ±0.03 ±0.04 ±0.05 ±0.07 ±0.08 ±0.16 ±0.19 ±0.24 ±0.29 50 0.00 –0.01 –0.02 –0.03 –0.05 –0.06 –0.08 –0.11 –0.14 –0.17 –0.32 –0.41 –0.51 –0.63 ±0.00 ±0.00 ±0.01 ±0.01 ±0.02 ±0.02 ±0.03 ±0.04 ±0.05 ±0.06 ±0.10 ±0.13 ±0.16 ±0.19 60 –0.01 –0.02 –0.03 –0.04 –0.05 –0.07 –0.09 –0.11 –0.13 –0.16 –0.27 –0.34 –0.43 –0.53 ±0.00 ±0.01 ±0.01 ±0.02 ±0.02 ±0.03 ±0.04 ±0.05 ±0.07 ±0.08 ±0.07 ±0.10 ±0.12 ±0.14 70 –0.01 –0.01 –0.02 –0.03 –0.04 –0.06 –0.08 –0.10 –0.12 –0.15 –0.23 –0.30 –0.37 –0.45 ±0.01 ±0.01 ±0.01 ±0.02 ±0.02 ±0.03 ±0.05 ±0.06 ±0.08 ±0.09 ±0.06 ±0.07 ±0.09 ±0.11 77 0.00 –0.01 –0.02 –0.02 –0.04 –0.05 –0.07 –0.09 –0.11 –0.13 –0.21 –0.27 –0.34 –0.41 ±0.01 ±0.01 ±0.01 ±0.01 ±0.02 ±0.03 ±0.04 ±0.06 ±0.07 ±0.08 ±0.05 ±0.06 ±0.07 ±0.09 80 0.00 –0.01 –0.01 –0.02 –0.03 –0.04 –0.06 –0.08 –0.10 –0.12 –0.20 –0.26 –0.33 –0.40 ±0.01 ±0.01 ±0.01 ±0.01 ±0.02 ±0.02 ±0.03 ±0.04 ±0.06 ±0.07 ±0.05 ±0.06 ±0.07 ±0.09 90 0.00 0.00 –0.01 –0.02 –0.02 –0.03 –0.04 –0.05 –0.06 –0.07 –0.18 –0.23 –0.29 –0.34 ±0.00 ±0.00 ±0.01 ±0.01 ±0.01 ±0.01 ±0.00 ±0.00 ±0.00 ±0.00 ±0.04 ±0.04 ±0.04 ±0.06 100 0.00 0.00 –0.01 –0.01 –0.02 –0.03 –0.02 –0.03 –0.03 –0.04 –0.15 –0.20 –0.25 –0.29 ±0.00 ±0.00 ±0.01 ±0.01 ±0.01 ±0.01 ±0.02 ±0.03 ±0.04 ±0.04 ±0.04 ±0.02 ±0.02 ±0.04 120 0.00 0.00 –0.01 –0.01 –0.02 –0.02 –0.01 –0.01 –0.01 –0.02 –0.09 –0.13 –0.17 –0.18 ±0.00 ±0.00 ±0.01 ±0.01 ±0.01 ±0.01 ±0.03 ±0.04 ±0.06 ±0.06 ±0.02 ±0.02 ±0.02 ±0.00 150 0.00 0.00 0.00 0.00 –0.01 –0.01 –0.02 –0.02 –0.03 –0.03 0.08 0.09 0.10 0.16 ±0.00 ±0.00 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.08 ±0.12 ±0.16 ±0.17 200 0.00 0.00 0.00 0.00 –0.01 –0.01 –0.01 –0.01 –0.02 –0.02 0.27 0.35 0.41 0.56 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.02 ±0.17 ±0.25 ±0.33 ±0.35 250 0.00 0.00 0.00 0.00 0.00 –0.01 –0.01 –0.01 –0.01 –0.01 0.47 0.60 0.72 0.96 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.02 ±0.26 ±0.37 ±0.49 ±0.53 300 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.67 0.85 1.02 1.35 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.02 ±0.35 ±0.50 ±0.65 ±0.71 a These temperature corrections were interpolated to even temperature values by using a cubic-spline fit to magnetoresistance data measured by B. L. Brandt, D. W. Liu, and L. G. Rubin (1999), Rev. Sci. Instrum. 70, 104–110. A5.7 Temperature-controller tuning with the Ziegler–Nichols method (Sec. 5.4.3) Temperature controllers do not always have an auto-tuning feature, and, even if they do, it does not work well for some applications. Here we describe a time-proven, relatively simple, step-by-step procedure for manually optimizing a proportional-integral-differential (PID) controller’s settings of gain G, integral time ti, and the derivative time td for a specific system. The settings for a proportional-integral (PI) controller and a simple proportional (P) controller are also given. The optimum settings enable a controller to react quickly to a change in heat demand without much overshoot, oscillation, or droop below the set point (Sec. 5.4.3). Before describing the tuning procedure, we first define a few terms. Definitions Cycle time (sometimes referred to as duty cycle): This denotes the time it takes an on–off or time-proportional controller to complete an on–off cycle, illustrated in Fig. 5.15. This applies only to time-proportional controllers and not to analog voltage or current controllers. Proportional band, and gain: Let Pb be the proportional band around the set point (see Fig. 5.16), usually expressed as a percent of full scale. This is also be referred to as the gain, G, which is the reciprocal of the proportional band; that is, G 1/Pb Integration-time constant (sometimes called reset): Let ti be the characteristic time constant for integration to eliminate the offset error. (The offset error is the steady-state difference between the system temperature and the set point, illustrated in Fig. 5.16). Derivative-time constant (sometimes called rate): Let td be the characteristic time constant to correct transient disturbances in the system with a minimum of overshoot or undershoot. (The terms proportional band, reset, and rate are those generally used in the field of industrial controls, whereas gain, integration-time constant, and derivative-time constant are the terms employed by physicists and the companies that sell controllers to them. Procedure 1. The tuning procedure is easier to observe with a recorder or scrolling data-acquisition display to monitor the process temperature, since the time constants for cycling may be as long as 30 min or more. 2. For time-proportional controllers only, adjust the cycle time to a short time so that the system will not be limited in its time response because of the duty cycle of the heater power. 3. Set the gain to a small value (proportional band to a large value) so the system is overdamped to start with. 4. Turn the integral (reset) and derivative (rate) controls off. 5. Enter the set point where control is desired and wait until the temperature is close to that point, or use the manual heater control (manual reset) to reach a temperature near the set point. When the sample temperature is close to the desired temperature, increase the gain (decrease the proportional band) until the system just becomes unstable and starts to oscillate. This is easiest to observe by looking directly at the output power to the heater. Let us denote this critical value of the gain as G’. 6. Measure the time period of the oscillations and let us denote this as t’. 7. Initial optimum values, G and ti, for optimum stable control are calculated from the following table by using the measured values of G’ and t’. These values are given for three types of controllers: proportional control (P), proportional-integral control (PI), and for proportional-integral-differential control (PID). 8. After entering these initial values into the controller and turning on the integral and derivative control, the parameters can be fine tuning. If overshoot occurs in response to step changes in the set-point temperature, it can be eliminated by decreasing the derivative (rate) time constant td. When changes are made in the derivative time constant, a corresponding change should also be made in the integral time constant to keep their ratio about the same as that given in the table. 9. For time proportional controllers only, increase the cycle time after satisfactory tuning has been achieved to extend the contactor life of the power-supply. Increase the cycle time as much as possible without causing the system to breaking into oscillation when the heater power cycles on and off. Calculation of control parameters for critical damping with the Ziegler–Nichols tuning formula.a,b Control Type c PID PI P Proportional gain G = 0.6 G’ G = 0.45 G’ G = 0.5 G’ Integral time (reset time constant) ti = 0.5 t’ ti = 0.85t’ Derivative time (rate time constant) td = 0.125 t’ a J. G. Ziegler and N. B. Nichols (1942), Trans. ASME 64, 759–768. b P. B. Deshpande and R. H. Ash (1981), “Computer process control,” ISA pub., USA. c P ≡ proportional control, PI ≡ proportional-integral control, and PID ≡ proportional-integral-differential control. For large time constants, where the Ziegler–Nichols method becomes time-consuming, a refinement that improves performance based on set-point weighting has been suggested by C. C. Hang, K. J. Astrom, and W. K. Ho (1991), IEEE Proc.-D 138, 111–118. If a computer is available to monitor the temperature, it is easy to implement an improved form of PID control with software described by C. K. Chan (1988), Rev. Sci. Instr. 59, 1001–1003. A good reference for understanding PID control at cryogenic temperatures is E. M. Forgan (1974), Cryogenics 14, 207–214. A6. Properties of solids at low temperature (ref. Chapter 6) Additional sources of materials data in the literature and on the Internet are given in the suggested reading and web sites listed in Secs. 6.7.1 and 6.7.2, respectively. A6.1 Elements: Physical properties at room temperature a For anisotropic elements, polycrystalline values are listed unless otherwise noted. Element Atomic Weight Crystal Structure Density (298 K) [g/cm3] Debye Temp. D b (295K) [K] Specific Heat (at const. press.) (298 K) [J/(g∙K)] Coef. of Thermal Linear Expansion (298 K) [10–6 K–1] Electrical Resistivity (295 K) [∙cm] Thermal Cond-uctivity (300 K) [W/(m∙K)] Magnetic Susceptibility e [10–6 SI] Supercon- ducting Transition Temperature f [K] Aluminum 26.98 f.c.c. 2.70 380 0.904 23.1 2.67 237 20.8 1.175 Antimony 121.76 rhombohedral 6.68 210 0.207 11.0 41.3 c 24.3 –68.3 Arsenic 74.91 d rhombohedral 5.73 c 290 0.329 5.6 c (293 K) 29 c 37 d –5.4 Barium 137.33 b.c.c. 3.62 110 0.205 20.6 33.5 18.4 0.1 Beryllium 9.013 h.c.p. 1.85 920 1.82 11.3 3.62 200 –23.1 0.026 Bismuth 208.98 rhombohedral 9.79 120 0.122 13.4 116 c 7.87 –165.0 Boron 10.81 hexagonal 2.535 1300 1.277 8.3 1012 d 30 d –19.7 Cadmium 112.41 h.c.p. 8.69 175 0.231 30.8 7.27 c 96.8 –19.0 0.517 Calcium 40.08 f.c.c. 1.54 210 0.646 22.3 3.38 98 d 19.4 Carbon: graphite diamond 12.01 12.01 hexagonal diamond 2.22 3.51 400 2000 0.709 0.4715 3 1.18 102–106 d 1012 d 200 990 –13.9 Cesium 132.91 b.c.c. 1.93 45 0.242 97 20.6 35.9 5.1 Chromium 52.00 b.c.c. 7.15 480 0.450 4.9 12.5 93.7 290.5 Cobalt 58.93 h.c.p. 8.86 380 0.421 13.0 5.80 c 100 ferro Copper 63.55 f.c.c. 8.96 310 0.385 16.5 1.69 401 –9.7 Gallium 69.72 orthorhombic 5.91 240 0.374 18 14.85 c 40.6 –23.2 1.083 Germanium 72.59 diamond 5.32 400 0.3219 6.1 5107 d 64 –10.7 Gold 196.97 f.c.c. 19.3 185 0.129 14.2 2.23 317 –34.5 Hafnium 178.49 h.c.p. 13.3 210 0.144 65.9 33.3 23 66.4 0.128 Indium 114.82 tetragonal 7.31 110 0.233 32.1 8.75 c 81.6 –8.2 3.408 Iridium 192.22 f.c.c. 22.5 290 0.131 6.4 5.07 c 147 36.8 0.112 Iron 55.85 b.c.c. 7.87 400 0.449 11.8 9.71 80.2 ferro Lead 207.20 f.c.c. 11.3 88 0.127 28.9 20.9 35.3 –15.8 7.196 Lithium 6.94 b.c.c. 0.534 360 3.57 46 9.36 84.7 13.6 Magnesium 24.30 h.c.p. 1.74 330 1.024 24.8 4.43 156 11.8 Manganese 54.94 cubic (complex) 7.43 410 0.479 21.7 144 7.82 869.7 Mercury 200.59 rhombohedral 13.534 110 (220 K) 0.139 60.4 95.9 c 83.4 –21.4 4.154 b Molybdenum 95.94 b.c.c. 10.2 380 0.251 4.8 5.39 138 96.2 0.915 Nickel 58.69 f.c.c. 8.90 390 0.445 13.4 7.01 90.7 ferro Niobium 92.91 b.c.c. 8.57 250 0.265 7.3 14.5 c 53.7 241.1 9.25 Osmium 190.23 h.c.p. 22.59 400 0.130 5.1 9.13 c 87.6 16.3 0.66 Palladium 106.42 f.c.c. 12.0 290 0.244 11.8 10.6 71.8 766.6 1.4 Platinum 195.08 f.c.c. 21.5 225 0.133 8.8 10.6 71.6 266.7 Potassium 39.10 b.c.c. 0.89 98 0.757 83.3 7.28 102.4 5.7 Rhenium 186.21 h.c.p. 20.8 275 0.137 6.2 18.6 c 47.9 94.9 1.697 Rhodium 102.91 f.c.c. 12.4 350 0.243 8.2 4.78 c 150 154.9 Rubidium 85.47 b.c.c. 1.53 61 0.364 90 12.9 58.2 3.8 Ruthenium 101.07 h.c.p. 12.1 450 0.238 6.4 7.37 c 117 59.4 0.49 Selenium (gray) 78.96 hexagonal 4.81 250 0.293 17.89 (|| c) 74.09 (┴ c) >1012 d
~107 d 0.45 (|| c)
0.13 (┴ c) –17.1
Silicon 28.09 diamond 2.328 700 0.702 2.49 >1010 d 124 –3.2
Silver 107.87 f.c.c. 10.5 220 0.235 18.9 1.60 429 –23.8
Sodium 22.99 b.c.c. 0.97 160 1.225 71 4.81 141 8.5
Strontium 87.62 f.c.c. 2.64 140 0.306 22.15 13.3 35.3 34.3
Tantalum 180.95 b.c.c. 16.4 230 0.140 6.3 13.2 57.5 177.5 4.47
Tellurium 127.61 hexagonal 6.23 180 0.197 18.0 d 0.4106 d 3.38 –23.4
Thallium 204.38 h.c.p. 11.8 94 0.129 29.9 16.4 c 46.1 –36.4 2.38
Thorium 232.04 f.c.c. 11.7 140 0.118 11.0 15 c 54.0 61.4 1.38
Tin 118.71 tetragonal 7.26 160 0.227 22.0 11.0 c 66.6 –28.9 3.722
Titanium 47.88 h.c.p. 4.51 360 0.522 8.6 43.1 c 21.9 178 0.40
Tungsten 183.84 b.c.c. 19.3 315 0.132 4.5 5.33 c 174 69.9 0.0154
Uranium 238.03 orthorhombic 19.1 160 0.116 13.9 25.7 c 27.6 411.3 0.2
Vanadium 50.94 b.c.c. 6.0 380 0.489 8.4 19.9 30.7 429.5 5.40
Zinc 65.39 h.c.p. 7.14 240 0.388 30.2 5.94 116 –12.6 0.85
Zirconium 91.22 h.c.p. 6.52 250 0.278 5.7 42.4 22.7 107.5 0.61
a Unless otherwise noted, data are from the CRC Handbook of Chemistry and Physics (2002), 83st edition, CRC Press, Boca Raton, Florida.
b Values of the Debye temperature D are from G. K. White (1987), Experimental Techniques in Low-Temperature Physics, Oxford University Press, determined from specific heat data in the range D/2–D. These data were compiled from the American Institute of Physics Handbook (1972), 3rd edition. McGraw-Hill; Touloukian et al., ed. (1970–1977), Thermophysical Properties of Matter, Plenum Press; Landolt–Börnstein, Springer-Verlag, Berlin, 1968, 1971, etc.; and K. A. Gschneidner (1964), Solid State Phys. 16, 275–476.
c American Inst. of Physics Handbook (1972), 3rd edition, coordinating ed. D. E. Gray, Table 9d, p. 9-39, McGraw Hill, NY.
d G. K. White. and P. J. Meeson. (2002), Experimental Techniques in Low-Temperature Physics, 4th edition, Oxford University Press.
e Magnetic susceptibility data were recalculated from molar susceptibilities given in the CRC Handbook of Chemistry and Physics (2000), 81st edition, CRC Press, Boca Raton, Florida; and from compilations given in Landolt–Börnstein, Numerical Data and Functional Relationships in Science and Technology, New Series, II/16 (1986); III/19, subvolumes a to i2 (1986–1992); and II/2, II/8, II10, II11, and II12a, (1966-1984), Springer-Verlag, Heidelberg; Tables de Constantes et Donnees Numerique (1957), Vol. 7, Relaxation paramagnetique, Masson, Paris.
f Superconducting critical temperatures are from B. W. Roberts (1978), “Properties of selected superconductive materials,” 1978 Supplement, NBS Technical Note 983, U.S. Government Printing Office, Washington, D.C.; tabulated in the CRC Handbook of Chemistry and Physics (2000), 81st edition, CRC Press, Boca Raton, Florida. Note that thin films of these elements generally have higher critical temperatures than those listed here for bulk materials (see the CRC handbook).
A6.2 Specific heat vs. temperature of technical materials (Sec. 6.1)
To convert these values of specific heat at constant pressure to volumetric heat capacity, multiply each value by the density of the material (densities of elements are given in Appendix A6.1).
Specific Heat CP [J/(gK)] [10-3 J/(kgK)]
Material 4 K 10 K 20 K 30K 50K 77 K 100 K 150 K 200 K 300 K
Metals
Al a 0.00026 0.00140 0.0089 0.032 0.142 0.336 0.481 0.684 0.797 0.902
Cu a, b 0.00009 0.00088 0.0070 0.027 0.097 0.192 0.252 0.323 0.356 0.386
Fe a 0.00038 0.00124 0.0045 0.012 0.055 0.144 0.216 0.323 0.384 0.447
In a 0.00095 0.0155 0.061 0.108 0.162 0.191 0.203 0.219 0.225 0.233
Nb a 0.00040 0.00220 0.0113 0.035 0.099 0.167 0.202 0.239 0.254 0.268
Ni a 0.00050 0.00162 0.0058 0.017 0.068 0.163 0.232 0.328 0.383 0.445
Si a 0.000017 0.00028 0.0034 0.017 0.079 0.177 0.259 0.425 0.556 0.714
Ti a 0.00032 0.00126 0.0070 0.025 0.099 0.218 0.300 0.407 0.465 0.522
W a 0.00004 0.00023 0.0019 0.008 0.033 0.068 0.089 0.114 0.125 0.136
Alloys
Al 2024 e — — — — — 0.478 0.534 0.639 0.736 0.855
Al-6061-T6 f 0.00029 0.00157 0.0089 0.033 0.149 0.348 0.492 0.713 0.835 0.954
Brass (65wt%Cu–35wt%Zn g (yellow brass) 0.00015e — 0.011 0.041 0.118 0.216 0.270 0.330 0.360 0.377
Constantan (60wt%Cu–40wt%Ni) a 0.00049 0.00169 0.0068 0.022 0.083 0.175 0.238 0.322 0.362 0.410
Inconel (77wt%Ni–15wt%Cr–7wt%Fe) e — — — — — 0.275 0.291 0.334 0.369 0.427
Stainless Steel 304L f 0.0017 0.0047 0.016 — — — — — — —
Stainless Steel 310 d 0.0020 0.0052 0.017 0.01 0.10 0.20 0.25 0.35 0.40 0.48
Ti–6wt%Al–4wt%V e — — — 0.007 0.098 0.217 0.300 0.410 0.477 0.529
Polymers & Composites
Epoxy (Stycast 2850FT) h 0.0005 0.0063 0.0226 0.042 0.083 0.154 0.240 — — —
Epoxy (CY221) c — 0.022 0.085 0.170 0.270 0.400 0.480 — 1.000 1.300
G-10CR f glass/resin 0.0020 0.0154 0.047 0.081 0.149 0.239 0.317 0.489 0.664 0.999
Glass/resin (S 901Glass/ NASA Resin 2) i 0.00064 0.0067 0.028 0.050 0.094 0.169 0.262 0.56 0.96 1.94
Plexiglas™ (PMMA) c — 0.017 0.080 0.147 0.280 0.420 0.550 — 0.920 —
Polyamide (NylonÔ) f 0.0016 0.020 0.100 0.200 0.380 0.574 0.717 0.984 1.21 1.62
Polyimide (Kapton) f 0.00079 0.0117 0.0579 0.116 0.224 0.338 0.414 0.537 0.627 0.755
Teflon (PTFE) c — 0.026 0.079 0.126 0.210 0.310 0.392 0.550 0.677 0.870
Ceramics and Nonmetals
AlN e — — — — — 0.074 0.139 0.305 0.471 0.739
Apiezon N f 0.00203 0.0243 0.0925 0.172 0.332 0.522 0.657 0.913 1.201 —
Carbon (diamond) a — 0.00002 0.0001 0.000 0.002 0.008 0.020 0.084 0.195 0.518
Ice a 0.00098 0.0152 0.114 0.229 0.440 0.689 0.882 1.230 1.570 —
MgO a — — 0.0022 0.006 0.024 0.101 0.208 0.465 0.680 0.940
Pyrex a 0.00020 0.0042 — — — — — — — —
Sapphire (Al2O3) e — 0.00009 0.0007 0.003 0.015 0.060 0.126 0.314 0.502 0.779
SiC e — — — — — 0.052 0.107 0.253 0.405 0.676
Silica glass (SiO2), Quartz Crystal (SiO2)a — 0.00070 0.0113 0.035 0.097 0.185 0.261 0.413 0.543 0.745
SrTiO3 e — — — — — 0.181 0.246 0.358 0.439 0.536
ZrO2 e — — — — — 0.100 0.153 0.261 0.347 0.456
a R. J. Corruccini and J. J. Gniewek (1960), National Bureau of Standards Monograph 21, U.S. Government Printing Office, Washington,. D.C.
b C. Y. Ho and A. Cezairliyan (1988), Specific Heat of Solids, Hemisphere Publishing Corp., New York.
c G. Hartwig (1994), Polymer Properties at Room and Cryogenic Temperatures, Plenum Press, New York,
d L. L. Sparks (1983), Chapter 2 in Materials at Low Temperatures, R. P. Reed and A. F. Clark, eds., ASM International, Metals Park, Ohio.
e Y. S. Touloukian and E. H. Buyco (1970), Specific Heat, Vols. 4 and 5, Plenum Press, New York.
f R. Radebaugh et al. (2003), http://www.cryogenics.nist.gov/ and the references listed therein.
g G. K. White and P. J. Meeson (2002), Experimental Techniques in Low-Temperature Physics, 4th edition, Oxford University Press.
h C. A. Swenson (1997), Rev. Sci. Instrum. 68, 1312–1315.
i E. W. Collings and R. D. Smith (1978), Adv. in Cryog. Eng. 24, 290–296.
