Low-Temperature Properties of Silver

Pure silver is used extensively in the preparation of high-temperature superconductor wires, tapes, films, and other configurations in which the silver not only shields the superconducting material from the surrounding materials, but also provides a degree of flexibility and strain relief, as well as stabilization and low-resistance electrical contact. Silver is relatively expensive, but at this stage of superconductor development, its unique combination of properties seems to offer the only reasonable means of achieving usable lengths of conductor. In this role, the low-temperature physical (electrical, thermal, magnetic, optical) and mechanical properties of the silver all become important. Here we present a collection of properties data extracted from the cryogenic literature and, to the extent possible, selected for reliability.


Introduction
Most applications of high-temperature superconductors involve the use of silver. In thin-film devices, it serves as an electrical contact with acceptably low contact resistance, on the order of 10 -8 ⍀иcm 2 . In large-scale devices the contact application is also important; the contact resistance can be higher, but the contact area is large. In conductor applications silver also provides (1) containment for the precursor powder mix during mechanical processing; (2) containment for the powder while allowing passage of oxygen during heat treatment to form the superconducting structure; (3) stabilization of the conductor in operation by providing an alternate current path and a thermal path to the coolant; (4) providing mechanical strength to the finished conductor; and (5) providing internal strain relief for the brittle superconductor when mixed as part of the powder. In low-temperature superconductors, most of these roles are carried out by copper. Silver, however, is required in high-temperature superconductors because of its unique ability to allow passage of diffusing oxygen. The lowtemperature properties of silver also make it an attractive choice in applications that are not directly related to superconductivity. These properties are its high thermal and electrical conductivities, diamagnetism, high reflectance, and low emittance.
Over the past two decades, we have produced two versions [1,2] of a wall chart presenting the cryogenic properties of copper. This chart has been popular with the low-temperature superconductor community. With the advent of high-temperature superconductors, we decided that a similar wall chart of cryogenic properties of silver would be valuable. This new chart, Cryogenic Properties of Silver , became available for distribution in January 1994 [3]. Limitations on the size of the chart did not allow us to present all the cryogenic-property data which were found in our review of the literature. Furthermore, the chart provided very little room for presentation of tabular data or commentary. Providing that additional information is the purpose of this paper.
Here we present figures and tables of the electrical, magnetic, mechanical, optical, thermal, and thermodynamic properties of pure silver, culled from more than 200 documents covering a period of over 60 years. While we concentrated on these properties, we have also included information in areas such as metal physics and effects of irradiation by high-energy particles on resistivity.
In general, we treated only pure silver. However, one paper on coin silver was included because this alloy is commonly available and might be of use at cryogenic temperatures. Also, mention is made (see Sec. 3.1,Ref. [137]) of a new material, a dispersion-hardened silver alloy which can provide additional strength and resistance to thermal shock, while maintaining acceptable thermal and electrical properties. The door to the properties of other alloys of silver was not cracked further open.
This work is not a complete survey of all the literature available on silver. While we made a strong effort to identify those papers of most use to workers interested in the cryogenic applications of silver, resources did not permit an exhaustive search of all the current literature. We believe that we located most, if not all, of the important sources of available data. Each bibliographic reference to a paper has been annotated by a capsule summary of the content specifically related to cryogenic applications of silver. Before the reader uses any data for engineering design or other critical applications, the reader is strongly urged to consult the original reference(s) for complete details of the conditions of measurement.
Some mention needs to be made regarding the techniques by which data were transferred from the graphs and plots in the original documents into the form presented here. Where tabular data were available there was no problem, and the plots were made directly. However, most of the documents provided data in graphical form, and often only as very small graphs and of quality less than that available now in the era of desktop publishing. In those cases, a combination of enlargement by photocopier, use of a graphics tablet, and commercial plotting software allowed extraction and smoothing of the data while maintaining the accuracy of the original plot. Because of the increased size of many of the plots, some curves show a minor lack of smoothness. All such variations lie within the accuracy of the data as plotted in the original publication.

Properties Data
In the following descriptions of tables and figures, a bullet (•) at the beginning of a paragraph denotes a description corresponding to a table or figure used in the wall chart, Cryogenic Properties of Silver [3]. Those tables or figures were selected for inclusion in the chart because of widest appeal or interest. For each graph or table in this paper there is always a corresponding bibliographic entry in Sec. 3, similarly marked with a dagger ( †) and index number. The additional documents in the bibliography, under the topics listed here, contain further information on these properties.

General
• The fundamental physical and chemical properties of silver are summarized in Table 1. It is fairly easy to find more than one secondary source for many of the data in the table, and there are usually small (within one or two percent) disagreements among the various values for a given property. Where multiple values were found, a representative (most frequently occurring) value was chosen for listing in this table.

Electrical Resistivity
• Matthiessen's rule assumes that the total electrical resistivity el is well approximated by a sum of two terms. The first is a temperature-dependent intrinsic term int (T ), which is zero at absolute zero. The second term is a residual term 0 due to the effect of impurities and crystal defects, and does not vanish at absolute zero. In Fig. 1 the total electrical resistivity of silver as a function of temperature is plotted for four different values of RRR (residual resistance ratio; a measure of impurity content), and a value for int at 0 ЊC is given. For metals in electrical applications, values for RRR are a more sensitive measure of purity than total content of chemical impurities. Above 80 K the resistivity of dilute alloys of silver is independent of impurity content, whereas below 80 K the resistivity is sensitive to impurity content. At room temperature the variation of resistivity with temperature is nearly linear.
• Table 3 lists the change in electrical resistivity per "atomic percent" (mole fraction) of alloying element for 20 different metallic elements in pure silver. The effect of alloying small amounts of impurities with silver is to increase its resistivity. The change, initially linear with very small increases in concentration, is nonlinear as larger amounts of alloying element are added. The transition point to nonlinear dependence on concentration is different for different impurities. Hence the values listed in the table should be used only as a guide, and then very cautiously. One can assume without large error that the effects of several impurities are additive in small quantities. • Magnetoresistance, the dependence of electrical resistivity on applied magnetic field, is given in Fig. 2 as a Kohler plot. For many polycrystalline metals, data covering wide ranges of temperature and metallic purity reduce to a single curve for the transverse case (field normal to current), and data for many different metals will fall on the curve. This is known as Kohler's rule. However, it is frequently not obeyed, and the silver data in particular seem not to obey the rule. Regardless, application of relatively modest magnetic fields can cause large changes in resistance at low temperatures. The longitudinal magnetoresistance, measured with the field parallel to the current, saturates for most metals at a relatively low value, as seen here. In single-crystal specimens, the magnetoresistance is a strong function of crystal orientation, and measurements of magnetoresistance are used to determine the detailed topology of the Fermi surface. Figure 3 plots the relative change in resistivity due to cold working (increasing the number of defects in the specimen; the reduction in the cross-sectional area of specimen accompanying the cold working is taken into account in computing the resistivity). This figure shows that the dependence on cold working varies somewhat with temperature. That is, Matthiessen's rule does not strictly apply. Cold working should affect only the number of geometrical defects such as vacancies and interstitials, which are independent of temperature (until annealing begins to occur, at temperatures well above room temperature); the ideal temperature-dependent intrinsic resistivity int (T ) due only to lattice vibrations should not be affected by cold working.
Electrical resistance depends weakly on applied pressure. The relative resistance R (P )R (P = 0), normalized to unity at "zero pressure," is plotted in Fig. 4. Both the relative resistance and the relative volume of silver, also normalized to unity at "zero pressure," are tabulated in Table 4. One assumes that "zero pressure" is, in practice, atmospheric pressure, about 100 kPa or 10 -4 GPa; the error in this assumption is minuscule.  Figure 5 plots the isochronal recovery of resistivity of silver wires after deformation under tension. After the deformation, carried out at -183 ЊC (90 K), the temperature of the wire was raised by a constant amount and held at that temperature for a fixed period of time. The holding time was the same ("isochronal") for all annealing temperatures. From the shape of the curve, the two values for energy of activation of recovery, 0.18 eV and 0.69 eV, were deduced.
Another study of isochronal annealing at 77 K gave "resistivity differential curves of fractional isochronal annealing" after torsional deformation, for silver and silver alloys. Figure 6 gives the plot for pure silver. The peaks in the curve at 110 K and 250 K are interpreted as major annealing stages. (We question whether the ordinate is correctly given as ⌬(⌬ /⌬ 0 ) /⌬T in the original paper, because o should not change in the given experiment. The ordinate should rather be ⌬(⌬ / 0 ) /⌬T, the change in fractional recovery of resistivity with temperature.) The use of superconducting magnets in applications involving high-energy particle beams motivated the inclusion of Fig. 7. This shows the relative change in resistivity per unit electron flux, as a function of average energy of the bombarding electrons. The bombardment was performed at specimen temperatures of 20.4 K.