A6.3 Debye model values of the molar heat capacity and molar internal energy as a function of temperature (Sections 6.1.2 and 6.1.3)
Tabulated values of the molar heat capacity are at constant volume, designated as CV . Molar internal energy U is obtained by integrating the heat capacity, tabulated here as (U–Uo)/T T–1 0T CV dT and plotted in Fig. 6.2.
Values of the Debye temperature D are tabulated for common elements in Appendix A6.1.
T/D D/T CV
[J/(mol∙K)] (U–Uo)/T T–1 0T CV dT
[J/(mol∙K)]
0.0 24.94 24.94
10 0.1 24.93 24.02
5 0.2 24.89 23.12
2.5 0.4 24.74 21.40
2.0 0.5 24.63 20.58
1.667 0.6 24.50 19.78
1.25 0.8 24.16 18.25
1.0 1.0 23.74 16.82
0.833 1.2 23.24 15.48
0.714 1.4 22.66 14.24
0.625 1.6 22.02 13.08
0.556 1.8 21.33 12.00
0.500 2.0 20.59 11.00
0.400 2.5 18.60 8.83
0.333 3.0 16.53 7.07
0.286 3.5 14.48 5.66
0.250 4.0 12.55 4.53
0.222 4.5 10.78 3.64
0.200 5.0 9.195 2.93
0.1667 6.0 6.625 1.94
0.143 7.0 4.760 1.31
0.125 8.0 3.447 0.912
0.111 9.0 2.531 0.654
0.100 10.0 1.891 0.481
0.0909 11.0 1.440 0.363
0.0833 12.0 1.117 0.281
0.0769 13.0 0.882 0.221
0.0714 14.0 0.707 0.177
0..0625 16.0 0.474 0.119
0.0556 18.0 0.333 0.083
0.0500 20.0 0.243 0.061
0.0400 25.0 0.124 0.031
0.0333 30.0 0.072 0.018
a From G. T. Furukawa, T. B. Douglas, and N. Pearlman (1972), Chapter 4e in American Institute of Physics Handbook, McGraw–Hill.
A6.4 Thermal expansion/contraction of technical materials (Sec. 6.2)
The total linear contraction from room temperature to the indicated temperature T is defined as
L/L (L293K – LT)/L293K.
Thecoefficient of linear expansion at room temperature is defined as
(1/L) dL/dT.
Since the thermal expansion/contraction is approximately linear above room temperature, the total contraction from an upper reference temperature Tu above room temperature (such as soldering temperature) to a low temperature T can be determined approximately from
L/LTu–T = L/L293K–T + (293K) (Tu – 293 K).
Data on Invar, glasses, ceramics, and other materials having a very low thermal contraction are given in Figs. 6.8 and 6.9. Thermal contraction data at 4 K and 77 K for a few additional materials are tabulated in Appendixes A7.4 and A7.5.
Thermal Expansion/Contraction of Technical Materials
Definitions: L/L (L293K – LT)/L293K ; (1/L) dL/dT
Material ∆L/L
at 4 K
[%] ∆L/L
at 40 K
[%] ∆L/L
at 77 K
[%] ∆L/L
at 100 K
[%] ∆L/L
at 150 K
[%] ∆L/L
at 200 K
[%] ∆L/L
at 250 K
[%]
at 293 K
[10–6 K–1]
Metals
Ag b 0.413 0.405 0.370 0.339 0.259 0.173 0.082 18.5 h
Al a 0.415 0.413 0.393 0.370 0.295 0.201 0.097 23.1 b
Au b 0.324 0.313 0.281 0.256 0.195 0.129 0.061 14.1
Be b 0.131 0.131 0.130 0.128 0.115 0.087 0.045 11.3 d
Cu a 0.324 0.322 0.302 0.282 0.221 0.148 0.070 16.7 i
Fe a 0.198 0.197 0.190 0.181 0.148 0.102 0.049 11.6 b
Hg b,* 0.843 0.788 0.788 0.592 0.396 0.176 * 57.2 *
In b 0.706 0.676 0.602 0.549 0.421 0.282 0.135 32.0
Mo b 0.095 0.094 0.090 0.084 0.067 0.046 0.022 4.8 d
Nb a 0.143 0.141 0.130 0.121 0.094 0.063 0.030 7.3 d
Ni a 0.224 0.223 0.212 0.201 0.162 0.111 0.053 13.4 d
Pb b 0.708 0.667 0.578 0.528 0.398 0.263 0.124 29
Ta b 0.143 0.141 0.128 0.117 0.089 0.059 0.028 6.6
Sn b (white) r 0.447 0.433 0.389 0.356 0.272 0.183 0.086 20.5
Ti a 0.151 0.150 0.143 0.134 0.107 0.073 0.035 8.3 b
W b 0.086 0.085 0.080 0.075 0.059 0.040 0.019 4.5
Alloys
Al-6061-T6 c 0.414 0.412 0.389 0.365 0.295 0.203 0.097 22.5
Brass (65%Cu–35%Zn) b
(yellow brass) 0.384 0.380 0.353 0.326 0.253 0.169 0.080 19.1 b
Constantan (50Cu–50Ni) b — 0.264 0.249 0.232 0.183 0.124 0.043 13.8 b
Cu–2%Be-0.3%Co (Beryll-
ium copper, Berylco 25) b 0.316 0.315 0.298 0.277 0.219 0.151 0.074 18.1 b
Fe–9%Ni a 0.195 0.193 0.188 0.180 0.146 0.100 0.049 11.5
Hastelloy C q 0.218 0.216 0.204 0.193 0.150 0.105 0.047 10.9 c
Inconel 718 a 0.238 0.236 0.224 0.211 0.167 0.114 0.055 13.0 k
Invar (Fe–36%Ni) a — 0.040 0.038 0.036 0.025 0.016 0.009 3.0 k
50%Pb–50%Sn solder a 0.514 0.510 0.480 0.447 0.343 0.229 0.108 23.4 d
Stainless Steel (AISI 304) b 0.296 0.296 0.281 0.261 0.206 0.139 0.066 15.1 l
Stainless Steel (AISI 310) b — — — 0.237 0.187 0.127 0.061 14.5
Stainless Steel (AISI 316) b 0.297 0.296 0.279 0.259 0.201 0.136 0.065 15.2 l
Ti–6Al–4V a 0.173 0.171 0.163 0.154 0.118 0.078 0.036 8.0 m
Superconductors
Bi-2212 a,b-axes u,y 0.152 0.150 0.139 0.132 0.106 0.074 0.036 8.3
Bi-2212 c-axis u,y 0.295 0.289 0.266 0.250 0.199 0.136 0.064 15.1
Bi (2223)/Ag tape g
(≥ 2nd cool-down) — 0.31 0.30 0.28 0.22 0.15 0.07 13
Bi-2223 a,b-axes z,u,y 0.15 0.15 0.14 0.13 0.11 0.07 0.04 8.3
Bi-2223 c-axis z,u,y 0.30 0.29 0.27 0.25 0.20 0.14 0.06 15
Bi-2223/61%Ag-alloy tapew,x 0.24
Nb3Sn a 0.16 0.16 0.14 0.13 0.095 0.065 0.03 7.6 t
Nb3Sn(10vol%)/Cu wire s 0.30 0.28
Nb–45 Ti a 0.188 0.184 0.169 0.156 0.117 0.078 0.038 8.2
Nb–Ti/Cu wire a 0.265 0.262 0.247 0.231 0.179 0.117 0.054 12.5
YBCO a-axis f — — 0.12 0.12 0.10 0.070 0.04 7.4
YBCO b-axis f — — 0.16 0.15 0.13 0.10 0.05 9.6
YBCO c-axis f — — 0.34 0.33 0.25 0.17 0.09 17.7
Polymers
Epoxy a 1.16 1.11 1.028 0.959 0.778 0.550 0.277 66
Epoxy (Stycast 2850FT) e 0.44 0.43 0.40 0.38 0.32 0.225 0.12 28
CTFE (Teflon) a 1.135 1.070 0.971 0.900 0.725 0.517 0.269 67 b
TFE (Teflon) a 2.14 2.06 1.941 1.85 1.600 1.24 0.750 250 n
PMMA (Plexiglas) a 1.22 1.16 1.059 0.99 0.820 0.59 0.305 75 o
Polyamide (Nylon) a 1.389 1.352 1.256 1.172 0.946 0.673 0.339 80
Polyimide (Kapton) c 0.44 0.44 0.43 0.41 0.36 0.29 0.16 46
Composites a
G-10CR epoxy/glass
( || glass fibers) 0.241 0.234 0.213 0.197 0.157 0.108 0.052 12.5
G-10CR epoxy/glass
(normal) 0.706 0.690 0.642 0.603 0.491 0.346 0.171 41 p
Ceramics & Nonmetals
Al N (|| a-axis) q — — 0.032 0.031 0.028 0.020 0.011 3.7
Al N (|| c-axis) q — — 0.025 0.025 0.022 0.017 0.009 3.0
C (diamond) b 0.024 0.024 0.024 0.024 0.023 0.019 0.011 1.0
Glass (Pyrex) 0.055 0.057 0.054 0.050 0.040 0.027 0.013 3.0 o
MgO b 0.139 0.139 0.137 0.133 0.114 0.083 0.042 10.2
Quartz (|| optic axis) b — — — 0.104 0.085 0.061 0.030 7.5
Sapphire (Al2O3) m (||c-axis) — 0.079 0.078 0.075 0.066 0.048 0.025 5.4 q
Si b 0.022 0.022 0.023 0.024 0.024 0.019 0.010 2.32
α-SiC (polycrystalline) q — — 0.030 0.030 0.029 0.024 0.013 3.7
Silica glass b -0.008 -0.005 -0.002 -0.0001 0.002 0.002 0.002 0.4
* For mercury, all data are referenced to its solidification temperature, 234 K.
a A. F. Clark (1983), Chapter 3 in Materials at Low Temperatures, ASM International, Materials Park, Ohio.
b R. J. Corruccini, and J. J. Gniewek. (1961), Thermal Expansion of Technical Solids at Low Temperatures, National Bureau of Standards Monograph 29, U.S. Government Printing Office, Washington, D.C.
c R. Radebaugh. et al. (2001), http://www.cryogenics.nist.gov/ and the references listed therein.
d CRC Handbook of Chemistry and Physics (2001), 82nd edition, CRC Press, Boca Raton, Florida.
e C. A. Swenson (1997), Rev. Sci. Instrum. 68, 1312–1315.
f Calculated from data by H. You, J. D. Axe, X. B. Kan, S. Hashimoto, S. C. Moss, J. Z. Liu, G. W. Crabtree, and D. J. Lam (1988), Phys. Rev. B38, 9213–9216.
g N. Yamada, K. Nara, M. Okaji, T. Hikata, T. Kanedo, and N. Sadakata (1998). Cryogenics 38, 397–399.
h V. J. Johnson, ed. (1961). Properties of Materials at Low Temperature, Phase 1, U.S. Government Printing Office.
i T. A. Hahn (1970). J. Appl. Phys. 41, 5096–5101.
j N. J. Simon, E. S. Drexler, and R. P. Reed (1992), Properties of Copper and Copper Alloys at Cryogenic Temperatures, NIST Monograph 177, U.S. Government Printing Office, Washington, D.C.; N. Cheggour and D. P. Hampshire, Rev. Sci. Instr. 71, 4521–4529 (2000).
k A. F. Clark (1968), Cryogenics 8, 282–289.
l Handbook on Materials for Superconducting Machinery (1974, 1976), National Bureau of Standards, U. S. Government Printing Office, Washington, D. C.
m V. Arp, J. H. Wilson, L. Winrich, and P. Sikora (1962), Cryogenics 2, 230–235.
n R. K. Kirby (1956), J. Res. Natl. Bur. Stand. 57, 91–94.
o H. L. Laquer and E. L. Head (1952). Low Temperature Thermal Expansion of Plastics. AECU-2161, Technical Information Service A.E.C., Oak Ridge, Tennessee.
p A. F. Clark, G. Fujii, and M. A. Ranney (1981), IEEE Trans. Magn. MAG-17, 2316–2319.
q Y. S. Touloukian, Thermal Expansion 12, 1248.
r Tin is anisotropic. Mean values were calculated as 1/3(║) + 2/3(┴), where (║) and (┴) signify the contraction parallel and perpendicular to the tetragonal axis. White tin is the ordinary ductile variety; it may transform to brittle grey tin (with a diamond-type lattice) at low temperatures, but usually it does not because of impurity stabilization. (See ref. b for more information.)
s L. F. Goodrich, S. L. Bray, and T. C. Stauffer (1990), Adv. Cryog. Eng. (Mater.) 36A, 117–124.
t D. S. Easton, D. M. Kroeger, W. Specking, and C. C. Koch (1980), J. Appl. Phys. 51, 2748.
u M. Okaji, K. Nara, H. Kato, K. Michishita, and Y. Kubo (1994), Cryogenics 34, 163.
v S. Ochiai, K. Hayashi, and K. Osamura (1991), Cryogenics 31, 959.
w E. Harley (2004), American Superconductor Corp., personal communication.
x J. P. Voccio, O. O. Ige, S. J. Young, and C. C. Duchaine (2001). IEEE Trans. Appl. Supercon. 11, 3070–3073.
y M. Mouallem-Bahout, J. Gaudé, G. Calvarin, J.–R. Gavarri, and C. Carel, (1994), Mater. Lett. 18, 181–185.
z Data are for Bi-2212 oriented crystals, but the atomic structures of the Bi-2223 and Bi-2212 phases are close enough that the Bi-2212 crystal data should approximately apply to both.
A6.5a Ideal electrical resistivity vs. temperature for pure metals (Sec 6.3)
The ideal resistivity i(T) is tabulated below for ideally pure metals. The total resistivity (T) of nearly pure metals is approximated by summing the temperature-dependent ideal resistivity i(T) and the temperature-independent residual resistivity res (that arises from defects). This is expressed as Matthiessen’s rule:
(T) res + i(T).
(Deviations from Matthiessen’s rule are briefly described in Sec. 6.3.4.)
In nearly pure metals, res is highly variable from specimen to specimen, because res depends on trace impurity levels and cold-work conditions. Therefore, it must be measured on an individual material basis (typically with a dip test in liquid helium) or estimated from such a measurement on a similar material. The total resistivity is then calculated from the above equation.
The Residual Resistance Ratio (RRR RRT/R4K = 295K/4K) is often used as an indicator of sample purity for pure metals [that is, the residual resistivity res ≈ 4K = i 295K/(RRR – 1)]. The higher the value of RRR, the lower res, and the more defect-free the metal. (Appendix A3.1 lists RRR values for common conductor materials, which can be used to estimate res; an example is given in Sec. 6.3.4.)
Values of the ideal resistivity i(T) tabulated below were determined experimentally by assuming the validity of Matthiessen’s rule and subtracting the measured value of res from precise measurements of the total (T) measured for very pure metals.
For convenience, the total resistivities of two oxygen-free copper (OFHC) samples are also listed, one with RRR @ 100, and the other 60% cold-drawn. Unlike the rest of the data, entries for these two material listings are not ideal resistivities and apply only to copper samples of comparable RRR or cold work.
Resistivity data at room temperature for additional elements are given in Appendix A6.1.
Ideal Resistivity i [Ω∙m 10–8 µΩ∙cm]
Pure Metal 10 K 20 K 50 K 77 K 100K 150 K 200 K 250 K 295 K
[RRR RT/4K]
Ag (RRR=1800) a 0.0001 0.003 0.103 0.27 0.42 0.72 1.03 1.39 1.60
Al (RRR=3500) a — 0.0007 0.047 0.22 0.44 1.01 1.59 2.28 2.68
Au (RRR=300) b 0.0006 0.012 0.20 0.42 0.62 1.03 1.44 1.92 2.20
Cu (RRR=3400) k — 0.0010 0.049 0.19 0.34 0.70 1.05 1.38 1.69
Cu(OFHC) (RRR@100)i (total ) 0.015 0.017 0.084 0.21 0.34 0.70 1.07 1.41 1.70
Cu (OFHC) (60 % cold drawn) i (total ) 0.030 0.032 0.10 0.23 0.37 0.72 1.09 1.43 1.73
Fe (RRR=100) c 0.0015 0.007 0.135 0.57 1.24 3.14 5.3 7.55 9.8
In (RRR=5000) d 0.018 0.16 0.92 1.67 2.33 3.80 5.40 7.13 8.83
Nb (RRR=213) e — 0.062 0.89 2.37 3.82 6.82 9.55 12.12 14.33
Ni (RRR=310) c — 0.009 0.15 0.50 1.00 2.25 3.72 5.40 7.04
Pb (RRR=14000, f
RRR=105) g — 0.53 f 2.85 f 4.78 f 6.35 g 9.95 g 13.64 g
17.43 g 20.95 g
Pt (RRR=600) c 0.0029 0.036 0.72 1.78 2.742 4.78 6.76 8.70 10.42
Ta (RRR=77) c 0.0032 0.051 0.95 2.34 3.55 6.13 8.6 11.0 13.1
Ti (RRR=20) h — 0.020 1.4 4.45 7.9 16.7 25.7 34.8 43.1
W (RRR=100) j 0.0002 0.0041 0.150 0.56 1.03 2.11 3.20 4.33 5.36
a R. S. Seth and S. B. Woods (1970), Phys. Rev. B2, 2961; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a, Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
b D. Damon, M. P. Mathur, and P. G. Klemens (1968), Phys. Rev. 176, 876; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
c G. K. White and S. B. Woods (1959), Philos. Trans. Roy. Soc. A251, 273; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a, Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
d G. K. White and S. B. Woods (1957), Rev. Sci. Instrum. 28, 638; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a, Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
e J. M. Abraham, C. Tete, and B. Deviot (1974), J. Less-comm. Met. 37, 181; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a, Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
f B. N. Aleksandrov and I. G. D’Yakov (1963), Sov. Phys. JETP (English Transl.) 16, 603–608; Zh. Eksp. Teor. Fiz. (1962) 43, 399; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a, Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
g J. P. Moore and R. S. Graves (1973), J. Appl. Phys. 44, 1174; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a, Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
h R. J. Wasilewski (1962), Trans. Met. Soc. AIME 224, 13; J. Bass, ed. (1982), Landolt–Börnstein, Vol. III/15a, Metals: Electronic Transport Phenomena, Springer-Verlag, Berlin.
j J. G. Hust (1976), High Temp.- High Press. 8, 377–390.
k J. S. Dugdale (1965), unpublished data, Univ. of Leeds, Leeds, UK.
A6.5b Total electrical resistivity vs. temperature for technical alloys and common solders a (Sec 6.3)
For alloys, values of the total resistivity (T) are tabulated [i.e., (T) res + i(T)], since there is little specimen-to-specimen variation in the residual resistivity contribution from defects.
Resistivities of solid alloys at room temperature are tabulated in Appendix A3.7.
Alloy total resistivity [Ω∙m 10–8 µΩ∙cm]
Alloy 10 K 20 K 50 K 77 K 100K 150 K 200 K 250 K 295 K
Al 1100–0 0.08 0.08 0.16 0.32 0.51 1.07 1.72 2.37 2.96
Al 5083–0 3.03 3.03 3.13 3.33 3.55 4.15 4.79 5.39 5.92
Al 6061–T6 1.38 1.39 1.48 1.67 1.88 2.46 3.09 3.68 4.19
Hastelloy C 123 123 123 124 — — 126 — 127
Inconel 625 124 124 125 125 — — 127 — 128
Inconel 718 108 108 108 109 — — 114 134 156
Berylco 25 6.92 6.92 7.04 7.25 7.46 7.96 8.48 8.98 9.43
(Cu-2%Be-0.3%Co)
Phosphor Bronze A 8.58 8.58 8.69 8.89 9.07 9.48 9.89 10.3 10.7
Cartridge Brass (70%Cu–30%Zn) 4.22 4.22 4.39 4.66 4.90 5.42 5.93 6.42 6.87
CuNi 30 (67Ni–30Cu) (Monel) 36.4 36.5 36.6 36.7 36.9 37.4 37.9 38.3 38.5
Ti–6%Al–4%V — 147 148 150 152 157 162 166 169
Stainless Steel (304L) 49.5 49.4 50.0 51.5 53.3 58.4 63.8 68.4 72.3
Stainless Steel (310) 68.6 68.8 70.4 72.5 74.4 78.4 82.3 85.7 88.8
Stainless Steel (316) 53.9 53.9 54.9 56.8 58.8 63.8 68.9 73.3 77.1
Invar (Fe–36%Ni) 50.3 50.5 52.1 54.5 57.0 63.3 70.0 76.5 82.3
a Values were interpolated with a cubic spline fit to data obtained by A. F. Clark, G. E. Childs, and G. H. Wallace (1970), Cryogenics 10, 295–305.
A6.6 Superconductor properties (Sec. 6.3.6)
Property values of these high-field superconductors are representative because there is some variation with sample composition, inhomogeneities, and impurity levels.
Additional data on critical-temperature values of superconducting elements are included in the general table of Appendix A6.1
Superconductor Crystal Structure * Lattice Constants [Å] † Tc
[K] µ0Hc2 (0 K)
[T] GL‡
[nm] GLζ
[nm]
a b c
Low Tc
Nb–Ti e A2 9.3 j 13 300 4
V3Ga e A15 4.816 n — — 15 23 90 2–3
V3Si e A15 4.722 n — — 16 20 60 3
Nb3Sn e A15 5.289 n — — 18 23 65 3
Nb3Al o A15 5.187 n — — 18.9 32
Nb3Ga o A15 5.171 n — — 20.3 34
Nb3(Al75Ge25) b A15 20.5 41
Nb3Ge e A15 5.166 n — — 23 38 90 3
NbN e B1 16 15 200 5
V2(Hf,Zr) o C15 10.1 24
PbMo6S8 e Chevrel 15 60 200 2
MgB2 hexagonal 3.086 m — 3.521 m 39 ~16 (a,b) l
~2.5 (c) 140 k 5.2 k
High Tc a
La1.85Sr0.15CuO4– e I4/mmm 3.779 3.779 1.323 40 50 80 (a,b)
400 (c) ~4 (a,b)
0.7 (c)
YBa2Cu3O7– d
(YBCO) Pmmm 3.818 3.884 11.683 90 670 (a,b)
120 (c) 150 (a,b)
900 (c) ~2 (a,b)
0.4 (c)
Bi2Sr2CaCu2O8– d
(Bi-2212) A2aa 5.410 5.420 30.930 90 280 (a,b)
32 (c) 300 (a,b) ~3 (a,b)
0.4 (c)
(Bi,Pb)2Sr2Ca2Cu3O10+
(Bi-2223) Perovskite
(orthorhombic) 5.39 5.40 37 110
Tl2Ba2CaCu2O8+ d,p
(Tl-2212) I4/mmm 3.856 3.856 29.260 110 215 (a,b) 2.2 (a,b)
0.5 (c)
Tl2Ba2Ca2Cu3O10– d,p
(Tl-2223) I4/mmm 3.850 3.850 35.88 125 120 205 (a,b)
480 (c) 1.3 (a,b)
HgBa2Ca2Cu3O8+ a Pmmm 3.85 — 15.85 133 160 q 1.42 (a,b) q
Notation:
* Crystal structures for the low-Tc superconductors are listed here mostly by the Strukturbericht designation, whereas for the high-Tc materials they are mostly listed by the Space group designation. Tables of cross lists to different nomenclatures are given in the appendixes to the ASM Handbook (1992), Vol. 3, Alloy Phase Diagrams, ASM International, Materials Park, Ohio.
† (a, b) refers to magnetic field, penetration depth, or coherence length being coplanar with the a,b crystallographic direction or Cu–O planes (usually parallel to the flat faces of practical conductors); (c) refers to an orientation along the c-axis; that is, perpendicular to the Cu–O planes (usually perpendicular to the flat faces of most practical conductors).
‡ The penetration depth GL is the constant prefactor in the Ginzburg–Landau expression GLGLc
ζ The coherence length GL is the constant prefactor in the Ginzburg–Landau expression GLGLc
References:
a CRC Handbook of Chemistry and Physics (2002), 83rd edition., CRC Press, Boca Raton, Florida.
b G. Bogner (1977), “Large scale applications of superconductivity,” in Superconductor Applications: SQUIDS and Machines, eds. B. B. Schwartz and S. Foner, Plenum, New York.
c B. W. Roberts (1978), Properties of Selected Superconducting Materials, NBS Technical Note 983, U.S. Government Printing Office, Washington, D.C..
d T. Datta (1992), “Oxide superconductors: physical properties,” pp. 414–423 in Concise Encyclopedia of Magnetic & Superconducting Materials, J. Evetts, ed., Pergamon Press.
e R. J. Donnelly (1981), in Physics Vade Mecum, ed. H. L. Anderson, Am. Inst. of Physics; T. P. Orlando and K. A. Delin, (1991), Foundations of Applied Superconductivity, Addison-Wesley.
j L. F. Goodrich and T. C. Stauffer (2003), unpublished data, National Institute of Standards and Technology, Boulder, Colorado.
k D. K. Finnemore, J. E. Ostenson, S. L. Bud’ko, G. Lapertot, and P. C. Canfield (2001), Phys. Rev. Lett. 86, 2420–2422.
l P. C. Canfield and G. W. Crabtree (2003), Physics Today 56, 34–40.
m T. Vogt, G. Schmneider, J. A. Hriljac, G. Yang, and J. S. Abell (2001), Phys. Rev. B 63, 220505/1–3.
n M. Weger and I. B. Goldberg (1973), p. 3 in Solid State Physics, Vol. 28, eds. H. Ehrenreich, F. Seitz, and D. Turnbull, Academic Press, p. 3.
o J. W. Ekin (1983), Chapter 13 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio.
p E. Bellingeri and R. Flükiger (2003), in Handbook of Superconducting Materials, Vol. 1, D. A. Cardwell and D. S. Ginley, eds., Inst. of Phys. Publishing, pp. 993–1027.
q J. Schwartz and P.V.P.S.S. Sastry (2003), in Handbook of Superconducting Materials, Vol. 1, D. A. Cardwell and D. S. Ginley, eds., Inst. of Phys. Publishing, pp. 1029–1048.
A6.7 Thermal conductivity vs. temperature for selected metals, alloys, glasses, and polymers (Sec. 6.4).
Additional thermal-conductivity data for various amorphous solids (vitreous silica, germania, selenium) and amorphous materials (polystyrene and PMMA) are shown in Fig. 6.14. Thermal conductivity integrals are tabulated for selected cryostat construction materials in Appendix A2.1. Properties of metals with very high thermal conductivities are given in Appendix A3.1.
Thermal conductivity [W/(mK)]
Material 4 K 10 K 20 K 40 K 77 K 100 K 150 K 200 K 295 K
Metals & Alloys
Al 5083 a 3.3 8.4 17 33 55 66 85 99 118
Al 6061-T6 a 5.3 14 28 52 84 98 120 136 155
Beryllium–Copper g 1.9 5.0 11 21 36 43 57 72 95
Brass (UNS C36000) (61.5wt%Cu–35.4wt%Zn–3.1st%Pb) b 2.0 5.7 12 19 29 40 47 64 86
Brass (68wt%Cu–32wt%Zn) d 3.0 10 22 38 53 — — — —
Copper OFHC (RRR≈100) a 630 1540 2430 1470 544 461 418 407 397
Inconel 718 a 0.46 1.5 3.0 4.7 6.4 7.1 8.1 8.7 9.7
Invar b 0.24 0.73 1.7 2.6 4.2 6.2 7.6 10 12
Manganin (Cu–12wt%Mn–3wt%Ni) d 0.44 1.4 3.2 6.8 11 — — — —
Soft-Solder
(Sn–40wt%Pb) d 16 43 56 53 53 — — — —
Stainless Steel 304,316 a 0.27 0.90 2.2 4.7 7.9 9.2 11 13 15
Ti (6%Al–4%V) a — — 0.84 1.9 3.5 3.8 4.6 5.8 7.4
Polymers
G-10CR (Normal) a 0.072 0.11 0.16 0.22 0.28 0.31 0.37 0.45 0.60
G-10CR (Warp) a 0.073 0.14 0.20 0.27 0.39 0.45 0.57 0.67 0.86
HDPE c 0.029 0.090 — — 0.41 0.45 — — 0.40
Kevlar 49 a 0.030 0.12 0.29 0.59 1.0 1.2 1.5 1.7 2.0
PMMA (Plexiglas) c 0.033 0.060 — — — 0.16 0.17 0.18 0.20
Polyamide (Nylon) a 0.012 0.039 0.10 0.20 0.29 0.32 0.34 0.34 0.34
Polyimide (Kapton) a 0.011 0.024 0.048 0.083 0.13 0.14 0.16 0.18 0.19
Polyethylene terepthalate (Mylar) b 0.038 0.048 0.073 0.096 0.12 — — — —
PVC c 0.027 0.040 — — — 0.13 0.13 0.13 0.14
PTFE (Teflon) a 0.046 0.10 0.14 0.20 0.23 0.24 0.26 0.27 0.27
Ceramics and Nonmetals
Alumina (Al2 O3 , sintered) d 0.49 5.6 24 80 157 136 93 50 —
Macor™ e 0.075 0.25 0.60 — — — — — —
MgO (crystal) d 82 1130 2770 2160 507 294 135 91 61
Pyrex glass d 0.10 0.12 0.15 0.25 0.45 0.58 0.78 0.92 1.1
Sapphire (Al2 O3 , synthetic crystal) d, f 230 2900 15700 12000 1100 450 150 82 47
α-SiC (single crystal,
┴ to c-axis) d 27 420 2000 4700 4000 3000 1500 950 510
SiO2 crystal (avg. of ║ and ┴ to c-axis) d 185 1345 545 134 43 30 18 13 9
a Cryogenic Materials Properties Program CD, Release B-01 (June 2001), Cryogenic Information Center, 5445 Conestoga Ct., Ste. 2C, Boulder, CO 80301-2724, Ph. (303) 442-0425, Fax (303) 443-1821.
b R. Radebaugh. et al. (2003), http://www.cryogenics.nist.gov/ and the references listed therein.
c G. Hartwig (1994), Polymer Properties at Room and Cryogenic Temperatures, Plenum Press, New York.
d Y. S. Touloukian and E. H. Buyco (1970), Thermal Conductivity, Vols. 1 and 2, Plenum Press, New York.
e W. N. Lawless (1975), Cryogenics, 15, 273–277.
f For sapphire, the direction of heat flow is 60 degrees to the hexagonal axis; values are thought to be accurate to within 10 % to 15 % at temperatures above 60 K, but highly sensitive to small physical and chemical variations below 60 K.
g D. E. Gray, ed. (1972), Thermal Conductivity of Alloys, Am. Inst. of Physics Handbook, McGraw Hill, Table 4g–9.
A6.8a Magnetic mass susceptibility from 1.6 K to 4.2 K of materials commonly used in cryostat construction (Sec. 6.5)
Mass susceptibility is useful for small samples or irregularly shaped parts where the mass of the sample is more easily determined than its volume. It is not difficult, however, to convert between the two with the relation
(mass susceptibility (volume susceptibility density in kg/m3)
Mass susceptibility is defined as /where is the magnetic moment per unit mass and H is the magnetic intensity.
The coefficients B and C tabulated below (fourth and fifth columns) are used to calculate mass susceptibility in the temperature range 1.6 K to 4.2 K through the Curie law
/ B + (C/T).
Mass susceptibility has been evaluated at 4.2 K in the third column.
Values are tabulated in SI units (mks). To convert to cgs units, divide the values in this table by 4 10–3 to get / in cm3/g; see Appendix A1.4.
Magnetic mass susceptibility /
Material
Supplier /
at 4.2 K
[10–8 m3/kg] B
at 1.6K to 4.2 K
[10–8 m3/kg] C
at 1.6K to 4.2K
[10–8 m3K/kg]
Dielectric Structural Materials
Alumina a Alcoa 2.8 1.0 0.8 7.5 2
Alscobond Y-725 and catalyst c Alloy Supply Co. –9.4 –3.3 0.1 –25.5
Bakelite, type 950 c Thiokil Chemical Co. 0.3 0.7 0.16 –1.8
Epibond 100A b Furane Plastics, Inc. –0.5 –0.5 0.1 0.1 0.2
Epibond 104 a “ 60 30 3 160 10
Epibond 121 a “ 0.4 0.1 1 1 3.1
Glass
Plate 7740 a Corning Glass Co. 14 3.8 1 44 5
Test tubes, Pyrex a “ 7.3 2.5 1 20 4
Tubing 7740 a “ 5.6 2.0 1 15 4
Lava, grade A c American Lava Corp. –21.1 –4.2 0.7 –71
Nylon 101, type 66 b Polypenco Ltd. –0.81 –0.79 0.01 –0.08 0.03
PTFE (Teflon) b “ –0.40 –0.41 0.01 0.06 0.01
Resin 3135 w/catalyst 7111 c Crest Products Co. 1.24 1.44 0.2 –0.86
Resin 3170 w/catalyst 7133 c “ 0.53 0.59 0.017 –0.25
Resin 7343 w/catalyst 7139 c “ 1.70 2.24 0.18 –2.24
Quartz a 3 4 1.9 –4.4 4.4
Silica
No washing a Fisher Scientific Co. 1.2 1.8 0.5 –2.3 2.3
Acid washing a –0.1 –0.3 0.9 0.6 1.9
Stycast 2850GT w/catalyst No. 9 a 27 19 5 33 9
Tufnol, Carp brand b Tufnol Ltd. 0.5 –0.3 0.3 3.0 0.5
Fibrous Materials
Absorbent cotton a New Aseptic Labs, Inc. 570 380 40 800 100
Felt a McMaster–Carr 60 42 16 75 50
Pyrex Wool a Corning Glass Co. 240 150 30 380 50
Thread (white) a Coates and Clark Co. –15.3 –19 16 15 38
Fluids
Apiezon “J” oil a James G. Biddle Co. –0.19 –0.04 0.18 –0.6 4
Apiezon “N” grease a “ –0.5 0.1 1.5 –2.5 0.4
Aquadag a Colloids Corp. 15 15 2 0.1 5
RTV-102 adhesive c General Electric 0.79 1.2 0.5 –1.62
Silicone-oil 50 cs a Dow–Corning 1.7 1.6 1.6 0.1 4
GE 7031 Varnish-toluene
(1:1 mixture) a General Electric 0.4 3 2 –9 4
Metals
Brass a Central Steel & Wire Co. –235 –226 25 –38 13
Copper magnet wire
Formex insulated a Anaconda Copper Co. 0.04 0.04 1.3 0.01 1.9
Sodereze insulated a Phelps Dodge Inc. 0.1 0.6 0.9 –2.0 2.1
Evanohm
Double silk covered a Wilbur B. Driver Co. 86.0 –3.8 7.5 377 38
Formex insulated a “ 155 11 18 603 75
Manganin, enamel insulated a Driver–Harris Inc. 147 151 13 –16 4
Phosphor bronze c Central Steel & Wire Co. 2.1 2.9 1.1 –3.3
Beryllium copper Berylco 25 c Meier Brass & Aluminum 254 254 1.9 0.0
Stainless steels
303 a Central Steel & Wire Co. 148 144 19 13 13
304 a “ 134 136 11 –7.2 6
316 a “ 361 392 31 –130 25
321 a “ 133 126 13 31 13
347 a “ 215 250 120 –150 190
Sheets and Tapes
Kapton H film c Du Pont 0.8 1.3 0.7 –2.1
Mylar a Du Pont 63 63 9 1.9 8
Paper
White a W. M. Morgan Putnam Co. 4 –0.3 3 18 9
Black (photographic) a “ 3 1 4 6 8
Tapes
Cellophane a Minn. Mining & Mfg. Co. 0.7 0.1 2 2 6
Glass #27 a “ 30 9 2 88 10
Masking (Tuck Tape) a Technical Tape Corp. 11 10 2 3 5
Special
Cupro-Nickel (70–30) a Superior Tube Co. 480 480 100
Eccosorb LS-22 foam sheet c Emerson and Cuming Inc. –0.6 1.2 0.9 –7.7
Germanium resistor c Cryocal Inc. 6330 6380 48 –215
Inconel a Superior Tube Co. 2.6105 (3.3 ± 0.25)105 (–2.9 ± 0.25)105
SC-13 flexible silver micropaint a Microcircuits Co. 1.5104 (1.3 0.9)104 (0.87 0.09)104
Rubber (Neoprene) a Microdot Inc. 35 16 24 78 56
a G. L. Salinger and J. C. Wheatley (1961), Rev. Sci. Instrum. 32, 872–874.
b R. J. Commander and C. B. P. Finn (1970), J. Phys. E: Sci. Instrum. 3, 78–79.
c D. M. Ginsberg (1970), Rev. Sci. Instrum. 41, 1661–1662.
A6.8b Magnetic volume susceptibility at 295 K, 77 K, and 4.2 K of structural materials commonly used in cryostat construction (Sec. 6.5)
Volume susceptibility is useful for structural parts with well defined shapes (such as tubes, rods, sheets, and blocks) where the volume of the part is readily determined. (In contrast, mass susceptibility is utilized in situations where the mass is easier to determine, such as with small or irregularly shaped parts.) The two quantities are simply related by
(mass susceptibility (volume susceptibility density in kg/m3)
Volume susceptibility is defined by where M is the magnetic moment per unit volume and H is the magnetic intensity.
Values in the table below are tabulated in SI units (mks). To convert to cgs units, divide these values by 4 (from Appendix A1.4).