Thermal Properties
• 2.3.1 Thermal Conductivity Thermal conductivity is the measure of steady-state conductive flow of heat. Figure 8 gives the thermal conductivity of silver for various values of RRR (residual resistance ratio, a measure of impurity content). The expression 1 / = W th = W i + W 0 is the thermal analog of Matthiessen's rule; here W th is the total thermal resistance, W i is the ideal temperature-dependent resistance, and W 0 is the resistance due to impurities. The Lorenz number L = el /(W th иT ), where el is electrical resistivity and T is absolute temperature, is theoretically a constant for all metals at absolute zero. Theory predicts that should vary linearly with temperature at very low temperatures (below the three conductivity peaks near 10 K). The slope immediately above the peak is observed empirically to vary approximately as T -2.3 . This gives the slope at which the conductivity would continue to rise with decreasing temperature in the absence of any physical defects or chemical impurities (thermal resistance decreasing to zero at absolute zero). The conductivity is observed empirically to become independent of impurity content above about 70 K.
Thermal conductivity can vary with magnetic field, due to the effect of the field on the conduction electrons. This magneto-thermal resistance is shown in Fig. 9 in the form of thermal resistance W , normalized to the zero-field resistance W 0 = W (B = 0), as a function of magnetic induction B . The four curves are for different values of W 0 and specimen orientation (longitudinal or transverse to the thermal current); the temperatures of the measurement were also slightly different for each experiment.
• 2.3.2 Thermal Diffusivity Thermal diffusivity a is a measure of the transient flow of heat through a material. (The symbol a is chosen here for diffusivity to avoid confusion between diffusivity and thermal expansion, both of which are conventionally represented by the symbol ␣ ). The relation a = /(C p и ) defines the thermal diffusivity in terms of the thermal conductivity , the specific heat C p and the density . Fig. 10 shows the diffusivity as a function of temperature.
• 2.3.3 Thermal Expansion The linear thermal expansion coefficient measures the one-dimensional expansion of a material with changing temperature. It is defined by ␣ = (dL /dT ) /L 0 = dln(L ) /dT, where L 0 is the length of the specimen at a reference temperature, conventionally taken to be 293 K (20 ЊC). For silver it varies with temperature as shown in Fig. 11. Because ␣ varies by several orders of magnitude between liquid-helium temperature and room temperature, the curve is given as a log-log plot. The volumetric expansion coefficient, ␥ = dln(V ) /dT, measures the three-dimensional variation of volume with temperature, and is well approximated by ␥ =3␣ . The variation of density with temperature is given by where m is the mass of the specimen. The areal, or two-dimensional, thermal expansion coefficient is ␤ = dln(A ) /dT and is well approximated by ␤ = 2␣.
• 2.3.4 Thermoelectric Power The absolute thermoelectric power S = dE /dT is a measure of the rate of change of absolute thermoelectric emf E with temperature. An electrical circuit involving two junctions of silver (of thermoelectric power S a ) with another metal (of thermoelectric power S b ) will generate an output emf per degree equal to S a -S b when the junctions are at different temperatures. Because of its strong dependence on temperature, it is given on a log-log plot in Fig.  12.

Hall Coefficient
The Hall coefficient -R H (T ) is plotted in Fig. 13 as a function of temperature for silver in a magnetic induction B = 1.5155 T. (Because electrons are the charge carriers in silver, the Hall coefficient for silver is negative; so -R H is plotted vertically upward).
The low-field dependence with temperature of the relative Hall coefficient R H (T ), normalized to its value at 77 K, is given in Fig. 14.
In Fig. 15, the temperature dependence of the same normalized (relative) Hall coefficient is graphed for two different values of applied magnetic induction (solid lines: B = 0.5145 T ; dashed lines: B = 8.5 mT ). For each value of magnetic field, the normalized coefficient is graphed for three different values of RRR (residual resistance ratio, a measure of impurity content), values of 510, 3250, and 3550). Figure 16 displays the relative Hall coefficients at 4.2 K, normalized to the value at 77 K, as a function of magnetic induction B and RRR value. These are the same specimens (and same RRR values) as in Fig. 15. The relative Hall coefficients for the specimens having RRR = 3250 (specimen 1) and 3550 (specimen 2) show saturation at magnetic inductions above 0.2 T , while that for the low-purity (specimen 3: RRR = 510) did not saturate. The authors could offer no explanation for the occurrence of minima in the R H curves for specimens 2 and 3 and the apparent absence of a minimum in R H for specimen 1 (intermediate purity); possibly this specimen has a minimum at B = 0 T.

Anisotropy Factor
The anisotropy factor NP / BP is a measure of relaxation time on the "neck" (N) of the Fermi surface (FS) normalized by the relaxation time for the FS "belly" (B), for electron-phonon (P) scattering. Fig. 17 plots the anisotropy factor for pure silver. Figure 18 shows how normal spectral emissivity of silver at 295 K depends on (visible and infrared) wavelength.

Absorptance
The absorptance of electromagnetic radiation from the near ultraviolet (220 nm) through the visible (400 nm to 700 nm), to the near infrared (5000 nm), is tabulated in Table 5 and plotted in Fig. 19. Values are given for freshly evaporated film, for bulk metal, and for a polished, chemically deposited surface. The latter two sets are probably more realistic for modeling the optical properties of silver-sheathed superconducting wire in practical applications.

Angular Reflectivity
The angular reflectivity of silver, for visible light of wavelength = 546 nm, is given in Fig. 20 as a function of angle of incidence. Table 6 lists the real and imaginary components, n and k , of the complex optical index of refraction N = n -ik , along with the reflectance R , for the visible and near infrared parts of the spectrum. The values are also plotted in Fig. 21. Optical reflectance plus absorptance for a metal sum to unity. It follows that surface films of oil, grease or dust, compromising the cleanliness of the silver and increasing its optical absorptance, easily reduce its reflectance.

Index of Refraction
2.6 Thermodynamic Properties 2.6.1 Density From the temperature dependence of the linear thermal expansion coefficient ␣, plotted in Fig. 11, the volumetric expansion coefficient can be calculated, and then the dependence of density with temperature. The results of this calculation is given in Fig. 22; a value of 10.492 g /cm 3 (see Table 1) was assumed for the value of density at 20 ЊC.
2.6.2 Specific Heat, Entropy, Enthalpy, and Gibbs Energy For the low-temperature range (temperatures from 3 K to 30 K) the temperature dependence of specific heat C p is tabulated in Table 7. The low-temperature specific heat is graphed in Fig. 23 on a linear scale to show the high-temperature behavior, and in Fig.  24 on a log-log scale to emphasize the low-temperature behavior.  Table 8 gives the temperature dependence over the range of temperatures from 0 K to 300 K for specific heat C p , entropy S , enthalpy H Њ T , Gibbs energy G Њ T , enthalpy function (H Њ T -H Њ 0 ) /T, and Gibbs energy function Ϫ(G Њ T -H Њ 0 ) /T. The enthalpy function H Њ T is the enthalpy at 101 kPa (1 atm) and temperature T , and similarly for the Gibbs energy. The reference function H Њ 0 is the enthalpy at 101 kPa and 0 K.
• Specific heat capacity C p and enthalpy, H Њ T -H Њ 0 , are both plotted as functions of temperature in Fig. 25 for ease of comparison. The entropy S and specific heat C p are both plotted in Fig. 26 for ease of comparison. Entropy is the integral over temperature of the specific heat function divided by temperature. Consequently, as the specific heat approaches a constant value at high temperatures (law of Dulong and Petit) the entropy tends to increase linearly with temperature.  From the specific heat (calorimetric) measurements of Table 7, the low-temperature Debye characteristic temperature ⌰ cal = ⌰ D as a function of temperature was also obtained, and values are listed in Table 7; the values are plotted in Fig. 29 for temperatures 0 K < T < 30 K. From a different set of specific heat measurements, Debye characteristic temperatures were obtained and plotted in Fig. 30 for temperatures 0 K < T < 100 K. From the specific heat measurements of Table 8, values for the Debye characteristic temperature ⌰ D were obtained and are plotted in Fig. 31 for temperatures 0 K < T < 300 K. These three different plots of ⌰ D cover somewhat different, but equally useful, ranges of temperature.

Mechanical Properties
• 2.7.1 Hardness, Tensile Strength Figure 32 displays Rockwell F hardness and tensile strength for the same specimens of pure silver, subjected to varying degrees of reduction by cold rolling at room temperature.
• 2.7.2 Ultimate Tensile Strength Ultimate tensile strength U of annealed silver is plotted in Fig. 33 for temperatures from 0 K to 300 K. The effects of duration and temperature of annealing were not explored in this work (applies also to Figs. 34 and 44).
Ultimate tensile strength (UTS ) is plotted in Fig. 34 as a function of temperature. The UTS has been normalized by dividing it by the fatigue strength, defined as the peak stress for fracture in 10 5 cycles. The fatigue strength rises to a peak as the temperature is lowered below room temperature to about liquid-nitrogen temperature (78 K), but then falls off at temperatures below 78 K. The effects of duration and temperature of annealing were not explored in this work (applies also to Figs. 33 and 44).
• 2.7.3 Tensile Stress vs Strain Figure 35 gives tensile stress as a function of percent strain for coldworked polycrystalline silver. The apparent intercept of the curve with the stress axis at 40 MPa is only an artifact; the original figure as published was quite small and the curve could not be resolved from the axis below this value. Figure 36 displays stress as a function of strain for different values of stress rate and overstrain, all for the same same specimen. The source paper gives no information on the rate of stress for curves 1 and 2. Tables V and VI of the original paper give the following information: for curve 1, the limit of proportionality was 17.9 MPa (1.3 ton /in 2 ), the 0.01 % proof stress was 45.5 MPa (3.3 ton /in 2 ), the elastic modulus was 69.7 GPa (10.1 Mpsi), the maximum applied stress was 51.9 MPa (3.76 ton /in 2 ), the strain at maximum stress was 10.0ϫ10 -4 , and the permanent strain was 1.34ϫ10 -4 . For curve 4: the maximum applied stress was 13.8 MPa (1 ton /in 2 ), the strain at maximum stress was 8.42ϫ10 -4 , and the permanent strain was 6.4ϫ10 -4 .

Elastic Moduli
Adiabatic elastic coefficients, or moduli, (commonly termed "constants" even though they vary with temperature and pressure) are listed in Table 9. Of the six quantities listed in the table, only three are independent, and these are usually taken to be C 11 , C 12 and one adiabatic elastic shear modulus, C 44 . Another elastic shear modulus is defined as C' = (C 11 -C 12 ) /2, the bulk modulus is B = (C 11 +2C 12 ) /3, and the longitudinal elastic modulus is C L = (C 11 +C 12 +C 44 ) /2. The elastic anisotropy is defined as C 44 /C ' = 2C 44 /(C 11 -C 12 ).  both time and log(time), but corresponding plots are not related by just a single mapping (linear to logarithmic). Careful study raises many unanswered questions, so the reader is cautioned to be careful in using the information from Fig. 1 of Ref. [128].

Flow Stress
The ratio T /G T of flow stress to shear modulus G , normalized to unity at T = 0, ( T /G T ) /( 0 /G 0 ), is plotted in Fig. 43 as a function of temperature. The subscripts T denote temperature dependence.
• 2.7.8 Fatigue Tension-compression fatigue is plotted in Fig. 44 as a function of number of cycles to fracture for test temperatures of 4.2 K, 20 K, 90 K, and 293 K. The effects of temperature and duration of annealing were not explored in this work (applies also to Figs. 33 and 34).