Magnetic volume susceptibility
Material Condition Density
[103 kg/m3] (293 K)
[SI units] (77 K)
[SI units] (4.2 K)
[SI units]
Aluminum alloys
pure c 2.70 2.0710–5 2.5210–5
2014 c 2.79 1.8010–5 1.7210–5
Copper alloys
99.999% pure Cu b cold drawn, etch, and annealed –9.3410–6 –9.1810–6 –8.6710–6
99.96+% pure Cu b as formed –7.3210–6 –8.5510–6 –9.3210–6
Oxygen-free Cu b –9.1410–6 –9.3210–6 –8.9310–6
ETP copper c 8.92 3.2210–5 2.5310–5
Beryllium copper c
(Cu–2%Be) 8.33 1.5610–3 1.8210–3
Phosphor Bronze A c
(94.8Cu–5Sn–0.2P) 8.95 –5.8610–6 –5.5610–6
90Cu–10Ni b 1.6910–5 1.5710–5 2.2110–5
Brass, plain (cartridge) c
(70%Cu–30%Zn) 8.52 –3.4810–6 –6.1410–5
Brass, free-cutting c (61.5%Cu–35.4%Zn–3.1%Pb) 8.52 1.1210–2 –1.410–2
Manganin j
(83Cu–13Mn–4Ni) 2.710–3 1.2610–2
Constantan j
(Cu–45Ni) 0.45 4.3
Titanium alloys:
Ti 4.51 f 1.7810–4 e
Ti–6%Al–4%V 4.41 c 1.8010–4 c –8.2710–6 c
Stainless Steels:
304 7.86 g 2.710–3 h 5.510–3 h
304L fully softened 7.86 g 2.610–3 h 4.910–3 h
304N 7.86 g 2.610–3 h 5.210–3 h 4.810–3 d
309 a fully softened
sensitized 2.110–3
2.610–3 6.210–3
6.710–3 2.410–2
2.110–2
310 a fully softened
sensitized 7.85 g 2.210–3
2.310–3 8.310–3
1.210–2 F
F
310S 7.85 g 2.610–3 h 9.510–3 h
316 7.97 g 3.010–3 c, h 7.710–3 h 1.610–2 d
316L 7.97 g 3.010–3 h 8.010–3 h
316LN a fully softened
sensitized 7.97 g 2.610–3
3.510–3 6.910–3
7.210–3 1.110–2
1.110–2
Nickel alloys: i Maximum susceptibility
at: value:
Inconel 718-1153 19 K 13
Inconel 718-1094 16 K 3.2
Inconel 718-1 15 K 3.8
Inconel 625 < 5 K 0.0032
Nichrome (Ni–20Cr) j 5.210–4 5.7510–3
Polymers:
Acrylic c 1.05 –6.9810–6 –2.6510–6
Nylon c 1.15 –9.0410–6 –7.4610–6
Composites: c
G10CR 1.83 2.6310–6 5.3410–4
G11CR 1.90 2.5910–6 4.5810–4
Linen Phenolic 1.35 –4.2610–6 2.9310–6
Miscellaneous: c
Hardwood 0.63 6.0910–6 1.2210–5
Quartz 2.21 –1.0310–5 –9.2710–6
F ≡ Ferromagnetic at this temperature.
a D. C. Larbalestier and H. W. King (1973), Cryogenics 13, 160–168.
b Handbook on Materials for Superconducting Machinery (1977), MCIC-HB-04,. Battelle, Columbus, Ohio.
c F. R. Fickett (1992), Adv. Cryog. Eng. (Mater.) 38, 1191–1197.
d E. W. Collings and R. L. Cappelletti (1985), Cryogenics 25, 713–718.
e Landolt–Börnstein (1986), New Series, II/16, Diamagnetic Susceptibility, Springer-Verlag, Heidelberg; Landolt–Börnstein (1986–1992), New Series, III/19, Subvolumes a to i2, Magnetic Properties of Metals, Springer-Verlag, Heidelberg; CRC Handbook of Chemistry and Physics (2000), 81st edition, CRC Press, Boca Raton, Florida.
f Metals Handbook (1961), Vol. 1, Properties and Selection of Materials, 8th edition., ASM International, Materials Park, Ohio.
g H. I. McHenry (1983), Chapter 11 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio
h E. W. Collings and S. C. Hart (1979), Cryogenics 19, 521–530. (The coefficients given in Table 6 of this reference should be multiplied by 4 to correctly give mass susceptibility in units of m3/kg–1.)
i I. R. Goldberg, M. R. Mitchell, A. R. Murphy, R. B. Goldfarb, and R. J. Loughran (1990), Adv. Cryog. Eng. (Mater.) 36, 755–762.
j M. Abrecht, A. Adare, and J. W. Ekin (2007), Rev. Sci. Inst. 78, 046104.
A6.8c Ferromagnetic traces at 4.2 K induced by welding and cyclic cooling of austenitic stainless steels a (Sec. 6.5)
Austenitic stainless steels are paramagnetic, but most become unstable below room temperature and partially transform into a martensitic phase, which is ferromagnetic. The transformation depends critically on the exact chemical composition and heat treatment of the alloy, as evidenced by the difference in data below for various samples of the same type of steel, indicated by (a), (b), and (c). Welding can also induce ferromagnetic behavior. Only 316LN and X6CrNi 1811 show no ferromagnetic traces on cooling or welding.
Stainless Steel Alloy Ferromagnetic Traces
on first cooling on welding on cyclic cooling
303 + + not tested
304 (a) + + not tested
304 (b) – + not tested
304 (c) + + not tested
304N + – +
310 + + +
310S + + –
316 (a) – + –
316 (b) – – –
316 (c) – – –
316L + + –
316LN – – –
316Ti + + –
X6CrNi 1811 – – –
321 (a) + + –
321 (b) + + +
+ ≡ detected
– ≡ not detected
a Data from K. Pieterman, A. Ketting, and J. C. Geerse (1984), J. Phys., 45, C1-625–C1-631.
A6.9 Composition of austenitic stainless steels, nickel steels, and aluminum alloys (Sec. 6.6)
Compositions of austenitic stainless steels a
These are Fe–Cr alloys with sufficient Ni and Mn to stabilize the f.c.c. austenitic phase so they retain their strength, ductility, and toughness at cryogenic temperatures.
Temperature-dependent mechanical and physical properties of AISI b 304, 310, and 316 are tabulated in Appendix A6.10.
Composition [weight percent]
AISI b
Type No. Cr Ni Mn C, max N Other
201 16–18 3.5–5.5 5.5–7.5 0.15 0.25, max 1.00 Si max
10.060 P max
0.030 S max
202 17–19 4–6 7.5–10 0.15 0.25, max 1.00 Si max
10.060 P max
0.030 S max
301
16–18 6–8 0.15
302
17–19 8–10 0.15
304
18–20 8–12 0.08
304 L
18–20 8–12 0.03
304 N 18–20 8–10.5 2.0 max 0.08 0.10–0.16 1.00 Si max
10.060 P max
0.030 S max
304 LN 18–20 8–12 2.0 max 0.03 0.10–0.16 1.00 Si max
10.060 P max
0.030 S max
305
17–19 10.5–13 0.12
309
22–24 12–15 0.20
310
24–26 19–22 0.25 1.5 Si max.
310 S
24–26 19–22 0.08 1.5 Si max.
316
16–18 10–14 0.08 2–3 Mo
316 L
16–18 10–14 0.03 2–3 Mo
316 N 16–18 10–14 2.0 max 0.08 0.10–0.16 1.00 Si max
10.060 P max
0.030 S max
316 LN 16–18 10–14 2.0 max 0.03 0.10–0.16 1.00 Si max
10.060 P max
0.030 S max
321
17–19 9–12 0.08 (5 %C)
Ti, min.
347
17–19 9–13 0.08 (10 %C)
Nb+Ta, min.
ASTM c XM-10 19–21.5 5.5–7.5 8–10 0.08 0.15–0.40 1.00 Si max
10.060 P max
0.030 S max
ASTM XM-11 19–21.5 5.5–7.5 8–10 0.04 0.15–0.40 1.00 Si max
10.060 P max
0.030 S max
ASTM XM-14 17–19 5–6 14–16 0.12 0.35–0.50 1.00 Si max
10.060 P max
0.030 S max
ASTM XM-19 20.5–23.5 11.5–13.5 4–6 0.06 0.20–0.40 0.10–0.30 Nb, 1.00 Si max
10.060 P max
0.030 S max
ASTM XM-29 17–19 2.25–3.75 11.5–14.5 0.08 0.20–0.40 0.10–0.30 V
1.5–3.0 Mo
1.00 Si max
10.060 P max
0.030 S max
a From H. I. McHenry (1983), Chapter 11 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio.
b AISI: American Iron and Steel Institute, a designation system for steels.
c ASTM: Formerly known as the American Society for Testing and Materials, now ASTM International, an organization that provides a global forum for consensus standards for materials, products, systems, and services.
Composition of nickel steels a
These Fe–Ni alloys have a predominantly b.c.c. crystal structure that undergoes a ductile-to-brittle transition as temperature is reduced; the transition temperature decreases with increasing nickel content.
Temperature-dependent mechanical and physical properties of 3.5 Ni, 5 Ni, and 9 Ni alloys are tabulated in Appendix A6.10.
Composition [weight percent]
ASTM b
Specifi-cation Alloy Minimum
Service
Temp.[K] C
max Mn P
max S
max Si Ni Mo Cr
A203
Grade D
3.5 Ni 173 0.17 0.7 max 0.035 0.040 0.15–0.30 3.25–3.75
A203
Grade E
3.5 Ni 173 0.20 0.7 max 0.035 0.040 0.15–0.30 3.25–3.75
A645
5 Ni 102 0.13
0.30–0.60 0.025 0.025 0.20–0.35 4.75–5.25 0.20–0.35
A645
5.5 Ni 77 0.13 0.90–1.50 0.030 0.030 0.15–0.30 5.0–6.0 0.10–0.30 0.10–1.00
A553
Type II
8 Ni 102 0.13 0.90 max 0.035 0.040 0.15–0.30 7.5–8.5
A553
Type I
9 Ni 77 0.13 0.90 max 0.035 0.040 0.15–0.30 8.5–9.5
a From H. I. McHenry (1983), Chapter 11 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio.
b ASTM: Formerly known as the American Society for Testing and Materials, now ASTM International, an organization that provides a global forum for consensus standards for materials, products, systems, and services.
Composition of aluminum alloys a
Aluminum alloys have an f.c.c. crystal structure and thus retain their strength, ductility, and toughness at cryogenic temperatures. Temperature-dependent mechanical and physical properties of type 1100, 2219, 5083, and 6061 alloys are tabulated in Appendix A6.10.
Composition [weight percent]
Type Si
max Fe
max Cu Mn Mg Cr Zn
max Ti Other
elements
1100
1.0 Si (+Fe) 0.05–0.20 0.05 max — — 0.10 — 99.00 Al min
2219
0.2 0.3 5.8–6.8 0.20–0.40 0.02 max — 0.10 0.02–0.10 0.05–15 V
3003
0.6 0.7 0.05–0.20 1.0–1.5 — — 0.10 — 0.10–0.25 Zr
5083
0.4 0.4 0.1 max 0.40–1.0 4.0–4.9 0.05–0.25 0.25 0.15 max
6061
0.4–0.8 0.7 0.15–0.40 0.15 max 0.8–1.2 0.04–0.35 0.25 0.15 max
7005
0.35 0.40 0.10 0.2–0.7 1.0–1.8 0.06–0.20 4.0–5.0 0.01–0.06 0.08–0.20 Zr
a From H. I. McHenry (1983), Chapter 11 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio.
A6.10 Mechanical properties of structural materials used in cryogenic systems (Sec. 6.6)
The following four tables give the mechanical and physical properties of austenitic stainless steels, nickel steels, aluminum alloys, and other selected metal alloys and polymers.
Mechanical and Physical Properties of Austenitic Stainless Steels
Compositions of these and other stainless-steel alloys are tabulated in Appendix A6.9; a plot of the temperature dependence of the yield strength of AISI 304 with various cold-work conditions is given Fig. 6.17.
Alloy
Temperature Density a
[g∙cm–3] Young’s
Modulusa
[GPa] Shear
Modulusa
[GPa] Poisson’s
Ratio a Fracture Tough-
ness b
[MPa∙m0.5] Thermal
Conduc-
tivity b
[W/(m∙K)]
Thermal
Expansion b
(mean)
(K–110–6) Specific
Heat b
[J/(kg∙K)] Electri-
cal
Resis-
tivity c
[∙cm] Magnetic
Permea-
bility d
(initial) 0.2 % Yield Strength,
annealed e
[MPa]
AISI f 304
295 K 7.86 200 77.3 0.290 14.7 15.8 480 70.4 1.02 240
77 K 214 83.8 0.278 7.9 13.0 — 51.4 — —
4 K 210 82.0 0.279 0.28 10.2 1.9 49.6 1.09 —
AISI 310
295 K 7.85 191 73.0 0.305 150 11.5 15.8 480 87.3 1.003 275
77 K 205 79.3 0.295 220 5.9 13.0 180 72.4 — —
4 K 207 79.9 0.292 210 0.24 10.2 2.2 68.5 1.10 —
AISI 316
295 K 7.97 195 75.2 0.294 350 14.7 15.8 480 75.0 1.003 240
77 K 209 81.6 0.283 510 7.9 13.0 190 56.6 — —
4 K 208 81.0 0.282 430 0.28 10.2 1.9 53.9 1.02 —
Major data source: Compilation by H. I. McHenry (1983), Chapter 11 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio.
a H. M. Ledbetter, W. F. Weston, and E. R. Naimon (1975), J. Appl. Phys. 46, 3855–3860.
b D. B. Mann, ed. (1978), LNG Materials and Fluids, National Bureau of Standards, U.S. Government Printing Office, Washington, D. C.
c A. R. Clark, G. E. Childs, and G. H. Wallace (1970), Cryogenics 10, 295–305.
d K. R. Efferson and W. J. Leonard (1976), Magnetic Properties of Some Structural Materials Used in Cryogenic Applications, ORNL-4150, p. 126, Oak Ridge National Laboratory, Oak Ridge, Tennessee.
e Metals Handbook, Vol. 1, Properties and Selection of Materials (1961), 8th edition, ASM International, Materials Park, Ohio.
f AISI: American Iron and Steel Institute; a designation system for steel alloys.
Mechanical and Physical Properties of Nickel Steels
Compositions of these and other nickel-steel alloys are tabulated in Appendix A6.9; a plot of the temperature dependence of the yield strength of quenched and tempered 9% Ni steel is given in Fig. 6.17.
Alloy
Temperature Minimum Service Temp.c
[K] Densitya
[g∙cm–3] Young’s
Modulusa
[GPa] Shear
Modulusa
[GPa] Poisson’s
Ratioa Fracture Tough-
ness b
[MPa∙m0.5] Thermal
Conduc-
tivity b
[W/(m∙K)] Thermal
Expansion b
(mean)
[K–110–6] Specific
Heat b
[J/(kg∙K)]
3.5 Ni 173
295 K 7.86 204 79.1 0.282 190 35 11.9 450
172 K 210 81.9 0.281 210 29 10.2 350
5 Ni 102
295 K 7.82 198 77.0 0.283 210 32 11.9 450
111 K 208 81.2 0.277 200 20 9.4 250
76 K 209 81.6 0.277 90 16 8.8 150
9 Ni 77
295 K 7.84 195 73.8 0.286 155 28 11.9 450
111 K 204 77.5 0.281 175 18 9.4 250
76 K 205 77.9 0.280 170 13 8.8 150
Major data source: Compilation by H. I. McHenry (1983), Chapter 11 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio.
a W. F. Weston, E. R. Naimon, and H. M. Ledbetter (1975), pages 397–420 in Properties of Materials for Liquefied Natural Gas Tankage. ASTM STP 579, American Society for Testing and Materials, Philadelphia.
b D. B. Mann, ed. (1978), LNG Materials and Fluids, National Bureau of Standards, U.S. Government Printing Office, Washington, D.C.
c The minimum service temperature arises because of the ductile-to-brittle-phase transition that occurs at low temperatures in the nickel steels.
Mechanical and Physical Properties of Aluminum Alloys
Compositions of these and other aluminum alloys are tabulated in Appendix A6.9; a plot of the temperature dependence of the yield strength of various aluminum alloys is given in Fig. 6.18.
Alloy
Temperature Density
[g∙cm–3] Young’s
Modulus
[GPa] Shear
Modulus
[GPa] Poisson’s
Ratio 0.2% Yield Strength g
[MPa] Thermal
Conductiv-
ity
[W/(m∙K)] Thermal
Expansion
(mean)
[K–110–6] Specific
Heat
[J/(kg∙K)] Electrical
Resistivity
[∙cm]
1100-0
295 K 2.75 69 < 35
Precipitation-hardened
2219-T6
295 K 2.83 77.4 d 29.1d 0.330 d 393 120 c 23 f 900 f 5.7 b
77 K 85.1d 32.3 d 0.319 d 56 c 18.1f 340 f
4 K 85.7 d 32.5 d 0.318 d 3 c 14.1f 0.28 f 2.9 b
Annealed 5083
295 K 2.66 71.5 a 26.8 a 0.333 a 145
(half hard: 228) 120 b 23 b 900 b 5.66 c
77 K 80.2 a 30.4 a 0.320a 55 b 18.1b 340 b 3.32 c
4 K 80.9 a 30.7 a 0.318 a 3.3 b 14.1b 0.28 b 3.03 c
Precipitation-hardened
6061-T87
295 K 2.700 70.1b 26.4 b 0.338 b 275 23 b 900 c 3.94 c
77 K 77.2 b 29.1b 0.328 b 18.1b 340 c 1.66 c
4 K 77.7 b 29.2 b 0.327 b 14.1b 0.28 c 1.38 c
Major data source: Compilation by H. I. McHenry (1983), Chapter 11 in Materials at Low Temperatures, eds. R. P. Reed and A. F. Clark, ASM International, Materials Park, Ohio.
a E. R. Naimon, H. M. Ledbetter, and W. F. Weston (1975), J. Mater. Sci. 10, 1309–1316.
b D. B. Mann, ed. (1978), LNG Materials and Fluids, National Bureau of Standards, U.S. Government Printing Office, Washington, D.C.
c A. R. Clark, G. E. Childs, and G. H. Wallace (1970), Cryogenics 10, 295–305.
d R. P. Read and H. M. Ledbetter (1977), J. Engineering Materials and Technol. 99, 181–184.
e G. E. Childs, L. J. Ericks, and R. L. Powell (1973), Thermal Conductivity of Solids at Room Temperature and Below, NBS Monograph 131, U.S. Government Printing Office, Washington, D.C.
f Assumed to be the same as that of 5083 and 6061.
g Compiled by R. Radebaugh et al. (2001), http://www.cryogenics.nist.gov/ and the references listed therein.
Mechanical Properties of Metal Alloys and Polymers
All data are at room temperature, unless noted otherwise by three consecutive values corresponding to 295K/76K/4K.
Mechanical property data for additional materials are available in the literature (Sec. 6.7.1) and on the Internet (Sec. 6.7.2).
Material
Density
[g∙cm–3] Young’s
Modulus
[GPa] Yield Strength
[MPa]
Metal Alloys
Beryllium S-200F a 1.86 290 240
Copper, oxygen-free (annealed) a 8.95 117 70
Cu–2%Be (UNS C17200-TH 04) b, c 8.23 119 1030
Inconel 625 d 8.44 f 195/207/207 500/720/810
Inconel 718 a 8.20 200 1060
Hastelloy C-276 d,e 8.9 192/209/205 480/700/810
Ni (annealed) d 8.9 60/70/91 60/70/80
Ni–13at%Cr d 8.7 111/112/119 120/160/190
Ni–5at%W d 10.4 118/128/134 180/260/280
Titanium 3Al–2.5V f (various shapes) 4.5 100 830
Titanium 6Al–4V a (sheet form) 4.4 114 830
Polymers a
G-10 Fiberglass epoxy 1.65 28 —
KaptonÔ (film) 1.43 3.4 210
MylarÔ 1.38 3.8 70
NylonÔ 1.14 3.4 —
TeflonÔ 2.2 0.3 14
a R. Radebaugh et al. (2001), http://www.cryogenics.nist.gov/ and the references listed therein.
b Metals Handbook (1961), Vol. 1, Properties and Selection of Materials, 8th edition, ASM International, Materials Park, Ohio.
c N. J. Simon, E. S. Drexler, and R. P. Reed (1992), Properties of Copper and Copper Alloys at Cryogenic Temperatures, NIST Monograph 177. National Institute of Standards and Technology, U.S. Government Printing Office, Washington, D.C.
d C. C. Clickner, J. W. Ekin, N. Cheggour, C. L. H. Thieme, Y. Qiao, Y.-Y. Xie, and A. Goyal (2006), “Mechanical properties of pure Ni and Ni-alloy substrate materials for Y-Ba-Cu-O coated conductors,” Cryogenics, to be published.
e Values averaged for three different batches of Hastelloy C-276
f www.matweb.com
A7. (i) Specialized resistivity measurement methods (ref. introduction to Part II)
A7.1 Sheet-resistance measurement of unpatterned films
When film properties are initially screened, it is convenient to measure the sheet resistance of the films without the need for patterning. This can be done by a four-probe contact method, wherein four equally spaced in-line probes (such as pogo pins) are pressed against the film near its middle, as illustrated by the insets in Fig. A7.1 for both circular- and rectangular-shaped samples. The usual four-terminal technique is used (described in detail in Sec. 7.2), wherein a small current I from a constant-current supply is passed through the outer two probes, and the voltage V is measured across the inner two probes. The sheet resistance Rs is given by Rs = (V/I) C [/sq], (A7.1) in units of ohms per square, and C is a correction factor given by the curves in Fig. A7.1, valid for a thin film having a thickness d that is much less than the sides or diameter of the chip (A or D in the insets of Fig. A7.1). In the limit of small-probe spacing s << D, C = /ln2 = 4.53. The bulk resistivity of the film is related to the sheet resistance by = Rs d [∙cm]. (A7.2)
FIG. A7.1 Correction factor C used in Eq. (A7.1) for determining the sheet resistance of an unpatterned film using four, equally spaced, in-line probes. Adapted from Smits (1958) and Anner (1990).
This in-line probe technique is also insensitive to in-plane anisotropy between the a-axis and b-axis crystallographic directions, with
Rs = (Rs a-axis ∙ Rs b-axis)0.5.
Other probe/sample configurations are treated in Smits (1958) and also in Wasscher (1961).
References
Anner, G. E. (1990). Planar Processing Primer, p. 585, Van Nostrand Reinhold.
Smits, F. M. (1958), “Measurements of sheet resistivities with four-point probe,” Bell Syst. Tech. J., 37, 711–718.
Wasscher, J. D. (1961). “Note on 4-point resistivity measurements on anisotropic conductors”, Philips Res. Rep. 16, 301–306.
A7.2 van der Pauw method for measuring the resistivity and Hall mobility in flat isotropic samples of arbitrary shape
The van der Pauw method (van der Pauw 1958) is particularly useful for measurements on materials that are not easily fabricated into a long, uniform, bar shapes: the type of configuration that is usually required for common transport measurements. The method works for electrically isotropic samples of arbitrary shape, such as that shown in Fig. A7.2a. All that is required is that they have a uniform thickness and be flat. Thus, this method is well suited to transport measurements of isotropic crystals or brittle materials where it is difficult to cut out bridge-shaped samples without fracturing the narrow arms. (For electrically anisotropic materials, use the Montgomery method, described in the next section, Appendix A7.3.) The sample needs to be deposited or grown flat, or to be capable of being polished flat. Also, it cannot have any holes in it (sorry, a slice of Swiss cheese wouldn’t work). For Hall-effect measurements, the van der Pauw method also has the advantage that clover-shaped samples (such as that shown in Fig. A7.2c) can be used, which give a larger Hall effect for the same amount of heat dissipation compared with the usual bridge-shape samples. This can be a significant advantage for materials with low electron mobility.
FIG. A7.2 (a) Arbitrarily shaped, flat sample with four small contacts at arbitrary places on the periphery, which can be used to measure the sample’s resistivity and the Hall effect. (b) The resistivity measurement is simplified to one resistivity measurement if the sample has a line of symmetry; two of the contacts are situated along the symmetry line and the other two symmetrically with respect to this line. (c) Clover-shaped sample where the influence of the contact size and placement is significantly reduced. For Hall-effect measurements, the clover shape also gives a larger signal for the same amount of heat dissipation, which can improve the measurement sensitivity for materials with low electron mobility. (Adapted from van der Pauw 1958.)
Here, we present a practical description of how to use this method. For details of the derivation, please refer to van der Pauw (1958).
The method consists of attaching small contacts to the sample at its periphery, as illustrated in Figs. A7.2a, b, and c. The diameter of the contact should be small compared with the overall sample size or diameter D. Also, the contacts should be made right at the outer edge of the sample. For disk-shaped samples, the error in resistivity that results from using contacts of size or located a distance away from the sample’s periphery is given approximately by (van der Pauw 1958) ∆/ ≈ (/D)2, (A7.3)
whereas the error in the Hall coefficient RH is given roughly by
∆RH/RH ≈ /D. (A7.4)
The accuracy of the technique is improved if the contacts are spaced apart around the periphery of the sample, as illustrated in Fig. A7.2a. Use of a clover-shaped sample shown in Fig. A7.2c can help minimize the error due to the finite size and placement of the contacts.
After instrumenting the sample in this way, the resistivity of the sample is then determined by measuring two resistances RAB,CD and RBC,DA. Here, RAB,CD is defined as the resistance calculated from the potential difference VD – VC measured between contacts D and C in Fig. A7.2a, divided by the current entering contact A and leaving contact B. The other resistance RBC,DA is correspondingly defined. The sample’s resistivity is then given by (van der Pauw 1958)
= d (2 ln 2)–1 (RAB,CD + RBC,DA) f(RAB,CD/RBC,DA), (A7.5)
where f is a function only of the ratio RAB,CD/RBC,DA. Figure A7.3 gives f as a function of RAB,CD/RBC,DA. Notice from Eq. (A7.5) that there are no measurements of the sample’s shape that enter into the determination of , only the sample’s thickness d.
FIG. A7.3 The function f(RAB,CD/RBC,DA) in Eq. (A7.5) used to determine the resistivity of an arbitrarily shaped flat sample (from Van Der Pauw 1958).
The measurement is particularly straightforward if the sample has a line of symmetry. In this case, contacts A and C can be placed along this symmetry line, and B and D can be placed symmetrically with respect to this line (Fig. A7.2b). Then, from the reciprocity theorem for passive four poles (interchange of current and voltage contacts), we have, generally, that RAB,CD = RBC,DA. Thus, a single measurement of resistance is sufficient.
The van der Pauw method is also well suited for measurements of the Hall coefficient. The Hall coefficient is determined by measuring the change of the resistance RAC,BD before and after a uniform magnetic field B is applied perpendicular to the plane of the sample. In this case, current is applied to an arbitrary contact A and removed from contact C (not contact B as with the resistivity measurement described above). The sample’s Hall coefficient RH is then given by (van der Pauw 1958)
RH = (d/B) ∆RAC,BD, (A7.6)
where ∆RAC,BD is the change in the resistance RAC,BD produced by the magnetic field B.
Footnote on reverse-field reciprocity method: For Hall-coefficient or magnetoresistance measurements, the magnetic field is usually reversed and the resistance data averaged to correct for sample inhomogeneities or voltage-terminal misalignment (in the case of a Hall-bar-geometry). The magnetic-field reversal can take a significant amount of time, especially in the case of high magnetic fields, and the extra time can present a problem for some measurements because of temperature drifts, for example.
For such cases, we call attention to the reverse-field reciprocity method by Sample et al. (1987). This method states that the equivalent of the reverse-field resistance measurement can be made by interchanging voltmeter and current sources, without the need to reverse the magnetic field. This is sometimes quite useful since, with computer-controlled data collection and switching, the second resistance measurement can be performed in hundredths of a second, whereas reversing the applied magnetic field can take several minutes.
References
Sample, H. H., Bruno, W. J., Sample, S. B., and Sichel, E. K. (1987). “Reverse-field reciprocity for conducting specimens in magnetic fields,” J. Appl. Phys. 61, 1079–1084.
Van der Pauw, L. J., (1958). “A method of measuring specific resistivity and Hall effect of discs of arbitrary shape,” Philips Res. Rep. 13, 1–9.
A7.3 Montgomery method for measuring the resistivity of anisotropic materials
The Montgomery method facilitates resistivity measurements of anisotropic crystals, providing an easier experimental technique of determining the various components of resistivity along their principal axes. (For isotropic materials, the van der Paaw method is better suited; see Appendix A7.2.) The Montgomery method is especially useful for anisotropic materials with two independent components of resistivity. In this two-component case, two samples are usually needed if the conventional four-terminal method is used, with each sample fabricated into long-bar shapes cut along the principal axes. However, with the Montgomery method, both components can be determined from one sample. The method is thus extremely useful for the common case where only one sample is available, or if both resistivity components must be measured simultaneously as, for example, when measuring changes in resistivity through a phase transition in a temperature-drift experiment.
Here, my aim is to present a clear, step-by-step description of the procedure for using this method, as well as some practical guidelines on its limitations. The derivation of the method based on the transformation of anisotropic sample coordinates into an equivalent isotropic space is given in detail in Montgomery (1971), Logan et al. (1971), and the background references cited therein.
Crystalline types appropriate for the application of this method are summarized in the Table A7.1 in order of increasing complexity. Here, we denote the various resistivity components of these crystal structures as 1, 2, and 3.
Table A7.1 Application notes for Montgomery method.
Crystal structure Number of independent resistivity components required to characterize the resistivity properties Application comments
Trigonal
Tetragonal
Hexagonal
Two:
2 along the c-axis;
1 = 3, mutually perpendicular to each other and to 2.
1 and 2 can be obtained from a single sample face, which is oriented with one edge along the c-axis and the other in any direction perpendicular to the c-axis.
Orthorhombic
Three:
1, 2, and 3 orthogonal components along the principal crystal axes.
1, 2, and 3 can be obtained from two sample faces with edges along the three principal crystal axes.
Monoclinic
Triclinic
Four to six tensor components:
Three components are sufficient if specially oriented, but the required orientation cannot be determined from the crystal structure alone.
The Montgomery method is usually too cumbersome for these crystal systems.
The insets to Fig. A7.4 show the face of a sample with a typical contact arrangement for applying the Montgomery method. If the resistivities 1 and 2 along sides l1′ and l2′ are expected to be vastly different, it helps to cut the sample face so that l2′/l1′ ≈ (1/2)1/2. This customized shaping avoids an extreme mismatch in the voltages measured along the two directions, which might otherwise make the smaller voltage difficult to measure.
FIG. A7.4 Measured resistance ratio versus the sample-dimension ratio l2/l1. (Unprimed coordinates denote the equivalent isotropic coordinates; see text. The actual physical dimensions of the sample are indicated by primed coordinates l1′ and l2′ as shown in the insets.) (Adapted from Montgomery 1971.)
From a practical standpoint, the Montgomery method is easiest to implement if the sample is relatively thin. Thicker samples can be accommodated, but the method becomes more cumbersome, as described below under the heading “Thick samples.” The simpler thin-sample formulas can be applied to thicker samples, however, if the voltage and current electrodes are extended along the edge of the sample perpendicular to the principal face.
Trigonal, tetragonal, and hexagonal crystal systems
Procedure for determining 1 and 2:
(1) After connecting instrumentation leads to the sample’s principal face (as shown in the insets to Fig. A7.4), measure l1′, l2′, l3′, R1, and R2. The quantities R1 and R2 are defined in the insets to Fig. A7.4. [Primes are used to indicate the actual physical dimensions li′ of the anisotropic crystal; unprimed quantities denote the transformed sample dimensions in equivalent isotropic space. We have used this convention to keep the same notation as in Montgomery’s original equations (Montgomery 1971). The figures shown in the first part of Montgomery’s article are for the equivalent isotropic case, and then, later in the article, the real sample dimensions are transformed back to the isotropic case, which can be a bit confusing. The reason this works is that the quantities R1 and R2 are the same in either real or equivalent space; that is, R1 = V1/I1 = V1′/I1′ and R2 = V2/I2 = V2′/I2′. The figure insets shown here have been modified from Montgomery’s article to help clarify the practical application of this method.]
(2) Determine the ratio l2/l1 in the equivalent isotropic space from the experimentally measured ratio R2/R1, by using the curves in Fig. A7.4.
(3) From the value determined for l1/l2 (equivalent space) and the value measured for l1′/l2′ (the actual dimensions of the anisotropic crystal), calculate
(2/1)1/2 = (l2/l1) (l1′/l2′). (A7.7)
(4) From the ratio l1/l2 (equivalent space), determine the dimensionless quantity H from the curve in Fig. A7.5.
Thin samples:
(5) For thin samples [that is, l3/(l1l2)1/2 <~ 0.5, where l3 is the thickness of the sample in equivalent space determined from Eq. (A7.9) below], calculate (12)1/2 = H l3′ R1, (A7.8) (where l3′ is the thickness of the sample in real space). (6) Finally, from the quantitative values for (2/1)1/2 and (12)1/2 [Eqs. (A7.7) and (A7.8)], calculate 1 and 2. FIG. A7.5 The quantity H [used in Eq. (A7.8)] versus the sample-dimension ratio (in equivalent isotropic coordinates). (Adapted from Montgomery 1971.) Thick samples: Steps 1 through 4 are the same as above. (5) For thick samples [ l3/(l1l2)1/2 >~ 0.5], instead of step 5 above, calculate
l3/(l1l2)1/2 = (1/2)1/4 l3′/(l1′l2′)1/2 , (A7.9)
where the numerical value of (1/2)1/4 is calculated from the value of (2/1)1/2 determined in step 3, and l3′/(l1′l2′)1/2 is calculated from the sample dimensions measured in step 1. [In this step we have taken 3 and 1 to be the equivalent resistivity directions (with 3 defined as the resistivity perpendicular to the sample face); if, on the other hand, 3 = 2 , interchange subscripts 1 and 2 in Eq. (A7.9) above and in Eq. (A7.10) below.]
(6) Use the value of l3/(l1l2)1/2 from Eq. (A7.9) to determine E/(l1l2)1/2 from Fig. A7.6 (where E is defined as the effective sample thickness in equivalent isotropic space).
(7) Calculate E′ (the effective sample thickness in anisotropic real space) from
E′/(l1′l2′)1/2 = (2/1)1/4 E/(l1l2)1/2, (A7.10)
where the value of 2/1 was determined in step 3.
(8) Calculate
(12)1/2 = H E′ R1, (A7.11)
(9) Finally, from the quantitative values for (2/1)1/2 and (12)1/2 [Eqs. (A7.7) and (A7.11)], calculate 1 and 2.
FIG. A7.6 Normalized effective thickness E vs. normalized sample thickness l3 for various ratios of the sample dimensions l2/l1 (in equivalent isotropic coordinates). (The ratio l2/l1 was determined from Fig. A7.4 in Step 2 above.) (From Montgomery 1971.)
The numerical limit given in step (5) [i.e., l3/(l1l2)1/2 >~ 0.5] for the thin and thicker cases is my subjective estimate for the validity of these two approximations. That is, the thin-sample case requires that E ≈ l3, which from Fig. A7.6 would appear to be valid for l3/(l1l2)1/2 < ~ 0.5 [or in real space, (1/2)1/4 l3′/(l1′l2′)1/2 < ~ 0.5 from Eq. (A7.9)].
Practical Note: To simplify the application of the method when many calculations need to be made for a single sample (such as in a temperature-drift experiment), calculate the multiplying factors to convert R1 to 1 and 2 for several ratios of R1/R2, and plot these versus R1/R2 for easier reduction of the data. Nearly straight-line relations are usually obtained on a log–log plot.
Orthorhombic crystal systems
Procedure for determining 1, 2, and 3:
With three unknown components of resistivity, the Montgomery method requires measurements on a second sample with its face cut normal to that of the first sample and also cut thin enough that the thin-sample condition is met [i.e., l3/(l1l2)1/2 <~ 0.5 as described above for the two-component case]. Then, the procedure for evaluating the data for the second sample face is the same as that for the thin-sample case given above, with corresponding new measured values assigned to l1′, l2′, and l3′. This works because all the equations are independent of 3 for the thin-sample case. For thicker orthorhombic crystals [l3/(l1l2)1/2 >~ 0.5], there appears to be no straightforward procedure for applying the Montgomery method unless one assumes a value for 3 and uses an iterative approach (Montgomery 1971). Again, the best procedure in this case would be to mimic the thin-sample case by extending the electrode contacts along the edge of the sample perpendicular to the principal face.
References
Logan, B. F., Rice, S. O., and Wick, R. F. (1971). “Series for computing current flow in a rectangular block,” J. Appl. Phys. 42, 2975–2980.
Montgomery, H. C. (1971). “Method for measuring electrical resistivity of anisotropic materials,” J. Appl. Phys. 42, 2971–2975.
(ii) Sample-holder material properties (ref. Chapter 7)
A7.4 Sample-holder materials: Thermal contraction on cooling to liquid-helium and liquid-nitrogen temperatures (Sec. 7.3.2)
The total linear contraction from room temperature to the indicated temperature T is defined as
L/L293K–T (L293K – LT)/L293K.
Thecoefficient of linear expansion at room temperature is defined as
293K (1/L) dL/dT.
Since the thermal expansion/contraction is approximately linear above room temperature, the total contraction from an upper reference temperature Tu above room temperature (such as soldering temperature) to a low temperature T can be determined approximately from
L/LTu–T = L/L293K–T + (293K) (Tu – 293 K).
Tabulated values are generally arranged within each material group by the magnitude of the thermal contraction to facilitate finding sample-holder materials with a thermal contraction which matches that of a given sample.
Additional thermal-expansion data for other materials and temperatures are tabulated in Appendix A6.4.
Thermal contraction of sample-holder materials
Material L/L293K–4K
[%] L/L293K–77K
[%] 293K
[10–6 K–1]
Metals
Niobium 0.143 c 0.130 m 7.1 s
Titanium c 0.151 0.143 m 8.5
Iron c 0.198 0.190 m 11.5
Nickel c 0.224 0.212 m 12.5
Copper b 0.324 0.302 m 16.7
Silver q 0.412 0.370 c 18.5
Aluminum c 0.415 0.393 m 22.5
Alloys
Fe–36Ni f,m ~0.037 0.038 3.0
Fe–9Ni d 0.195 0.188 11.5
Ti–6%Al-4%V d 0.173 0.163 m 8.0
Ti–5%Al–2.5%Sn n 0.20 0.17 8.3
Hastelloy C t 0.218 0.216 10.9 u
Inconel 718 f 0.24 0.22 13.0
Monel, S (67Ni–30Cu) d 0.25 0.24 m 14.5
SS 304 r 0.29 0.28 b 15.1
SS 304L d 0.31 0.28 15.5
SS 316 r 0.30 0.28 b 15.2
Cu–2%Be (UNS C17200
-TH 04) c 0.31 0.30 18.1
Brass 70/30 d 0.37 0.34 m 17.5
Bronze (Cu–5wt%Sn) n 0.33 0.29 15.0
Bronze (Cu–10wt%Sn) n 0.38 0.35 18.2
Bronze (Cu–13.5wt%Sn) k 0.40 0.36 18.8
Aluminum 2024-T86 f 0.396 0.374 21.5
Aluminum 7045-T73 f 0.419 0.394 23.5
Soft-Solder 50/50 c 0.514 0.480 m 25.5
Insulators
Pyrex j 0.055 0.054 3.0
G-11 (warp) a 0.21 0.19 11
G-11 (normal) a 0.62 0.55 37
G-10CR (warp direction) a 0.24 0.21 m 12.5
G-10CR (normal direction) a 0.71 0.64 m 41
Phenolic, Cotton (warp) f 0.26 0.24 m 15
Phenolic, Cotton (normal) f 0.73 0.64 m 42
Stycast 2850 FT s 0.44 0.40 28
CTFE c 1.14 0.97 m 67
Epoxy g 1.16 1.03 m 66
Plexiglas j 1.22 1.06 m 75
Nylon c 1.39 1.26 m 80
TFE (Teflon) i 2.14 1.94 m 250
a A. F. Clark, G. Fujii, and M.A. Ranney (1981), IEEE Trans. Magn. MAG-17, 2316–2319.
b T. A. Hahn (1970), J. Appl. Phys. 41, 5096–5101.
c R. J. Corruccini and J. J. Gniewek (1961), Thermal Expansion of Technical Solids at Low Temperatures. Monograph 29, National Bureau of Standards, U.S. Government Printing Office, Washington, D.C.
d V. Arp, J. H. Wilson, L. Winrich, and P. Sikora, P. (1962), Cryogenics 2, 230–235.
f A. F. Clark (1968), Cryogenics 8, 282–289.
g K. Dahlerup–Peterson and A. Perrot (1979). Properties of Organic Composite Materials at Cryogenic Temperatures. ISR-BOM/79-39, CERN, Geneva, Switzerland.
i R. K. Kirby (1956), J. Res. Natl. Bur. Stand. 57, 91–94.
j H. L. Laquer and E. L. Head (1952), Low Temperature Thermal Expansion of Plastics. AECU-2161, Technical Information Service Atomic Energy Commission., Oak Ridge, Tennessee.
k G. Rupp (1980), in Multifilamentary 15 Superconductors, eds. M. Suenaga and A. F. Clark, pp. 155–170, Plenum Press.
l D. S. Easton, D. M. Kroeger, W. Specking, and C. C.Koch, (1980), J. Appl. Phys. 51, 2748–2757.
m A. F. Clark (1983), Chapter 3, in Materials at Low Temperatures, pp. 96–97, ASM International, Materials Park, Ohio.
n Handbook on Materials for Superconducting Machinery (1977), MCIC-HB-04. Battelle, Columbus, Ohio.
o Compilation by A. F. Clark (1983), Chapter 3 in Materials at Low Temperatures, pp. 96–97, ASM International, Materials Park, Ohio.
p N. J. Simon, E. S. Drexler, and R.P. Reed (1992), Properties of Copper and Copper Alloys at Cryogenic Temperatures, U.S. Government Printing Office, Washington, D.C.; N. Cheggour and D. P. Hampshire (2000), Rev. Sci. Instrum. 71, 4521–4529.
q V. J. Johnson, ed, (1961), Properties of Materials at Low Temperature, Phase 1, U.S. Air Force.
r Handbook on Materials for Superconducting Machinery (1974, 1976), National Bureau of Standards, U.S. Government Printing Office, Washington, D.C.
s C. A. Swenson (1997), Rev. Sci. Instrum. 68, 1312–1315.
t Y. S. Touloukian (1975), Thermal Expansion, 12, 1248.
u R. Radebaugh et al. (2001), http://www.cryogenics.nist.gov/ and the references listed therein.