Yield Strength
Yield strength (defined as stress at 0.005 strain) is plotted as a function of temperature for specimens of high-purity (0.9997+) silver in Fig. 45. The specimens, of three different grain sizes (0.017 mm, 0.040 mm, and 0.250 mm), had been annealed at temperatures of 700 ЊC, 800 ЊC, and 900 ЊC, respectively.
In Fig. 46, tensile strength is plotted as a function of temperature and compared to the yield strength, for the same three grain sizes (related to annealing conditions) of Fig. 45.
• 2.7.10 Velocity of Dislocations Fig. 47 shows the velocities of dislocations moving in a <110 >{111} glide system as a function of stress , for specimen temperatures of 77 K and 300 K. There is a crossover at a stress of about 7 MN /m 2 ; above this stress the dislocation velocity is greater at 77 K, but below this crossover stress the velocity is greater at 300 K.

Displacement Cross Section
The displacement cross section as a function of electron energy required to displace a silver atom from a lattice site (in bulk silver) is graphed in Fig. 48. The cross sections were normalized by dividing by the displacement cross section at 1.35 MeV, the largest electron energy used. The associated threshold energy for displacement was 28 eV.

Miscellaneous
For the convenience of those working with cryogenic fluids, some useful thermodynamic properties are listed in Table 10. For commonly used cryogenic liquids at normal boiling points and for vapors at 101 kPa (1 atm), density, enthalpy, and specific heat are tabulated. Table 9. Adiabatic elastic coefficient for silver a Shear constants: C = C44 C ' = (C11 -C12)/2 Longitudinal modulus: CL = (C11 + C12 + 2C44)/2 Bulk modulus: B = (C11 + 2C12)/3 Young's modulus: E • The two elastic stiffness moduli C 11 and C 12 are plotted in Fig. 37 as functions of temperature (the changes in slope at low temperature are artifacts of the finite number of data points). Fig. 38 displays two adiabatic elastic shear moduli, C = C 44 and C ' = (C 11 -C 12 ) /2, plotted together as functions of temperature. The bulk modulus B and the longitudinal elastic modulus C L are plotted together in Fig. 39 as functions of temperature. All the elastic moduli are extremal at liquid-helium temperatures. Figure 40 shows the elastic anisotropy coefficient 2C 44 /(C 11 -C 12 ) as a function of temperature. It is minimal at 0 K.

Young's Modulus
Young's modulus E and its temperature derivative dE /dT are plotted in Fig. 41 as functions of temperature. The values for E and dE /dT were calculated from elastic-coefficient data.
• 2.7.6 Creep The creep coefficient ␣ is plotted in Fig. 42 as a function of temperature for an applied stress of 5.9 MPa. The creep strain is given by the expression ⑀ = ␣иln(␥t+1), where ␥ is a "time proportionality constant" and t is time. Values for ␥ were not given in the source paper. In Fig. 1   3.1 General Reviews [1]. Butts, Allison, and Coxe, Charles D., Silver: Economics, Metallurgy and Use, Robert E. Krieger, Malabar FL (1967).
Reviews history, sources and markets; extractive and refining processes; physical, chemical, mechanical properties (including bearings and electrical contacts) and behavior of silver bimetals and alloys; metallography of silver; silver batteries, brazing, catalysts, electroplating, and mirrors; silver in dentistry, medicine and photography; fabrication of articles (flatware, holloware) from silver and alloys; powder metallurgy, and migration of silver.
[2]. American Institute of Physics Handbook, Gray, Dwight E., Coordinating Editor, McGraw-Hill, NY (1957). †18 Sections on mechanics, acoustics, heat, electricity †19 and magnetism, optics, atomic and molecular †21 physics, and nuclear physics summarize physical †T5 properties of materials (for the vast majority of †T6 cases, by tabulation). Tensile strength and elongation (assume room temperature) are plotted as a function of amount of cold work (0 % to 74 %) and as a function of annealing temperature (100 ЊC to 700 ЊC).
The historical and experimental foundations of solid mechanics for metals, plastics and elastic materials are reviewed.
General physical and structural properties of the metallic elements are tabulated, along with information on processing, testing and inspection. This is a practical review of the elementary theories of specific heat, thermal expansion, and electrical and thermal conductivities, and of the properties of metals, alloys, and insulators at low temperatures.
The band structure of silver was calculated using the relativistic augmented-plane-wave (RAPW) method. A plot of density of states versus energy from 0 Ryd to 1.5 Ryd (1 Ryd = 13.6 eV) is given. The Fermi energy is approximately 0.444 Ryd above the muffin-tin zero, or 0.551 Ryd above the band bottom (⌫ 6 + ). The electronic density of states per atom at the Fermi level, N (E F ), is 3.4556 Ryd -1 . The band structure has been checked against optical experiments and overall agreement is very good.
A new, lower value for the optical mass m 0 of conduction electrons in pure silver is reported. The value m 0 = 0.85 m e was obtained for silver evaporated onto a sapphire substrate in an ultra-high vacuum of 6.7ϫ10 -8 Pa (5ϫ10 -10 Torr). After exposure of the silver film to air for 16 hours the optical mass was found to have increased to 0.87.
The increase is attributed to the development of a film of silver sulfide on the surface of the silver.
The theory of electrons in metals is reviewed. Electrical conduction and magnetic (Hall) effects in thin films and wires are discussed and compared with experimental results for Na wires. The anomalous skin effect is treated and the experimental high-frequency surface resistance of silver at low temperatures is graphed, as well as the theoretical absorption coefficient of silver at 4.2 K.
The de Haas-van Alphen frequencies for the symmetry-direction orbits ( <111 > belly, <111 > neck, <100 > belly, <100 > rosette, and <110 > dogsbone) and the turning point of the (110) planebelly turning point were measured for Ag, Au, and Cu. The data are estimated to have an "absolute precision" of 20 ppm (parts in 10 6 ) and a "relative precision" of nearly 1 ppm. Comparisons with other measurements from the literature are tabulated.