A7.5 Superconductor materials: Thermal contraction on cooling to liquid-helium and liquid-nitrogen temperatures (Sec. 7.3.2)
The total linear contraction from room temperature to the indicated temperature T is defined as
L/L293K–T (L293K – LT)/L293K.
Thecoefficient of linear expansion at room temperature is defined as
293K (1/L) dL/dT.
Since the thermal expansion/contraction is approximately linear above room temperature, the total contraction from an upper reference temperature Tu above room temperature (such as soldering temperature) to a low temperature T can be determined approximately from
L/LTu–T = L/L293K–T + (293K) (Tu – 293 K).
Tabulated values are generally arranged within each material group by the magnitude of the thermal contraction.
Thermal contraction of superconductor materials
Material L/L293K–4K
[%] L/L293K–77K
[%] 293K
[10–6 K–1]
High-Tc Superconductors
YBCO polycrystal i 0.23 0.21 11.5
YBCO a-axis i 0.13 0.12 7.4
YBCO b-axis i 0.18 0.16 9.6
YBCO c-axis i 0.38 0.34 17.7
YBCO a,b-plane avg. i 0.16 0.14 8.5
Bi-2212
a,b-axes o 0.15 0.14 8.3
c-axis o 0.30 0.27 15.1
Ag u 0.41 0.37 18.5
Bi-2223/61%Ag alloy a,b 0.24
Bi-2223 a,b-axes o,p 0.22
Ag q 0.41 0.37 18.5
(Bi-2223) / 75vol%Ag wire l
1st cool-down
2nd cool-down (difference due
to Ag yielding)
0.22
0.29
0.23
0.30
16
13
Tl-2223
Low-Tc Superconductors
(N–67wt%Ti)/64vol%Cu wire c 0.26 0.25 j 12.5
Nb–67wt%Ti 0.13 j,h 5.8 j
Nb–45wt%Ti 0.19 h 0.17 j 8.2 j
Nb3Sn wire (10vol%Nb3Sn) 0.30 g,m 0.28 g,m
Nb3Sn wire (20vol%Nb3Sn 0.28 g,m 0.26 g,m
Nb3Sn ~0.16 j 0.14 j 7.6 d
Bronze (Cu–13.5wt%Sn) f 0.40 0.36 18.8
Cu 0.32 k 0.30 j 16.7 e,d
Nb 0.14 e 0.13 j 7.3 r
Ta 0.14 e 0.13 e 6.3 e,d
a J. P. Voccio, O. O. Ige, S. J. Young, and C. C. Duchaine (2001), IEEE Trans. Appl. Supercond. 11, 3070–3073.
b E. Harley (2004), American Superconductor Corp., personal communication.
c A. F. Clark (1968), Cryogenics 8, 282–289.
d D. S. Easton, D. M. Kroeger, W. Specking, and C. C. Koch, (1980), J. Appl. Phys. 51, 2748.
e R. J. Corruccini, and J. J. Gniewek (1961) Thermal Expansion of Technical Solids at Low Temperatures. Monograph 29, National Bureau of Standards, U.S. Government Printing Office, Washington, D.C.
f G. Rupp (1980), pp. 155–170 in Multifilamentary 15 Superconductors, eds. M. Suenaga and A. F. Clark), Plenum Press, New York.
g L. F. Goodrich, S. L. Bray, and T. C. Stauffer (1990), Adv. Cryog. Eng. (Mater.) 36A, 117–124.
h F. J. Jelinek and E. W. Collings (1975), “Low-temperature thermal expansion and specific heat properties of structural materials,” in Materials Research in Support of Superconducting Machinery—IV, eds. A. F. Clark, R. P. Reed, and E. C. Van Reuth. Fourth Semi-Annual Technical Report, National Bureau of Standards, U.S. Government Printing Office, Washington, D.C.
i Calculated from data by H. You, J. D. Axe, X. B. Kan, S. Hashimoto, S. C. Moss, J. Z. Liu, G. W. Crabtree, and D. J. Lam (1988), Phys. Rev. B38, 9213–9216.
j Compilation by A. F. Clark (1983), Chapter 3 in Materials at Low Temperatures, ASM International, Materials Park, Ohio.
k T. A. Hahn (1970), J. Appl. Phys. 41, 5096–5101.
l N. Yamada, K. Nara, M. Okaji, T. Hikata, T. Kaneko, N. Sadakata (1998), Cryogenics 38, 397–399.
m K. Tachikawa, K. Itoh, H. Wada, D. Gould, H. Jones, C. R. Walters, L. F. Goodrich, J. W. Ekin, and S. L. Bray (1989), IEEE Trans. Magn. 25, 2368-2374.
o M. Okaji, K. Nara, H. Kato, K. Michishita, and Y. Kubo (1994), Cryogenics 34, 163–165.
p S. Ochaia, K. Hayashi, and K. Osamura (1991), Cryogenics 31, 954–961.
q R. J. Corruccini and J. J. Gniewek (1961), Thermal Expansion of Technical Solids at Low Temperatures, National Bureau of Standards Monograph 29, U.S. Government Printing Office, Washington, D.C.
r CRC Handbook of Chemistry and Physics (2001), 82nd edition, CRC Press, Boca Raton, Florida.
A7.6 Thin-film substrate materials: Thermal conductivity and thermal contraction (Sec. 7.4.1 and Sec. 7.4.2)
Material Thermal Conductivity Thermal Expansion
(4K)
[W/(m·K)]
(77K)
[W/(m·K)]
(295K)
[W/(m·K)] L/L
(293K–4K)
[%] L/L
(293K–77K)
[%]
(293K)
[10–6 K–1]
Al N (|| a-axis) j
(|| c-axis) j — — — — 0.032
0.025 3.7
3.0
Sapphire (Al2O3) l (|| c-axis) 451 e 10300 e 0.079 0.078 5.4 j
Beryllia ~1 i ~1000 i
C (diamond) 0.024 a 0.024 a 1.0 a
LaAlO3 12.6 b
MgO 82 d 507 d 61 d 0.139 a 0.137 a 10.2 a
NdGaO3 7.8 b
Ni 90.7 f 0.224 c 0.212 c 13.4 g
Quartz (|| optic axis) 420 32 0.104 a 7.5 a
Si 124 m 0.022 a 0.023 a 2.32 a
α-SiC (: crystal, ┴ to c-axis)
(L/L: polycrystal avg.) 27 k 4000 k 510 k – 0.030 j 3.7 j
SrTiO3 60 f 11.1 b
Y-stabilized Zirconia (YSZ) 10.3 b
Cu (OFHC) (for reference) 630 h 544 h 397 h 0.324 c 0.302 c 16.7 c
a R. J. Corruccini and J. J. Gniewek (1961), Thermal Expansion of Technical Solids at Low Temperatures, National Bureau of Standards Monograph 29, U.S. Government Printing Office, Washington, D.C.
b Shinkosha Co., Ltd. Tokyo, Japan
c A. F. Clark (1983), Chapter 3 in Materials At Low Temperatures, ASM International, Materials Park, Ohio.
d Y.S. Touloukian and E. H. Buyco, E.H. (1970), Specific Heat, Vols. 1 and 2, Plenum Press, New York.
e R. Radebaugh et al. (2003), http://www.cryogenics.nist.gov/ and the references listed therein.
f Compiled by M. Paranthaman, Oak Ridge National Laboratory, Oak Ridge, Tennessee; and from J. Evetts, University of Cambridge, UK
g CRC Handbook of Chemistry and Physics (2001), 82nd edition, CRC Press, Boca Raton, Florida.
h Cryogenic Materials Properties Program CD, Release B-01 (June 2001), Cryogenic Information Center, 5445 Conestoga Ct., Ste. 2C, Boulder, CO 80301-2724, Ph. (303) 442-0425, Fax (303) 443-1821
i Lake Shore Cryotronics, Inc. (2002), Temperature Measurement and Control , Westerville, Ohio.
j Y. S. Touloukian (1977), Thermal Expansion: nonmetallic solids, Vol. 13, IFI/Plenum, New York.
k Y. S. Touloukian (1970), Thermal Conductivity: metallic elements and alloys, Vol. 2, IFI/Plenum, New York.
l For the thermal conductivity data, the heat flow is 60 degrees away from the hexagonal axis; values are thought to be accurate to within 10 % to 15 % at temperatures above 60 K, but highly sensitive to small physical and chemical variations below 60 K. Thermal linear-expansion data are parallel to the c-axis.
m CRC Handbook of Chemistry and Physics (2002), 83st edition, CRC Press, Boca Raton, Florida.
A7.7 Ultrasonic wire-bond material combinations (Sec. 7.4.3)
Wire-bond lead
Substrate Metal Film Material Diameter or
Thickness Range
[mm]
Glass Aluminum
Aluminum
Nickel
Nickel
Copper
Gold
Gold
Tantalum
Chromel
Chromel
Nichrome
Platinum
Gold–Platinum
Palladium
Silver
Copper on silver Aluminum wire
Gold wire
Aluminum wire
Gold wire
Aluminum wire
Aluminum wire
Gold wire
Aluminum wire
Aluminum wire
Gold wire
Aluminum wire
Aluminum wire
Aluminum wire
Aluminum wire
Aluminum wire
Copper ribbon 0.05 to 0.25
0.08
0.05 to 0.5
0.05 to 0.25
0.05 to 0.25
0.05 to 0.25
0.08
0.05 to 0.5
0.05 to 0.25
0.08
0.06 to 0.5
0.25
0.25
0.25
0.25
0.7
Alumina Molybdenum
Gold–platinum
Gold on molybdenum–lithium
Copper
Silver on molybdenum–manganese Aluminum ribbon
Aluminum wire
Nickel ribbon
Nickel ribbon
Nickel ribbon 0.08 to 0.13
0.25
0.05
0.05
0.05
Silicon Aluminum
Aluminum Aluminum wire
Gold wire 0.25 to 0.5
0.05
Quartz Silver Aluminum wire 0.25
Ceramic Silver Aluminum wire 0.25
From Welding Handbook (1991), 8th edition, Vol. 2, Chapter 25, pp. 784–812, American Welding Society, Miami, Florida; G. G. Harmon (1997), Wire Bonding in Microelectronics: Materials, Processes, Reliability, and Yield, p. 7, McGraw-Hill.
A8. Sample contacts (ref. Chapter 8)
A8.1 Overview of contacts for low-Tc and high-Tc superconductors. (Secs. 8.3 and 8.4)
Contact Type Specific Contact Resistivity c *
[∙cm2] Common Usage/Comments
Low-Tc Superconductors (copper sheathed)
Cu / 63Sn–37Pb / Cu a
4 10–9 Copper-to-copper joint soldered under light pressure
High-Tc Bi–Sr–Ca–Cu–O oxide superconductors
Silver sheath/BSCCO interface d <<10–8 Copper connections to the silver sheath can be soldered with standard eutectic Pb–Sn solder and have c values comparable to those of the copper-to-copper joints listed above.
High-Tc Y–Ba–Cu–O oxide superconductors
Silver or gold deposited on YBCO:
In-situ c deposited; no anneal b 10–9 to 10–7 Contacts to oxide superconductor films, typically for electronic applications
Ex-situ c deposited; oxygen annealed e 10–9 to 10–6 Contacts for high current applications, including “coated conductors”
Ex-situ deposited; no oxygen anneal f 10–5 to 10–2 Applications where oxygen annealing is precluded by other sensitive materials or processing steps. The lowest values of c obtained when the superconductor surface is ion milled or sputter etched just prior to contact deposition.
Soldered Y–Ba–Cu–O oxide superconductors
Indium-solder connections to silver or gold pads deposited on YBCO 10–1 to 10–6 High-current coated-conductor applications. The lowest values of c are obtained when the gold or silver pad thickness is at least 7 µm to 10 µm thick (see the subtopic on soldering in Sec. 8.3.3)
Indium solder applied directly on YBCO g 10–2 to 10–1 Soldered voltage contacts for bulk oxide superconductors
* For low-Tc superconductors, the contact resistivity was measured at 4.2 K. For high-Tc superconductors, the contact resistivity does not appreciably change with temperature below Tc and thus applies to the entire temperature range from liquid-helium to liquid-nitrogen temperatures.
a L. F. Goodrich and J. W. Ekin (1981), IEEE Trans. Magn. 17, 69–72.
b M. Lee, D. Lew, C–B. Eom, T. H. Geballe, and M. R. Beasley (1990), Appl. Phys. Lett. 57, 1152–1154.
c “Ex-situ” and “in-situ” contacts refer to whether the superconductor surface is exposed to air before the noble-metal contact pad is deposited, as described in Sec. 8.4.2 on superconductor-film contact techniques.
d Y. S. Cha, M. T. Lanagan, K. E. Gray, V. Z. Jankus, and Y. Fang (1994), Appl. Supercond. 2, 47–59.
e J. W. Ekin, T. M. Larson, N. F. Bergren, A. J. Nelson, A. B. Swartzlander, L. L. Kazmerski, A. J. Panson, and B. A. Blankenship, B. A. (1988), Appl. Phys Lett. 52, 1819–1821.
f J. W. Ekin, A. J. Panson, and B. A. Blankenship (1988), Appl. Phys. Lett. 52, 331–333.
g J. W. Ekin, unpublished data, National Institute of Standards and Technology, Boulder, Colorado.
A8.2 Contact methods for voltage and current connections to bare YBCO superconductors. (Secs. 8.3.1, 8.3.2, 8.3.3, and 8.4.2)
Contact methods are ordered within each category by the magnitude of contact resistivity c, with the best (lowest c) listed first.
Any of the current contact methods can also be used for voltage contacts, but they are more complex to fabricate than the simple techniques listed for voltage contacts.
Contact materials that do not work well with the oxide superconductors are included at the end of the table for pedagogical reasons (described in Sec. 8.3.3).
Contact Method Procedure c a
[∙cm2] Comments
Voltage contacts
In–3wt.%Ag solder
In–48wt.%Sn solder For these solders to wet YBCO surfaces, lightly scratch the sample surface under the molten solder with the soldering-iron tip, or use an ultrasonic soldering iron; see “Soldered…” and “Wetting…” in Sec. 8.3.2. 10–2–10–1
“ Tmelt = 143 oC; eutectic
Tmelt = 118 oC; eutectic; beware that Sn dissolves thin silver or gold films
Spring contacts Beryllium copper or other conducting spring stock is used to contact the sample; see “Pressure contacts” in Sec. 8.3.2. Silver or gold pads deposited on the test sample lower the contact resistivity
Silver paint 10–1–100 Weak connection, but sometimes needed for delicate samples; c can be improved by oxygen annealing; solvent carrier in paint can damage thin films
Current contacts
In-situ gold or silver pad deposited on superconductor, no oxygen anneal
Descriptions of in-situ vs. ex-situ deposition techniques are given in Sec. 8.4.2.
10–9–10–7 Produces the lowest c
In-situ contacts are mainly amenable to HTS film (not bulk) fabrication techniques
Gold is more expensive than silver contact pads, but does not tarnish as readily
Ex-situ gold or silver pad deposited on superconductor, with oxygen annealing See Sec. 8.4.2 and “Fabrication” in Sec. 8.3.3. 10–9–10–6
Ex-situ gold or silver pad deposited on superconductor, no oxygen annealing See Sec. 8.4.2 and “Fabrication” in Sec. 8.3.3. 10–6 –10–2 Used for applications where oxygen annealing is not possible, or where very low c is not needed
c depends on how well the surface is cleaned
Indium-solder connection to silver or gold pad Make silver or gold pad thickness at least 7 m to 10 m. See “Soldering to noble-metal contact pads” in Sec. 8.3.3. 10–1–10–6 Used for connecting high-current bus bars or wires to the sample-contact pads
c depends strongly on the soldering technique used (see text)
Failures
Copper pad deposited on superconductor Sputter deposited 10–2 c no better than indium solder, and a lot more complex to fabricate.
Au–Cr pad deposited on superconductor Sputter deposited 10–1 Contact commonly used for semiconductors, but terrible for superconductors.
Pb–Sn solder no bond
a The contact resistivity does not appreciably change with temperature below Tc, so the same approximate c values apply at liquid-helium and liquid-nitrogen temperature.
A8.3 Optimum oxygen-annealing conditions for silver and gold contacts to Y-, Bi-, and Tl-based high-Tc superconductors. (Sec. 8.3.3 and 8.4.2)
Annealing times are about 30 min to 60 min (at full temperature) for contacts to bulk superconductors, 30 min or less for thin-film superconductors.
Contacts to Bulk High-Tc Superconductors Annealing Temperature
Ag/YBCO a
500 oC in O2
Au/YBCO a
600 oC in O2
Ag/BiPbSrCaCuO
~400 oC in O2
Ag/TlCaBaCuO
500 oC in O2
Contacts to Film YBCO Superconductors Annealing Temperature
Ag(<1µm)/YBCO film b
400 oC in O2
Au(<1µm)/YBCO film c
450 oC to 500 oC in O2
Annealing is carried out in oxygen at atmospheric pressure, flowing at a rate of about 2 × 10–6 m3/s (~0.3 scfh, standard cubic feet per hour) by using a furnace such as that shown in Fig. 8.9.
For YBCO, the contacts were cooled in oxygen by ramping the furnace temperature down at a slow rate, ~2.5 oC/min for the bulk sintered superconductors used in these tests; rates for thin films should be kept below 50 oC/min to allow time for the crystal structure to take up oxygen as it cools and to minimize oxygen disorder. (Further information is given in B. H. Moeckly, D. K. Lathrop, and R. A. Buhrman, (1993), Phys. Rev. B47, 400–417.)
For silver-contact pads on films, the silver pad will “ball up” at oxygen annealing temperatures higher than about 400 oC if the pad is thin (>1 µm), the optimum annealing temperature can be slightly higher (A. Roshko, R. H. Ono, J. Beall, J. A. Moreland, A. J. Nelson, and S. E. Asher. (1991), IEEE Trans. Magn. 27, 1616–1618).
a J. W. Ekin, T. M. Larson, N. F. Bergren, A. J. Nelson, A. B. Swartzlander, L. L. Kazmerski, A. J. Panson, and B. A. Blankenship (1988), Appl. Phys Lett. 52, 1819–1821.
b J. W. Ekin, C. C. Clickner, S. E. Russek, and S. C. Sanders (1995), IEEE Trans. Appl. Supercond. 5, 2400–2403.
c Y. Xu, J. W. Ekin, C. C Clickner, and R. L. Fiske (1998), Adv. Cryog. Eng. (Mater.) 44, 381–388.
A8.4 Bulk resistivity of common solders, contact-pad materials, and matrix materials (Sec. 8.5.2)
The bulk resistivity values listed below are useful for estimating the effective contact resistivity in conjunction with Eq. (8.5).
Additional properties of solders are tabulated in Appendix A3.7.
Material 4K
[cm] 77K
[cm] 295K
[cm]
Solder (compositions in wt%)
52In–48Sn a (eutectic) (Tmelt=118 oC) SC 18.8
97In–3Ag a (eutectic) (Tmelt=143 oC) 0.02 1.8 9.7
90In–10Ag a 0.03 1.8 9.1
Indium a (Tmelt=157 oC) 0.002 1.6 8.8
63Sn–37Pb a (standard eutectic soft-solder) (Tmelt=183 oC) SC 3.0 15
91Sn–9Zn a (Tmelt=199 oC) 0.07 2.3 12.2
Contact Pad Material
Silver (pure: evaporated, sputtered, or plasma-sprayed) variable 0.27 1.6
Gold (pure: evaporated or sputtered) variable 0.43 2.2
Low-Tc Superconductor Matrix Materials
Copper variable
(~0.017 typical) ~0.2 1.7
Bronze (Cu–13wt%Sn) ~2 ~2
High-Tc Superconductor Matrix Materials
Silver variable 0.27 1.6
Silver dispersion strengthened with 1at%Mn b 2.2 2.7 4.0
Silver dispersion strengthened with 2at%Mn b ~4.1 ~4.6 ~6.0
SC ≡ Superconducting (see Appendix A3.9).
a C. Clickner (1999), unpublished data, National Institute of Standards and Technology, Boulder, Colorado.
b M. Putti, C. Ferdeghini, G. Grasso, A. Manca, and W. Goldacker, (2000), Physica C 341-348, 2585-2586.
A8.5a Argon ion milling rates of elements* (Sec. 8.4.2)
Values of argon ion milling rates of elements are tabulated in nm/min, at a current density of 1 mA/cm2, and incident argon ion energies of 200 eV and 500 eV.
Rates at other energies can be estimated from Fig. 8.11. An example is given in Sec. 8.4.2 under Cleaning etch.
Element 200 eV
[nm/mm] 500 eV
[nm/min]
Ag 100 220
Al 29 73
Au 71 170
Be 5.2 17
C 1.3 4.4
Co 26 55
Cr 33 58
Cu 53 110
Dy 58.0 110
Er — 98
Fe 26 53
Gd 55.0 110
Ge 49 100
Hf 31 66
Ir 26 60
Mo 24 54
Nb 18 44
Ni 31 66
Os 20 51
Pd 60 130
Pt 39 88
Re 23 52
Rh 31 74
Ru 24 61
Si 16 38
Sm 51.0 110
Sn 85.0 180
Ta 20 42
Th 41 82
Ti 16 38
U 34 74
V 17 37
W 18 38
Y 45 96
Zr 27 62
* From a compilation by H. R. Kaufman and R. S. Robinson (1987), Operation of Broad-Beam Sources. Commonwealth Scientific Corp., Alexandria, Virginia, from data by G. K. Wehner et al. (1962), General Mills Report 2309, General Mills Electronic Division, Minneapolis, Minnesota, published by P. R. Puckett, S. L. Michel, and W. E. Hughes, (1991), p. 760 in Thin Film Processes II, eds. J. O. Vossen and W. Kern, Academic Press, Boston.
A8.5b Argon ion milling rates of compounds* (Sec. 8.4.2)
Values of argon ion milling rates of compounds are tabulated in nm/min, at a current density of 1 mA/cm2, and incident argon ion energies of 200 eV and 500 eV.
Rates at other energies can be estimated from Fig. 8.11. An example is given in Sec. 8.4.2 under Cleaning etch.
Compound 200 eV
[nm/min] 500 eV
[nm/min]
CdS a 110 230
GaAs (110) a 78 160
GaP (111) a 69 160
GaSb (111) b 90 190
InSb a 76 150
LiNbO3 (Y-cut) c — 40
MgO e — 16
Mo2C a — 29
PbTe a 160 380
SiC (0001) a — 35
SiO2 c — 40
YBa2Cu3O7 e — 45
* From a compilation by H. R. Kaufman and R. S. Robinson (1987), Operation of Broad-Beam Sources. Commonwealth Scientific Corp., Alexandria, Virginia, published by P. R. Puckett, S. L. Michel, and W. E. Hughes (1991), p. 760 in Thin Film Processes II, eds. J. O. Vossen and W. Kern, Academic Press, Boston; and from J. W. Ekin (1992), unpublished data, National Institute of Standards and Technology, Boulder, Colorado.
a J. Comas, J. and C. B. Cooper (1966), J. Appl. Phys. 37, 2820–2822.
b S. P. Wolsky, D. Shooter, and E. J. Zdanuk (1962), pp. 164–168 in Transactions of the 9th National Vacuum Symposium, Pergamon Press, New York
c H. L. Garvin (1971), Bull. Am. Phys. Soc., Ser. II 16, 836.
d J. L. Vossen and E. B. Davidson (1972), J. Electrochem. Soc. 119, 1708–1714.
e J. W. Ekin (1990), unpublished data, National Institute of Standards and Technology, Boulder, Colorado.
A10. Critical-current analysis parameters (ref. Chapter 10)
A10.1 Effective critical temperature Tc*(B) (Sec. 10.4.4)
Values of the effective critical temperature Tc* as a function of magnetic field B are tabulated below. They were determined by linearly extrapolating the Ic(T,B) curves of Figs. 10.17, 10.18, 10.19, and 10.21 to zero current.
The value of Tc*(B) at a given magnetic field B is useful when the critical current Ic(T1) is known at one temperature T1 and we wish to determine its approximate value Ic(T) at an arbitrary temperature T. For this purpose, a linear approximation of the Ic(T) characteristic works fairly well for a number of conductors [Eq. (10.14) in Sec. 10.4.4]:
Ic(T)/Ic(T1) = [Tc*–T]/[Tc*–T1] .
This linear relationship between the critical current and temperature usually breaks down at high temperatures (within about 10 % of Tc*) because of inhomogeneities in the superconductor, but the relationship is useful over most of the practical temperature range leading up to Tc* (ref. Figs. 10.17, 10.18, 10.19, and 10.21).
For materials that cannot be modeled with a linear relationship, the temperature-transformation method described in Sec. 10.6.3 is much more general in nature and quite accurate for nearby transformations. A summary of the method is also given in Appendix A10.2b below under the subsection entitled Temperature dependence of the critical current.
Values of Tc*(B) at Various Magnetic Fields for Selected Superconductors
Magnetic Field B
[T] Nb–Ti a
First number valid near Tc*; the second for the range 4.0 K to 4.5 K
[K] Nb3Sn b
[K] V3Ga d
[K] YBCO e
[K]
0 9.2 12.4 87
0.3 9.0 11.0
1 8.66 9.78
2 8.25 9.31 15.0
3 7.89 8.77
4 7.40 8.20 13.7 11.8
5 7.07 7.54 13.1 11.4
6 6.52 6.85 12.6
7 6.14 6.16
8 5.53 5.53 11.5 10.8
9 5.16 5.01
10 4.63 4.63 10.4 10.3
12 9.5
a For Nb–Ti, the first value of Tc* is for use over the high-temperature regime where T approaches Tc*. It is extrapolated from the slope of the data where Ic approaches zero. The second value of Tc* is for use over the liquid-helium temperature range. It is extrapolated from data only between 4.0 K and 4.5 K. Note that because of curvature in the Ic vs. T plot at low magnetic fields, the second value of Tc* (for use in the liquid-helium temperature range) can be significantly greater than the first (nominal) value. Data from L. F. Goodrich and T. C. Stauffer (2004), Adv. in Cryog. Eng. (Mater.) 50B, 338–345 (the source data are plotted in Fig. 10.17 of this textbook).
b Extrapolated from data by L. F. Goodrich, L. T. Medina, and T. C. Stauffer (1998), Adv. Cryog. Eng. (Mater.) 44, 873–880 (straight-line extrapolations are shown in Fig. 10.18 of this textbook).
c Extrapolated from data by S. A. Keys, N. Koizumi, and D. P. Hampshire (2002), Supercond. Sci. Technol. 15, 991-1010.
d Extrapolated from data by Y. Iwasa and B. Montgomery (1975), pp. 387–487 in Applied Superconductivity, Vol. 2, ed. V. L. Newhouse, Academic Press, (straight-line extrapolations are shown in Fig. 10.19).
e Extrapolated from data by R. Feenstra and D. T. Verebelyi (1999), unpublished, Oak Ridge National Laboratory, Tennessee (the source data are plotted in Fig. 10.21).
A10.2a Scaling parameters for calculating the magnetic-field, strain, and temperature dependence of the critical current of low-Tc superconductors (Secs. 10.3, 10.5, 10.6, and 10.7)
The scaling-parameter values listed below can be used for technological purposes to analytically model and transform the critical current of multifilamentary low-Tc superconductors as a function of magnetic field, strain, and temperature. These parameters are used in conjunction with the scaling relations summarized in Appendix A10.2b. The scaling relations are listed in Appendix A10.2b in order of increasing complexity, starting with the simplest scaling laws (appropriate for fixed strain or temperature) and working to the most complete (the unified scaling law suitable for variable magnetic field, temperature, and strain). The relations are mutually consistent and build on each other, so it is worth utilizing the simplest relation for the task at hand.
Values of the scaling parameters tabulated here were determined from data correlations for specific classes of superconductors. They show good consistency (± 10 % to 20 %) within each class, where sufficient data exist for meaningful statistical correlations to be made (e.g., binary Nb3Sn, V3Ga, and, to some extent, Nb3Al; see, for example, Figs. 10.30 and 10.31). These values will no doubt be refined and updated as more data are obtained for given classes of superconductors, such as specific types of ternary Nb3Sn. To that end, these standard values and the accompanying scaling relations provide a basic framework for systematizing additional data as they become available.
This ongoing task is aided by the separability of this parameter set into magnetic field, strain, and temperature parameters. Defined in this way, the parameter values show good consistency and are easily updated. A significant advantage of the separable parameter set used here is the independence of strain and temperature parameters, which enables their values to be determined from separate strain and temperature experiments. This offers flexibility and considerable time savings. Also, since the parameters are not commingled, the entire set does not have to be redetermined every time additional new strain or temperature data become available for a particular conductor. (Further information on the practical use of this parameter set is given in Sec. 10.7.1 under the subheading Separable form, and in Sec. 10.7.4.)
If strain and/or temperature data are available for a specific conductor, scaling parameters tailored to that conductor can be determined with the robust methods given in Sec. 10.7.4 and substituted for the standard parameters given in the table below entitled Scaling Parameters. In this case, the values below are needed only to fill in missing gaps in the scaling-parameter set (see Sec. 10.7.4 for more details). These standard values also have predictive utility in the initial design stage of superconducting magnets.
Limitations:
(1) These parameter values are valid specifically for conductors with solid filaments (not tubular filaments, or bundles of filaments that fuse into a tubular structure after reaction).
(2) They also depend on additive content and fabrication process (e.g., bronze process vs. internal tin). The tabulated additive concentrations are not the starting weight percent of additive in the conductor before reaction; rather, they are the atomic percent actually measured in the Nb3Sn reaction layer after fabrication, which can vary depending on the fabrication process. As more data become available, it is expected that consistent correlations of parameter values will be obtained for additional classes of conductors.
(3) The tabulated scaling parameters are for technological use over the moderate intrinsic-strain range (–0.5 % < 0 < +0.4 %), which is the range where most magnets are designed. For high compressive strains (0 < –0.5 %), four more parameters are needed, as described in Secs. 10.5.6 and 10.7.3 and summarized below in Eqs. (A10.9)–(A10.11) and (A10.30)–(A10.32); these high-compression parameters may not be so consistent.
Examples:
Practical examples for utilizing these parameters with the scaling relations are given in the following sections:
• Magnetic-field modeling Sec. 10.3.3
• Strain and magnetic-field modeling Sec. 10.5.7
• Simplified strain transformations Sec. 10.6.2
• Simplified strain-and-temperature transformations Sec. 10.7.5
Temperature-scaling parameters
Values of the temperature-scaling parameters {for use with the temperature scaling law [Sec. 10.6.3 and Eq. (A10.12) below] and the unified strain-and-temperature scaling law [Sec. 10.7 and Eq. (A10.18)]} are available mainly for Nb3Sn and not listed in the table below. The (dimensionless) temperature-scaling parameters n and w are nearly universal constants for the technological Nb3Sn conductors evaluated thus far, including binary and ternary Nb3Sn at both moderate and high intrinsic strains:
n = 1.5 for Nb3Sn, as shown by Figs. 10.37 and 10.38
w = 3.0 for Nb3Sn, as shown by Fig. 10.36.
Although not so nearly universal, the (dimensionless) temperature exponent η and the critical temperature at zero intrinsic strain Tc*(0can be effectively approximated for nearby temperature transformations by
η ≈ 2.5 for Nb3Sn
Tc*(0=0≈ 17 K for Nb3Sn.
With additional temperature correlations, values of these latter two parameters may become more standardized for individual classes of superconductors.
For Nb3Sn, the scaling parameter Bc2*(T=0,ε0=0) can be estimated by using Eq. (10.57) with the approximate values of Bc2*(4.2K,ε0=0) given in the table below; that is,
Bc2*(0,0≈Bc2*(4.2K,0) [1 – (4.2K/17.5K)n]–1 ≈ 1.14 Bc2*(4.2K,0).
Scaling Parameters [dimensionless, except for Bc2*]:
Superconductor Crystal
Structure Magnetic-Field Dependence of Ic Strain Dependence of Ic ‡ *
(–0.5% < 0 < +0.4%)
p † q † Bc2*
at 4.2K,0=0
[T] a– (0<0) a+ (0>0) s Ref.
Strain-dependent Superconductors
Nb3Al (RHQT) A15 0.5 ~2.0 26 370 ~2.5 a
V3Ga “ 0.4 1.0 21 450 650 1.4 b
Nb3Ge “ 0.6 1.9 25 500 — ~2 c
Nb3Sn* “ 0.5 2.0 21 900 1250 1 d
Nb3Sn +0.6at%Ti* “ 0.6 1.7 23 900 1250 1.1 e
Nb3Sn +1.85at%Ti* “ 0.5 1.5 25 1100 1450 1.2 e,f
Nb3Sn +0.6at%Ta* “ 0.5 1.4 24 900 1250 1.0 e
Nb3Sn +2.2at%Ta* “ 0.5 1.4 24 1350 1800 ~1 e,g
V3Si “ 0.5 1.7 16 3500 — ~1 a
PbMo6S8 Chevrel 0.3 6 63 — 1900 ~2 h
Strain-independent Superconductors
NbN B1 1.2 2.4 24 0 0 — i
NbCN “ 1.4 2.5 17 — 0 — j
V2(Hf,Zr) C15 0.7 0.6 20 — 0 — k
† Values of p and q are effectively independent of temperature and strain (Sec. 10.7.1).
‡ The strain-parameter values listed are valid only for the moderate intrinsic-strain range (–0.5% < 0 < +0.4%). To model Ic at high-compressive strains (0 < –0.5%), additional parameters are needed, as described in Sec. 10.5.6 for the strain-scaling law and Sec. 10.7.3 for the unified strain-and-temperature scaling law.
* The strain parameters a–, a+, and s are applicable to solid-filament (not tubular-filament) superconductors. The strain-sensitivity parameter a is defined by Eq. (10.21) of Sec. 10.5.5 for a power-law exponent u = 1.7. [This value of u is found to hold experimentally for all the A15 and Chevrel-phase superconductors listed in the above table (see Sec. 10.5.5). It is also given by the model of Markiewicz with no adjustable parameters (see Fig. 10.32).] The values of a–and a+ are dependent on the type and amount of additives, as shown for the various ternary Nb3Sn materials listed in the table. Atomic percentages given are those measured in the Nb3Sn layer after reaction. For example, 2.2at%Ta in the reaction layer was obtained with a starting filament alloy of Nb–7.5wt%Ta, but it can vary depending on conductor processing.
a Banno, N., Uglietti, D., Seeber, B., Takeuchi, T., and Flukiger, R. (2005), Supercond. Sci. and Technol. 18, S338 – S343. RHQT rapid heating, quenching and transformation process.
b J. W. Ekin (1981), IEEE Trans. Magn. MAG-17, 658–661; D. G. Howe, T. L. Francavilla and D. U. Gubser (1977), IEEE Trans. Magn. MAG-13, 815–817; Furukawa Corp. (1984), personal communication.
c J. W. Ekin (1981), IEEE Trans. Magn. MAG-17, 658–661.
d J. W. Ekin (1980), Cryogenics 20, 611–624.
e J. W. Ekin (1985), pp. 267–271 in Proc. International Symposium of Flux Pinning and Electromagnetic Properties of Superconductors, eds. K. Yamafuji and F. Irie, Matsukuma Press, Japan..
f Sample from G. Ozeryansky (1984), Intermagnetics General Corp.
g Sample from W. McDonald (1984), Teledyne Wah-Chang Corp.
h J. W. Ekin, T. Yamashita, and K. Hamasaki (1985), IEEE Trans. Magn., MAG-21, 474–477.
i J. W. Ekin, J. R. Gavaler, and J. Greggi (1982), Appl. Phys. Lett. 41, 996–998.
j J. W. Ekin, unpublished data, National Institute of Standards and Technology, Boulder, Colorado; sample from M. Dietrich (1984), Kernforschungszentrum Karlsruhe, Karlsruhe, Germany.
k H. Wada, H., K. Inoue, K. Tachikawa, and J. W. Ekin (1982), Appl. Phys. Lett. 40, 844–846.
A10.2b Summary of scaling relations for utilizing the scaling parameters in Appendix A10.2a
The relations summarized here are based on consistent scaling data of the critical current as a function of magnetic field, strain, and temperature. Used in conjunction with the scaling parameters tabulated in Appendix A10.2a, they provide analytic expressions for modeling, interpolating, and predicting the critical current of most practical low-Tc superconductors as a function of magnetic field, strain, and temperature for technological applications. Further discussion and examples of the application of these scaling relations are given in the main text sections referred to in parentheses with each summary heading. The scaling relations are mutually consistent and listed progressively from the simplest to the most general. Again, use the simplest relation for the task at hand.
Because the parameter set used here is separable into temperature and strain parameters (see Sec. 10.7.1), the relations, when assembled, become the general unified scaling law described at the end of the list below. The separable nature of this parameter set also has the benefit that limited numbers of scaling parameters determined from early limited data (i.e., an incomplete set of values) are not a wasted effort. Rather, they build on each other and generally do not need to be refit later as more data become available. As refined parameter values are measured for an individual conductor, they can be substituted for the “standard” values listed in the Scaling Parameters table in Appendix 10.2a. This building process also parallels the way data are usually obtained for a given conductor (Sec. 10.7.4).
Magnetic-field interpolations (Secs. 10.3.2 and 10.3.3)
The dependence of the critical current Ic on magnetic field B is given by [ref. Eqs. (10.9)–(10.11)]
Ic(B) = k B–1 [B/Bc2*]p [1 – (B/Bc2*)]q, (A10.1)
where k is a proportionality constant and Bc2* is the effective upper critical field. This is an empirical expression based on the general form of most pinning theories. Typical values of Bc2* and the exponential constants p and q for most of the common high-field low-Tc superconductors are given in columns 3 to 5 of the Scaling Parameters table in A10.2a. Values of the parameters p and q tailored to a specific conductor can be obtained from a single Ic(B) measurement at a fixed temperature and strain [preferably obtained at a temperature T << Tc*(0to minimize the effects of Bc2 inhomogeneity (Sec. 10.3.4), and at a strain not too far from e0 ≈ 0 to avoid Bc2 inhomogeneity effects as well as filament breakage at high tensile strains (Sec. 10.5.1)].
Strain dependence of the critical current (valid for fixed temperature T << Tc; e.g., at 4.2 K in Nb3Sn) (Secs. 10.5.4–10.5.7 )
The magnetic-field and strain dependence of the critical current of most low-Tc superconductors can be modeled at low temperatures with the strain scaling law [Eq. (10.18)]. The simplest parameterization of this law applies to the moderate intrinsic-strain range (–0.5 % < ε0 < +0.4 %, assuming εirr ≥ +0.4 %), which is also the strain range where most (but not all) magnets are designed mainly because the critical currents are highest in this regime. [Extended-strain parameters covering the high compressive strain range (0 < –0.5 %) are given in Secs. 10.5.6 and 10.7.3, and are summarized below in Eqs. (A10.9)–(A10.11) and (A10.30–A10.32).] This empirical scaling law is based on extensive correlations of strain data at 4.2 K in low-Tc superconductors, which show strain invariance of the shape of the pinning-force vs. magnetic-field characteristic (Sec. 10.5.4).
For the moderate strain range, the simplest and most consistent parameterization of the strain scaling law for technological purposes gives the following expression for the magnetic-field and strain dependence of the critical current Ic(B,0) [ref. Eqs. (10.20)–(10.22)]
Ic(B,0) = g(0) [1 – a01.7] s B–1 [B/Bc2*(0)]p {1 – [B/Bc2*(0)]}q, (A10.2)
valid for fixed temperature T << Tc (e.g., at 4.2 K in Nb3Sn). Here, g(0) is a proportionality constant and the scaling parameters p, q, a– (for 0<0), a+ (for 0>0), and s are tabulated in columns 3, 4, 6, 7, and 8, respectively, of the Scaling Parameters table in A10.2a. (The strain-sensitivity parameters a– and a+ are described more fully below.) The scaling parameter Bc2*(0) is the strain-dependent effective upper critical field; values at 4.2 K are tabulated in column 5 of Table A10.2a.
The variable 0 is the intrinsic strain, defined as
0 – m, (A10.3)
where is the axial strain applied to the superconductor and m is the axial strain at which Ic is maximum (e.g., Fig. 10.26). Negative values of 0 represent the compressive intrinsic strain in the superconductor, and positive values represent tensile intrinsic strain (Sec. 10.5.1). The upper strain limit for the validity of this strain scaling law is given by the irreversible strain limit e0irr, where the conductor is permanently damaged by axial strain. A typical value of e0irr in Nb3Sn is about e0irr ≈ +0.4 %, but it can be more or less than this amount (Sec. 10.5.1). (Lower values of e0irr usually occur in conductors with larger filament diameters or fused filament clusters, whereas higher values of e0irr usually occur in conductors with small filament diameters, such as Nb3Al or fine-filament Nb3Sn conductors.)