Fermi Surface
[23]. Cracknell, A. P., The Fermi Surfaces of Metals: A Description of the Fermi Surfaces of the Metallic Elements, Barnes & Noble, NY (1971) .
The neck radius of the Fermi surface of silver is 0.14 (with the radius of the spherical free-electron Fermi surface as the unit), and is smaller than the radii of the Fermi surfaces of both Cu and Au.
The independent-particle picture for electrons, simple equilibrium phenomena, dynamics of electrons in electric and magnetic fields, equilibrium phenomena in magnetic fields, simple transport phenomena, and transport phenomena in magnetic fields are reviewed in separate sections. A value of ␥ = 0.646 mJ /(K 2 иmol) for silver is given. Table 4 is a comprehensive list of references for band-structure calculations, the de Haas-van Alphen effect, the anomalous skin effect, galvanomagnetic effects, cyclotron resonance, size effects, and ultrasonics. There are 23 references for the above properties of silver.
[25]. Damask, A. C., and Dienes, G. J., Point Defects in Metals, Gordon and Breach, NY (1963). †48 This work reviews the general theory of: (1) thermodynamics, energies, mobility, and production of defects; (2) theory of annealing; (3) analysis of annealing curves; (4) influence of point defects on physical properties; (5) experimental investigations of the effects of quenching, irradiation, annealing and diffusion. The energy of formation E F of (monovacancy) defects is given as 1.09 eV. The change in electrical resistivity per electron flux (number of incident particles per cm 2 ) is plotted as a function of energy of incident electrons, and then replotted as the displacement cross section for moving a silver atom from a lattice site by incident electrons, again as a function of energy of incident electrons. The threshold energy E D for displacing a silver atom from its lattice site is given as 28 eV, and the resistivity change per unit of defect concentration in "atomic percent" (mole fraction) ⌬ F is 1.4 ⍀иcm /( at / o ). The damage rate under fastneutron irradiation at 20 ЊC at a flux of 7ϫ10 11 cm 2 иs -1 is 3.35ϫ10 -11 ⍀иcm /h. The neutron exposure required to produce 1 at / Њ of vacancy-interstitial pairs is 1.05ϫ10 20 nvt of fast neutrons (n is number density of neutrons, v is velocity, t is time; nvt is neutron fluence, number per second per unit area).
Broom, T., Lattice Defects and the Electrical Resistivity of Metals, Adv. Phys. 3 (9), 26-83 (1954) The annealing of quenched and cold-rolled silver (purities of 99.9 % and 99.999 %) was studied in the range of temperature from -40 ЊC to +250 ЊC. Distinct recovery processes were found. For quenched silver, a recovery stage attributed to migration of vacancies was found between 50 ЊC and 130 ЊC, with an activation energy of 0.88 eV. For cold-rolled silver, a recovery stage attributed to interstitial migration was found between -40 ЊC and -10 ЊC, with an activation energy of 0.60 eV. For the cold-rolled specimen, a recovery stage found above 70 ЊC was thought to be due to rearrangement and annihilation of dislocations. [28]. Nadgornyi, E., Dislocation Dynamics and Mechanical Properties of Crystals, Prog. Matls. Sci., Vol. 31, J. W. Christian, P. Haasen , and T. B. Massalski, eds., p. 319, Pergamon, Oxford (1988). †47 This work reviews dislocation dynamics theoretically and experimentally. The velocity of dislocations moving along a <110 >{111} glide system at 300 K and 77 K is plotted (Fig. 6.21), for silver and dilute silver-based alloys (Ag with In, Sb, Sn). For the source of this particular plot see the paper by H. Suga and T. Imura (Ref. 29).
The density, distribution, velocity, multiplication and mobility of dislocations is studied by changing solute content and test temperature. The stress exponent m for the velocity of dislocations in pure silver is 2.8 at room temperature. Four plots of data for dislocation parameters are given.
[30]. Takamura, S., and Kobiyama, M., Dislocation Pinning in Al and Ag Alloys after Low-Temperature Deformation, Phys. Stat. Sol. (a) 95, 165-172 (1986). †6 Specimens of Ag and Al (99.999 % pure), and of dilute alloys of each metal, were torsionally deformed at 77 K. The electrical resistivities 0 after isochronal annealing were measured at 4.2 K. In silver annealing stages were observed at 110 K and 230 K to 250 K. The lower-temperature stage is interpreted as being due to migration of interstitial clusters or vacancy-type defects near dislocations, and the higher-temperature stage, to long-range migration of vacancies and dissociation of vacancy clusters. The elastic modulus changes at 110 K and 240 K.
Electrical resistivities of commercially available high-purity silver (and of alloys of Ag with Au, Pd and Pt) were measured from 2 K to 20 K. Matthiessen's rule did not hold at all temperatures. Possible dependence of resistivity on temperature as T 5 was investigated; most specimens showed very nearly a T 4 dependence on temperature below 10 K.
[42]. Barnard, B. R., and Caplin, A. D., 'Simple' behavior of the low temperature electrical resistivity of silver?, Communications Physics 2, 223-227 (1977). Electrical resistivity for the temperature range 1.2 K < T < 9 K is plotted for large single crystals of silver. The resistivity varies with temperature as T 4 between 1.2 K and 4 K. From 4 K to 9 K the resistivity varies more slowly than T 4 .