Over the moderate intrinsic strain range (–0.5 % < e0 < e0irr), the effective upper critical field is well represented by a power law
= 1 – a01.7, (A10.4)
where Bc2*(0=0) is evaluated at the designated temperature T. This is an empirical parameterization that accurately and consistently represents the fundamental results of anharmonic strain theory over the moderate strain range (Sec. 10.5.5). Values of Bc2*(0=0) at 4.2 K are listed in column 5 of Table A10.2a. For Nb3Sn, values of Bc2*(0=0) at other temperatures can be estimated from the 4.2 K values by using Eq. (10.57) [i.e., Eq. (A10.19) below] and the temperature-scaling parameter values listed just before Table A10.2a.
The parameter a in Eq. (A10.4) is the intrinsic strain sensitivity and is a simple quantitative index of the sensitivity of a given class of superconductors to axial strain. For compressive strains (0 < 0), the strain sensitivity is denoted as a–, with values listed in column 6 of Table A10.2a. For tensile strains (0 > 0), the strain sensitivity is denoted a+, with somewhat greater values listed in column 7 of Table A10.2a. (Sec. 10.5.5 gives a discussion of the fundamental origin of this difference.) The compressive parameter a– (column 6) is usually the more important parameter from a technological standpoint, since it characterizes the strain sensitivity of a conductor over the moderate compressive strain range where most magnets are designed (–0.5 % < 0 < 0 %). Again, the tensile parameter a+ is valid only up to the irreversible strain limit e0irr where damage occurs in the superconducting filaments.
Limitations:
(1) It bears reiterating that the strain-scaling parameters a–, a+, and s in the Table A10.2a apply to the most common types of conductors, solid-filament multifilamentary wires, not to other filament shapes such as wires with tubular filaments or fused tubular clusters, where the strain sensitivity is enhanced by three-dimensional strain effects.
(2) Occasionally, particular conductors need to be characterized by unsymmetrical values of the parameter s in Eq. (A10.2) [s– for compressive intrinsic strain (0 < 0) and s+ for the tensile side (0 > 0)]. Further data correlations are needed to see if such unsymmetrical values of s are “standard” for certain types of superconductors.
(3) For high-compressive strains (0 < –0.5 %), a more general relationship must be used, given by Eqs. (10.23) and (10.24) in Sec. 10.5.6. This entails additional parameters that appear to be extrinsic in nature and, therefore, must be fitted on a conductor-by-conductor basis (Sec. 10.5.6).
Strain transformation method (Sec. 10.6.1)
The strain transformation method is a powerful technique for utilizing the strain scaling law to transform a single Ic(B,ε01) curve obtained at a strain ε01 to a curve Ic(B,ε02) valid at a different strain ε02, without the need to know the parameters p, q, Bc2*(0), or g(0) in Eq. (A10.2). The derivation of this transformation method is given in Sec. (10.6.1).
The transformation is independent of the parameterization scheme; it is illustrated here with the separable parameter set employed above because of its practical utility, but it can be used with any of the alternative parameterization schemes that have been proposed for the prefactor term g() in the strain scaling law [Eq. (10.18)]. Again, the strain-transformation method is limited to transformations carried out at fixed temperatures T << Tc, such as 4.2 K in Nb3Sn. (For combined strain and temperature transformations, see the unified-scaling transformation method summarized in the last subsection of this appendix.)
The application of the transformation consists of two steps. First, to obtain Ic(B,ε02) when ε01 and ε02 fall within the moderate strain range (–0.5 % < 0 < e0irr), multiply the magnetic field data of the Ic(B,ε01) data set by the constant to obtain a new set of magnetic-field values B [ref. Eq. (10.32)]
B = B. (A10.5)
Second, multiply the critical current data of the Ic(B,ε01) set by s–1 to obtain new critical current values corresponding to the new magnetic-field data; that is, [ref. Eq. (10.33)]
Ic(B,02) = s–1 Ic(B,01). (A10.6)
The constant in these two equations is given by [ref. Eq. (10.34)]
(A10.7)
(For an immediate clarification of this simple data-transformation procedure, please refer to the example given in Table 10.3 in Sec. 10.6.2.) Thus, all that’s needed to carry out the transformation are the scaling parameters a and s. Again, values of s are listed in column 8 of Table A10.2a, and values of the strain-sensitivity parameter a are given in columns 6 and 7 of Table A10.2a (a− for ε0 < 0, and a+ for ε0 > 0).
For nearby transformations, this method is quite accurate. For example, when transforming from ε0 = –0.3 % to 0 % in Nb3Sn conductors, an error of 10 % in the value of a results in an error of less than 0.5 % in the transformed B values and effectively no error in the Ic values.
High compressive strains: For transformations involving high-compressive strains (0 < –0.5 %), the magnetic-field transformation remains that given by Eq. (A10.5), but we use the more general transformation for the critical current [Eq. (10.30)]
(A10.8)
where
The functions Bc2*(0) and g(0) can be parameterized at high compressive strains by
= 1 – a01.7 + a10 – 0′a2 I(0 < 0′) (A10.9)
and = [1 – a01.7] s + g10 – 0.005g2 I(0 <0′) (A10.10)
with I(0 < –0.005) ≡ { . (A10.11)
Here, I(0 < 0′) is an indicator function, which is zero except at high-compressive strains: 0 < 0′, with 0′ ≈ –0.005 for Nb3Sn. Unlike transformations carried out over the intrinsic peak strain range, the extra parameters characterizing the extended strain range (a1, a2, g1, and g2) are probably extrinsic in nature and need to be determined on a conductor-to-conductor basis. The advantage of the parameterization given by Eqs. (A10.9)–A(10.10) is that the scaling parameters a and s characterizing the intrinsic peak region remain consistent and unaffected by those characterizing the extrinsic high compressive regime.
Temperature dependence of the critical current (valid for fixed strain) (Sec. 10.6.3)
The magnetic-field and temperature dependence of the critical current of most low-Tc superconductors can be modeled (at a fixed strain) with the temperature scaling law [Eq. (10.36)]. This empirical relation is based on correlations of temperature data at fixed strain, which show temperature invariance of the shape of the pinning force vs. magnetic-field characteristic (Sec. 10.6.3).
The simplest parameterization of the temperature scaling law for technological purposes is given by [ref. Eqs. (10.38)−(10.43)]
Ic(B,T) = h(0) [(1 − t v)]η B–1 [B/Bc2*(t)]p {1 – [B/Bc2*(t)]}q, (A10.12)
valid for fixed strain. Here, h(0) is a proportionality constant, p and q are the scaling constants given in columns 3 and 4 of Table A10.2a, and n and η are temperature-scaling constants listed for Nb3Sn just before Table A10.2a.
The variable t is the reduced temperature, defined as
t T/Tc*(0, (A10.13)
where T is the temperature of the superconductor and Tc*(0 is its effective critical temperature (at a fixed strain ε0). When t << 1 and |0| | – m| < ~0.4%, the better known strain-free value of Tc* at 0 = 0 can be used (since the strain dependence of Tc* is relatively gradual, as described for the unified-scaling relation below). For Nb3Sn, the strain-free Tc* is about 17 K, also listed with the temperature-scaling parameter values just before Table A10.2a.
The effective upper critical field in Eq. (A10.12) can be parameterized most simply by
Bc2*(tBc2*(0) (1 – t n), (A10.14)
where Bc2*(t=0) is evaluated at the designated strain ε0. For Nb3Sn, the strain-free value of Bc2*(t=0) (i.e., at 0 K and at 0 – m = 0) can be estimated from the values of Bc2*(4.2K,ε0=0) listed in column 5 of Table 10.2a by using the relation Bc2*(t=0,ε0=0)≈Bc2*(4.2K,ε0=0) [1 – (4.2K/17K)n]–1 ≈ 1.14 Bc2*(4.2K,ε0=0). (Here we have used the nearly universal value n = 1.5, which is appropriate for Nb3Sn.) The strain-free Bc2* also works for small values of intrinsic strain, since the strain dependence of Bc2* is relatively gradual compared with its temperature dependence.
The temperature-scaling parameters are the least well characterized at present. As indicated in the material just before the Scaling Parameters table A10.2a, the value n = 1.5 is fairly well established for Nb3Sn conductors, but the temperature-scaling parameter η is not yet, with values reported typically in the range 2 to 3.5 for different types of Nb3Sn superconductors. More standard values of η are expected to be determined as data correlations become available for specific classes of superconductors. In the meantime, a nominal value of η = 2.5 can be used at least for estimation purposes. This nominal value of η also serves well for temperature transformations, particularly if they are nearby transformations, as described next.
Temperature transformation method (Sec. 10.6.3)
Similar to the strain transformation summarized earlier, the temperature transformation method is a greatly simplified technique for utilizing the temperature scaling law to transform a single Ic(B,t1) curve, obtained at a temperature t1, to a curve Ic(B,t2) valid at a different temperature t2, without the need to know the parameters p, q, Bc2*(t=0), or h(0) in Eq. (A10.12).
The transformation is independent of the parameterization scheme. Again, it is illustrated here with the parameter set employed above because of its practical utility, but it can be used with any of the alternative parameterization schemes that have been proposed for the prefactor term h(T) in the temperature scaling law [Eq. (10.36)] The temperature-transformation technique is limited to transformations carried out at a fixed strain; for combined strain and temperature transformations, see the unified scaling law summarized in the next subsection.
As for strain transformations, the application of the temperature transformation method consists of two steps. First, to obtain Ic(B,t2), multiply the magnetic-field data of the Ic(B,t1) data set by the constant to obtain a new set of magnetic-field values B [ref. Eq. (10.39)]
B = B. (A10.15)
Second, multiply the critical-current data of the Ic(B,t1) set by η–1 to obtain values of the critical current corresponding to the new magnetic-field values; that is [ref. Eq. (10.41)]
Ic(B,t2) = η–1 Ic(B,t1). (A10.16)
The constant is given by [ref. Eqs. (10.42) and (10.43)]
(A10.17)
(Again, for a clarification of this simple data-transformation procedure, please refer to the example given for strain in Table 10.3 in Sec. 10.6.2.) All that is needed to carry out the transformation are the scaling parameters Tc*, v, and η.
As noted above, data correlations of the temperature-scaling parameters have not yet been made for many of the different classes of technological superconductors. The value n = 1.5 is fairly well established for Nb3Sn conductors and the strain-free Tc* at 0 = 0 is about 17 K for Nb3Sn, but the scaling parameter η is not so nearly universal, with values reported over the range 2 to 3.5 for a variety of Nb3Sn conductors. A nominal value of η = 2.5 can be used for estimation purposes with most Nb3Sn superconductors.
Nevertheless, this method is quite accurate if the transformations are nearby. For example, when making the very useful data transformation from 4.2 K to, say, 5.5 K in Nb3Sn conductors, even a large error of 20 % in the value of η (which covers the range 2 < η < 3) results in a difference of only about 3.5 % in the transformed Ic (and no error in B). For the same example, an error of 5 % in the parameter Tc* results in a difference of only 1.2 % in the transformed Ic, and only 0.6 % in the transformed B.
Unified strain-and-temperature dependence of the critical current (Sec. 10.7)
The combined magnetic-field, temperature, and strain dependence of the critical current of most low-Tc superconductors can be modeled with the unified scaling law [Eq. (10.45)]. The simplest parameterization of this scaling law applies to the moderate intrinsic-strain range (–0.5 % < ε0 < +0.4 %, assuming εirr ≥ +0.4 %), which is where most superconductor applications are designed (again, because in this regime the conductors are under the least stress and their critical currents are maximized). [Extended-strain parameters covering the high-compressive-strain range (0 < –0.5 %) are discussed below with the unified-transformation method.] The unified scaling law is based on data in low-Tc superconductors showing shape invariance of the pinning-force vs. magnetic-field characteristic with respect to both strain and temperature simultaneously.
For the moderate intrinsic-strain range, the unified scaling law can be parameterized with the separable parameter set, giving the combined magnetic-field, temperature, and strain dependence of the critical current Ic(B,T,ε0) [ref. Eqs. (10.56)–(10.58)]
Ic(B,T,ε0) = C B–1 (1 – a01.7) s (1 – t n) b p(1 – b) q (A10.18)
where
Bc2*(t,e0Bc2*(0,0) (1 – a01.7) (1 – t n) (A10.19)
and
(A10.20)
Here, C is a proportionality constant and the variables are defined by:
0 – m Intrinsic strain (where is the applied strain and m is the applied strain at the peak, all in absolute units, not percent)
Reduced magnetic field
Reduced temperature
Equations (A10.18) through (A10.20) utilize the separable parameter set, mentioned in the introduction to this appendix, wherein the scaling parameters are separated into magnetic-field, strain, and temperature-scaling parameters with consistent, independent values that are easily updated as additional strain or temperature data become available for a given conductor. With this set, parameter values can also be determined from independent strain or temperature data [rather than from a lengthy, full matrix of combined Ic(B,t,ε0) data], which can save a month or more of data acquisition per sample. Robust methods for determining values of the separable parameter set are described in detail in Sec. 10.7.4. Because this parameterization results in such consistent values, standard values of the parameters work for many purposes when values tailored to a particular conductor are not available. These parameters are discussed in the following paragraphs.
Magnetic-field scaling parameters. Standard values of the magnetic-field parameters Bc2*(T=4.2K,0=0), p and q are tabulated for most of the common low-Tc superconductors in columns 3 to 5 of the Scaling Parameters table in A10.2a. For Nb3Sn, the parameter Bc2*(0,0) can be estimated from the measured values of Bc2*(4.2K,0) by using Eq. (A10.19); that is, Bc2*(0,0≈Bc2*(4.2K,0) [1 – (4.2K/17K)n]–1 ≈ 1.14 Bc2*(4.2K,0). Here, we have used n = 1.5 and Tc*(0≈ 17 K for technical Nb3Sn conductors, as listed just before Table A10.2a.
Strain scaling parameters: Standard values of the strain parameters a–(0<0), a+(0>0), and s are tabulated in columns 6, 7, and 8, respectively, of Table A10.2a. The strain-sensitivity parameters a– and a+ are described more fully with the strain-scaling relation summarized above. The limitations of the validity of these standard strain-scaling parameter values are also summarized above (solid filaments, additive concentration dependence, and strain range). Note especially that they are limited to the moderate intrinsic strain range (–0.5 % < e0 < +0.4 %). The more general parameterization of the unified strain-and-temperature scaling law, valid for strains extending to high compression (0 < –0.5%), is given in Secs. 10.5.6 and 10.7.3, and is summarized in Eqs. (A10.30 – A10.32) below. As described in Sec. 10.5.6, this entails additional parameters that appear to be extrinsic in nature, and thus consistent parameter values cannot be tabulated; instead they must be fitted on a conductor-by-conductor basis.
Temperature scaling parameters. The temperature parameters and w have nearly universal values, at least for technical Nb3Sn superconductors; values are listed for these two parameters just before Table A10.2a. The temperature parameters Tc*(0 and are not yet well established. As noted for the temperature scaling law above, a value for Tc*(0) of about 17 K can be used for Nb3Sn, but this would benefit from further data correlations for given classes of superconductors. Likewise, values of have been reported anywhere from 2 to 3.5 for different types of Nb3Sn superconductors, but a nominal value of η = 2.5 can at least be used for estimation purposes and serves well for the transformation technique to nearby strains and temperatures (described just below).
Extrinsic parameters. This leaves the parameters εm and C, which are highly variable extrinsic parameters that mainly depend, respectively, on conductor geometry and heat-treatment conditions. Therefore, they must be determined on a conductor-by-conductor basis. Often εm can be approximated from measurements on similar conductors, or determined for a specific conductor from a single Ic(ε) measurement at any fixed magnetic-field. The proportionality constant C can be determined from a single Ic measurement on the conductor in question (at any fixed magnetic field, temperature, and strain).
Unified strain-and-temperature transformation method (Sec. 10.7.5)
Similar to the transformations described above, the transformation method is a powerful technique for utilizing the unified scaling law to transform a single Ic(B) curve measured at strain ε01 and temperature T1, to another combination of strain ε02 and temperature T2 without the need to know the parameters p, q, Bc2*(0), and C. The method is especially effective for nearby transformations, achieving high accuracy even with standard parameter values (see below). The transformation is independent of the parameterization scheme, but it is illustrated here with the separable parameter set because of its practical utility.
As shown in the example at the end of Sec. 10.7.5, the transformation is carried out in a spreadsheet program simply by multiplying a column of magnetic-field values and a column of corresponding critical-current values by two constant prefactors. That is, to transform a data set Ic(B,T1,ε01), which was obtained at temperature T1 and strain ε01, to a corresponding data set Ic(B,T2,ε02) valid at T2 and ε02, multiply the magnetic-field values in the first set by the constant
B = {} B , (A10.21)
and the critical-current values in the first set by the constant shown in brackets { }
(A10.22)
where
. (A10.23)
For the common case where ε01 and ε02 fall within the moderate strain range (–0.5 % < 0 < +0.4 %), these two transformation constants {the terms in brackets in Eqs. (A10.21) and (A10.22)} can be evaluated with the separable parameter set as
(A10.24)
and
(A10.25)
Again, the variables are defined by:
0 – m Intrinsic strain (where is the applied strain and m is the applied strain at the peak, all in absolute units, not percent)
Reduced temperature
with
(A10.26)
Thus, for the moderate strain range (–0.5 % < < +0.4 %), the task of transforming both strain and temperature with Eqs. (10.21) and (10.22) is reduced to evaluating these two transformation constants from the scaling parameters: n, w, , s, a–, a+, Tc*(0 and m. Again, the strain parameters are given in Table A10.2a and the temperature parameters are listed for Nb3Sn just above the table. This leaves εm as the one extrinsic parameter that must be determined on a conductor-by-conductor basis from a single Jc(ε) measurement at any fixed magnetic field and temperature, or it can be estimated from a single Jc(ε) measurement on a similar conductor.
For transformations where either ε01 or ε02 fall in the high-compressive-strain range (0 < –0.5 %), a more general form must be used for the two transformation constants because of the extrinsic nature of their strain dependence in this regime. The more general parameterization [which replaces Eq. (A10.24)] for the transformation constant is
(A10.27)
and the more general parameterization [which replaces Eq. (A10.25)] for the critical-current transformation factor is
(A10.28)
with
(A10.29)
Any parameterization of Bc2*(0,0) and g(0) will work with the general transformation factors given by Eqs. (A10.27)–(A10.29). The extended-range parameterizations discussed in Secs. 10.5.6 and 10.7.3 are suggested as a practical scheme that is easy to use because they preserve the consistency of the intrinsic parameter values a and s for the moderate strain range; that is,
= 1 – a01.7 + a10 – e0’a 2 I(0 < e0′), (A10.30)
= [1 – a01.7] s + g10 – e0’g2 I(0 < e0′). (A10.31)
Here, C is the same proportionality constant as in Eq. (A10.18), and the indicator function is defined as
I(0 < e0′) ≡ { , (A10.32)
with e0′ = –0.005 for Nb3Sn. [This function is readily programmed in spreadsheet programs with a conditional clause of the form: IF(0<–0.5%, 1 if true, 0 if false).] Again, for the many applications at moderate intrinsic-strain levels, the high-compression term is not needed. This happens seamlessly with Eqs. (A10.30) and (A10.31) because the indicator function automatically drops the extra term for moderate strains, reducing these relations to the simpler power-law expressions.
The simplicity of this transformation procedure becomes readily clear by referring to the example given in Table 10.4 at the end of Sec. 10.7.5. The entire process can be carried out in a few minutes with a spreadsheet program. It is also quite accurate. For example, a relatively large error of 20 % in the temperature parameter would result in an error in the Ic values of only about 3.5 % when transforming from the canonical temperature of 4.2 K to a difficult-to-measure temperature, such as 5.5 K. Likewise, when transforming from, say, ε0 = –0.3 % to 0 % in Nb3Sn conductors, a relatively large error of 10% in the value of the strain sensitivity parameter a would result in an error of less than 0.5% in the transformed B values and negligible error in the transformed Ic values. Again, with this method there is no need to determine the shape of the Ic–B curve, and it is independent of the extrapolation method used to determine Bc2*. The method, given in general form by Eqs. (A10.21) through (A10.23), relies only on the unified scaling law, Eq. (10.45), not on the separable parameter set used to illustrate it here. Thus, it would readily work with any other parameterization of the prefactor K(T,e0) in the unified scaling law, Eq. (10.45).