142.)
Electrical resistivity for the temperature range 1.2 K < T < 9 K is plotted for large single-crystal ingots of silver. Residual resistivity for Ag-Au alloys is plotted as a function of Au concentration, and for pure polycrystalline silver as a function of temperature for 40 K < T < 1200 K.
The observed T 4 behavior in the low-temperature (2 K to 6 K) resistivity of silver is shown to result from the scattering of electrons by both phonons and other electrons. The magnitude of the e-e scattering has been successfully calculated.
The unusual behavior of the resistivity of the noble metals is discussed in theoretical terms and compared with measurements of the resistivities of Ag, Au, and Cu for 2 K < T < 6 K.
The low-temperature electrical resistivity of Ag and Cu were measured over the temperature range 3 K < T < 16 K. These data together with other data from the literature are compared with two theories. The resistivity at low temperatures does not follow a well-defined power law; assuming ഠT n then n varies continuously from 6 to 4 as T varies from 1 K to 15 K. Electron-phonon (normal; Umklapp longitudinal and Umklapp transverse) contributions to the resistivity are tabulated for temperatures from 1 K to 15 K. Resistivities of Ag (RRR = 10 k) and Cu are plotted for this temperature range.
A condensed review of the theory of electronic conduction is followed by a survey of the dependence on temperature and composition of the electrical resistivity of binary alloys of 44 metals.
The contribution to electrical resistivity due to vacancies (extra-resistivity: ER) is calculated for Ag, Au, and Cu. The contributions to ER due to vacancies accompanying the presence of elements next to these three noble metals in the periodic table are tabulated. [56]. Khoshenevisan, M., Pratt, W. P., Jr., Schroeder, P. A., and Steenwyk, S. D., Low-temperature resistivity and thermoelectric ratio of copper and gold, Phys. Rev. B19 (8), 3873-3878 (1979).
Comparing this work with that of Barnard and Caplin (Refs. 42,43), and of Lawrence (Ref. 64), the authors verify that the resistivity of pure silver is given by: = 0 + BиT N , where N = 4, and B = 1.4ϫ10 17 ⍀иcm /K 4 . The crossover temperature, where scattering by electrons and phonons should make equal contributions to the resistivity, is 2.4 K. They conclude that there is strong evidence for a T 4 dependence in the low-temperature electrical resistivity of all the noble metals. Previous high-temperature measurements of the Lorenz function, combined with recent theory, confirm quantitatively the electron-electron scattering term seen by Khoshenevisan et al. (Ref. 56).
For pure polycrystalline silver the dependence of electrical resistivity on temperature follows -0 ϰ T 4 for T > 2 K, but at the lowest temperatures the resistivity obeys -0 ϰ T 2 и19 . The electrical resistivity is plotted for temperatures between 0.04 K and 7 K.
Electrical resistances of wires and strips of Ag and Au at 1.2 K < T < 4.2 K were measured, tabulated and plotted as reduced resistance values.
Accurate measurements of the electrical resistivity of silver from 3.5 K to 30 K are examined in the light of theoretical models of Klemens and Kamon, Mathur and Klemens based on phonon-assisted defect scattering. The low-temperature thermal resistivity is discussed in terms of the electrical-resistivity measurements and the Lorenz number. Two plots of thermal resistivity versus temperature are given for four sets of data.
Electrical resistivity and thermal conductivity were measured over the range from 2 K to 23 K and discussed in terms of theory. Results are tabulated and graphed.
With simultaneous torsion and extension applied to polycrystalline wires at 78 K, changes in specific resistivity as a function of strain are found to lie on a universal curve when average, rather than torsional, strain is taken as the independent variable. Results for different annealing temperatures are plotted and tabulated. [63]. Krsnik, R., and Babić, E., Influence of the anisotropy of electron-phonon scattering on the low temperature resistivity of normal metals, J. Phys. 39(8), Colloque C6-1052Colloque C6- -1053Colloque C6- (1978.
The phonon resistivities of Ag, Al, Au, Cu, Sn, and Zn are analyzed in terms of a T 5 variation with temperature and anisotropic electron-phonon scattering.
Electron-electron and electron-phonon contributions to the temperature-dependent resistivity of impure silver are graphed and tabulated. The experimental techniques for measuring electrical conductivity are reviewed and existing data are compared with theory.
The theory and experimental results of the electrical resistance of metals and alloys, and deformed and irradiated metals, as it varies with temperature, chemical impurities, physical state (vacancies, dislocations, or interstitials), applied pressure, magnetic field and dimensions of the specimen (size effect) are reviewed.
[70]. Nagata, S., Ogino, M., and Taniguchi Thin metal films, approximately 30 nm to 110 nm thick, of Ag, Au, Cu, Ni, Pd, and Pt were evaporated onto glass substrates at 77 K. The temperature dependences of the electrical resistivities of the films were measured and plotted for different processes of annealing at temperatures from 110 K to 330 K. Temperature coefficients of the resistivity are tabulated for those temperature ranges for which the resistivity changes are reversible.
The electrical resistivities of annealed and strained specimens of Ag, Al, and Pd were measured at temperatures below 40 K. The temperature-dependent resistivity component T is plotted as a function of 0 , on a log-log scale, for temperatures of 10 K, 13 K, 20 K, 30 K and 49 K.
Resistivity data from the literature for Ag and Cu are analyzed in terms of electron-electron Umklapp scattering processes.
The logarithmic derivative of resistivity is plotted as a function of temperature from 0.2 K to 1.5 K.
Previous data for the electrical resistivities of pure Ag, Au, and Cu have been reanalyzed to pin down the exponent(s) in the power-law dependence of the resistivities. Powers of 2, 4, or both, in the resistivity, were considered.
[76]. White, G. K., and Woods, S. B., Electrical and Thermal Resistivity of the Transition Elements at Low Temperatures, Phil. Trans. Roy. Soc. London A251, No. 995, 273-302 (1959) . Thermal and electrical resistivities were measured for 20 transition elements over the range from 2 K to 300 K. The ideal resistivities W i and i are tabulated and plotted. The three silver specimens were annealed, and of very high ("99.999 %") purity; the RRR values were 261,~1000, and~2000.
Theoretical results for the influence of the electron-phonon interaction on electrical resistivity are compared with data from the literature for pure Ag and Au and their dilute alloys at low temperatures.
Relative change of resistivity (RR ) with strain for wires of pure Ag, pure Cu, Ag-Cu alloys, Ag-Au alloys, and pure Au are graphed and tabulated. For pure Ag, RR increases as wires are strained at the liquid-air point to approximately 18 % extension; upon warming to room temperature, RR of the wires decreases by one third. Further strain at liquid-air temperature increases RR with a slope (RR vs strain) steeper than that before the wires were warmed.
The incremental change in electrical resistivity of silver from 1 K to 9 K, at various stages of cold work, is plotted. The value of the Lorenz number obtained is similar to the Sommerfeld value (2.45ϫ10 -8 (V/K) 2 ).
[80]. Barnard, B. R., Caplin, A. D., and Dalimin, M. N. B., The electrical resistivity of Ag and Ag-based alloys below 9 K, J. Phys. F: Metal Phys. 12, 719-744 (1982). Differential electrical resistivities of nominally pure silver and silver alloys from 1.2 K to 9 K, at various stages of cold work, are plotted. The Lorenz number is similar to the Sommerfeld value (2.45ϫ10 -8 (V/K) 2 ). Contributions to the low-temperature resistivity of silver, from impurities, dislocations, and surfaces, and from electron-phonon, electron-electron and Koshino-Taylor scattering, are estimated and tabulated.
[81]. Berghout, C. W., Increase of the resistivity of some face centered metals by cold-working, Physica 18(11), 978-979 (1952). †3 The areas of annealed wires of Ag, Au, Al, and Cu were reduced by cold working (drawing through dies at room temperature). The fractional increase in resistivities of Ag and Cu wires at room and liquid-air temperatures are plotted as a function of areal reduction to see whether change in resistivity due to cold working is independent of temperature (Matthiessen's rule applies). For Cu the change in resistivity is independent of temperature, but for silver deviations from Matthiessen's rule were found.
The theory and experiments dealing with the effect of lattice defects on electrical resistivity in pure metals and alloys are reviewed. Because deformed metals generally obey Matthiessen's Rule, the absolute change in resistivity ⌬ is generally independent of temperature. From an empirical point of view, however, it is convenient to use the relative change in resistivity ⌬ / 0 where 0 is the reference value of resistivity. The reference value is usually taken to be that measured at room temperature. For 10 % extension of silver, the relative change in resistivity at room temperature is 0.5 % (for a reference value 0 = 1.59 ⍀иcm at 20 ЊC).
After drawing of silver wire to a 98 % reduction in cross section, the relative increase in resistivity is 5 %. Values of ⌬ after irradiation of silver at -140 ЊC by 12 MeV deuterons are plotted as a function of deuteron flux.
Relative changes of resistivity with relative extension (strain) for Ag and Cu wires at 90 K are plotted; the silver wire was first drawn to 12 % extension, soft-annealed, and then further stretched to 17 %. Although the tension-deformation curve showed no discontinuity, the relative resistivity was reduced at the point where the deformation was halted for warming the specimen. The magnitude of the discontinuity in relative resistivity depends sensitively on the annealing temperature.
Damage in 14 high-purity metallic elements was caused by capture (98.2 % thermal in silver) of neutrons at 3.6 K, producing a change in electrical resistivity. Recovery was programmed with 60 heating pulses 5 min in duration. The isochronal recovery from damage by thermal-neutron capture is plotted as a function of temperature. The isochronal recovery for silver shows a largest peak at 30 K, a second-highest peak at 300 K, and two smaller peaks at 170 K and 140 K attributed to the effects of impurities.
[86]. Lucasson, P. G., and Walker, R. M., Production and Recovery of Electron-Induced Radiation Damage in a Number of Metals, Phys. Rev. 127(2), 485-500 (1962). †7 The residual electrical resistances of pure specimens of Ag, Al, Au, Cu, Fe, Mo, Ni, Ti, and W, at temperatures of 20 K, were changed by high-energy (0.5 MeV to 1.4 MeV) electron bombardment. The resistivity change per unit electron flux is plotted as a function of average electron energy for Ag, Al, Cu, Fe, and Ni.
Normalized recovery of electrical resistivity ⌬ / ⌬ 0 after electron bombardment is plotted as a function of reduced temerature for seven metals: Ag, Al, Au, Cu, Fe, Ni, and Mo. A table compares results from neutron damage with the results of electron bombardment, for the seven metals studied here.

Hall Coefficient
[88]. Alderson, J. E. A., Farrell, T., and Hurd, C. M., Hall Coefficients of Cu, Ag, and Au in the Range 4.2-300 ЊK, Phys. Rev. 174(3), 729-736 (1968). †13 The Hall coefficients for polycrystalline Ag, Au and Cu, two single crystals of Cu, and dilute alloys of Ag (with Au, Cd and Zn) were determined for 4.2 K < T < 300 K and in a magnetic field of 1.5 T. The temperature dependences of the Hall coefficients for each metal are plotted over the full temperature range of measurement.
The thermal and electrical conductivities of pure Al, Ag and Au were measured from 2 K to 31 K; the variation of Lorenz number with temperature was calculated from the data. [92]. Laubitz, M. J., Electron-Electron Scattering in the High-Temperature Thermal Resistivity of the Noble Metals, Phys. Rev. B2(6), 2252-4 (1970).
The electronic Lorenz function is plotted as a function of reduced temperature T /⌰ R for 1.3 <T / ⌰ R <5, where ⌰ R is the Debye temperature obtained from electrical-resistivity data. Reported deviations of the Lorenz function for the noble metals at high temperatures can be explained in terms of electron-electron scattering.  207-212 (1960) Resistance ratios for rhenium are tabulated and plotted. A Kohler plot ( ⌬ H, T / 0,T versus H /r T ) compares the magnetoresistance of Re as a function of reduced field to the magnetoresistances of 25 metals including silver.
The low-field magnetoresistances of single crystals of dilute Ag-and Cu-based alloys were measured at 1.5 K and 4.2 K. At low fields the fractional change in resistivity, ⌬ / , is given by Kи(B/) 2 , where K is independent of magnetic induction B and resistivity 0 . For longitudinal applied fields, K l = b + c + ld , where l depends only upon crystal orientation, b = 1.00, c = -0.97, and d = 1.67 (all three coefficients in units of (⍀иcm/T) 2 ϫ10 -16 ). For transverse fields, K t = b + t ()d , with the coefficients b and d taking the same values as those for the longitudinal fields; t () is a function of both crystal orientation and direction of B . The reduced Hall coefficient r , defined as the ratio of the actual Hall coefficient to that expected from the free-electron model, was calculated as 0.81, compared with an experimental value of 0.83.
Thin cylindrical films of 99.9999 % pure silver were formed by evaporating silver onto rotating glass fibers at room temperature. The magnetoresistance in the weakly localized regime was investigated in longitudinal magnetic fields (up to 0.3 T ) and at four different temperatures from 1.58 K to 4.2 K. At fields of less than 8 mT, oscillations in magnetoresistance were observed with a period of h /2e (the magnetic flux quantum), and confirms the predictions of a theory of Altshuler, Aronov and Spivak [Pis'ma Zh. Eksp. Teor. Fiz. 33, 101 (1981)]. This effect is due to quantum interference between pairs of multiply scattered electron waves. Plots (0 < H < 10 mT and 0 < H < 0.3 T) of magnetoresistance (R (H )-R (0))/R 2 (0) versus magnetic field H , are given for temperatures of 1.58 K, 2.13 K, 2.95 K and 4.2 K. (Note that the authors use a non-standard definition of magnetoresistance, and give units of Tesla for magnetic field H .) [96]. Dallaire, L., and Destry, J., Application of the Pippard-Klemens-Jackson theory of longitudinal magnetoresistance to a model of scattering on the Fermi surface of copper, silver and gold, Phys. Rev. B28(6), 2947-2956 (1983).
The ratio of "saturated" electrical resistivity to zero-field value, ϱ / 0 , as a function of temperature (0 K < T > 40 K) was calculated in the approximation of the small-angle scattering model. Calculated values for copper are shown in a plot to agree well with measurements by others. Using the present model, the resistivity ratios versus temperature were calculated for silver and gold, in the range 0 K < T > 40 K, for (100), (110), and (111) orientations, and plotted. There is a broad peak in the longitudinal magnetoresistances of silver at about 4 K for all three orientations.
The magnetoresistances of thin films of Ag, Au, and Fe-doped Au were measured at 1.35 K and 4.2 K in magnetic fields both parallel and perpendicular to the plane of the film. Magnetoresistance is plotted as a function of applied fields (parallel, perpendicular, and combined) from 0 T to 2 T. Guillon, F., Measurements of the low-temperature electrical resistivity of sintered silver powders by acoustic propagation in high magnetic fields, Can. J. Phys. 66, 963-8 (1988), see Ref. 51, Sec. 3.6 Electrical Resistivity.
Transport critical currents were measured as functions of magnetic fields applied parallel and perpendicular to the surfaces of superconducting composite tapes of high critical temperature, T C . Normalized resistances at 4.2 K are plotted as a function of applied field up to 9 T; this magnetoresistance is not presented as being definitive, but as a qualitatively useful result. The anisotropy factor, 1-I C (B//c )/I C (B Ќc ), and critical current densities are plotted as functions of parallel or perpendicular fields up to 2 T. The magnetoresistance and the magnetic-field dependence of the 1/f noise in thin films of silver (purity unspecified) were measured over the range of temperatures from 1 K to 25 K. Data were compared to theories of weak localization and of universal conductance fluctuations. Magnetoresistance is plotted for temperatures of 1.0 K, 4.4 K, 10 K and 25 K for magnetic fields of 10 -4 to 0.5 T.
Magnetoresistance of single crystals of Ag and Cu was measured in weak magnetic fields at 4.2 K. The relative change in resistance ⌬R /R is plotted as a function of the square of magnetic induction (B 2 ).
[103]. Pawlek, F., and Rogalla, D., The Electrical Resistivity of Silver, Copper, Aluminum and Zinc as a Function of Purity in the Range 4-298 K, Cryogenics 6(1), 14-20 (1966). Electrical and thermal magnetoresistances were determined at a maximum field of 4.46 T, over the range 80 K < T > 130 K. Results were interpreted in terms of "normal" and "magnetic" Wiedemann-Franz ratios for Ag, Au, and Cu. These "normal" and "magnetic" ratios were plotted. Parameters for the phonon conductivity are tabulated. Results were applied to data for conductivity of tungsten. The role of dislocations as scattering centers in the silver alloy system is interpreted with regard to the failure of silver to follow the universal curve of T versus 0 .
Deviations from Matthiessen's rule are comprehensively reviewed and sources of deviations are discussed: in substitutional alloys, after quenching, radiation damage, and plastic deformation, and in thin specimens.
Electrical resistivity at low temperatures is reviewed for both pure specimens and alloys of Ag, Al, Au, Cu, Mg, Pt, Sn and, Zn, in connection with deviations from Matthiessen's rule. The temperature-dependent part ( -0 ) of resistivity is plotted as a function of 0 , the residual resistivity, for variation of 0 by four to five orders of magnitude.

157(3), 552-560 (1967).
Relative departures ⌬/ 0 (⌬= tot -[ ph + ]) from Matthiessen's rule (MR) as a function of temperature were studied for strained Ag and Cu and for some dilute alloys of Ag and of Cu. At temperatures where the residual resistivity dominates, deviations from MR become as large as or greater than the ideal resistivity.
Deviations from Matthiessen's rule (DMR) are studied in the light of new measurements. The total DMR ⌬ has three parts: one (⌬ 1 ) due to impurity and phonon scattering, a second (⌬ 2 ) due to inelastic coherent scattering, and a third (⌬ 3 ) due to addition of charged impurities. Three plots of ⌬/ 0 versus temperature from 4 K to 300 K are given for strained silver, where the values of residual resistivity 0 were 0.009 ⍀иcm, 0.04 ⍀иcm, and 0.0259 ⍀иcm.
The temperature dependence of the electrical resistivity of strained and annealed specimens of "very pure" silver (values of RRR from 144 to 1600) was measured over the range 1.4 K < T > 295 K. The deviation from Matthiessen's rule was determined. A function for the ideal resistivity between 12 K and 23 K attributed to normal and umklapp scattering is given: i = 1.35ϫ10 -15 и T 4и88 и [1-0.008(⌰/T ) 5 ) и exp(⌰/T )]. Log ( i ) is plotted as a function of log (T ).
The electrical resistances of high-purity ("99.5" and "99.999 %") wires of Ag, Au, In, and Sn were measured at pressures up to 608 MPa (6 kbar), at temperatures between 4.2 K and 297 K. Results are graphed. The contributions from lattice vibrations and impurities to the total resistivity coefficient are obtained and analyzed in terms of Bloch-Grüneisen theory. At 4.2 K the resistance was measured up to a pressure of 810 MPa (8 kbar). Plots are given for resistance isobars as a function of temperature, resistance versus temperature, fractional change in resistivity per change in pressure versus temperature, fractional change in lattice resistivity per change in pressure versus temperature, and dln L /dlnV versus ( [113]. Hatton, J., Effect of Pressure on the Electrical Resistance of Metals at Liquid Helium Temperatures, Phys. Rev. 100(2), 681-684 (1955).
The effect of pressures up to "5000 kg/cm 2 " (490 MPa) on electrical resistances of polycrystalline "fine silver" (purity unknown), as well as Au, Cu, Pt, As, Sb, and Bi were measured at LHe temper-atures. All the metals investigated showed hysteresis in their resistance-pressure dependences. The resistance of silver rose nearly linearly with applied pressure (461 MPa max.) but due to hysteresis the resistance decreased approximately parabolically to a zero-pressure value P 01 slightly more than 0.5 % higher than the initial zero-pressure value P 00 . A second cycle to maximum pressure and back followed the same pattern of hysteresis.
The magnetic susceptibility of Ag(s) at 296 K is given as -0.181ϫ10 -6 emu/g.
The absolute magnetic mass susceptibility of silver is given as -0.1813ϫ10 -6 emu/g, at room temperature.
Gives the magnetic susceptibility of silver as -0.182ϫ10 -6 + 0.07ϫ10 -10 иT in unspecified units (by comparison to results of Henry and Rogers, one can assume that the units are mass susceptibility, emu/g).
A fracture-mechanism map for round bars of commercially pure silver tested in tension (nominal tensile stress/Young's modulus vs homologous temperature, T /T m ; T m = 1234 K) is given.
For polycrystalline Al, and single crystals of Ag, Cu, Al: a table and plot of influence of strain-rate and temperature on flow stress, plus plots of force versus strain for dislocations, are given.
[120]. Carreker, R. P., Jr., Tensile Deformation of Silver as a Function of Temperature, Strain Rate, and Grain Size, Trans. AIME, J. Metals 9, 112-115 (1957). †45 For silver of three grain sizes (0.014 mm, 0.040 †46 mm and 0.250 mm) and of "0.9997 + purity": yield strength, tensile strength, and percent elongation are plotted as functions of temperature from 0 K to approximately 1173 K; true stress is plotted as a function of true plastic strain for eight different temperatures: 20 K, 78 K, 195, K 205 K, 299 K, 473 K, 673 K and 873 K, after annealing temperatures of 973 K, 1073 K and 1173 K; flow stress at selected strains is plotted as a function of temperature for silver annealed at 973 K; and strain-rate sensitivity is plotted over the range 20 K to 1173 K. Correlations are also given with grain size.
Contrary to earlier belief that the yield point in mild steel should be attributed to its body-centered cubic (bcc) lattice structure, this work shows that non-ferrous metals or alloys, such as Ag, Ni, Mn-Ni, Be-Cu, or Duralumin, with face-centered cubic (fcc) structure, can show yield points. The yield strain of "standard silver" (of undefined composition) was found to be 1.5 % for a specimen quenched from 750 ЊC, strained 6 %, then tempered for 1 h at 175 ЊC. Specimens quenched and tempered, but not strained, showed no yield point.
Stress is plotted as a function of elongation to 28 %.
Change of electrical resistivity with tensile strain, for different annealing temperatures; torsional stress-strain curves for three different temperatures, and torsional shear stress versus square root of strain are plotted.
[128]. Koval, V. A., Osetski, A. I., Soldatov, V. P., and Startsev, V. I., Temperature Dependence of Creep in F.C.C. and H. C. P. Metals at Low Temperature, Advances in Cryogenic Engineering, Vol. 24, Plenum, NY (1978) pp. 249-255. †42 Curves are given for strain versus time (creep) at low temperatures for Ag and Al, Cu, Cd, and Zn. Low-temperature strain ⑀ is related to a temperature-dependent coefficient ␣ :⑀ = ␣ ln (␥ It+1). The temperature dependence of ␣ is shown for temperatures from 1.2 K to 80 K. Creep strain versus log time is plotted for constant stress and different (undefined) temperatures. In the authors' Fig. 1, creep is plotted as functions of both time and log(time), but corresponding plots are not related by just a single mapping (linear to logarithmic). Careful study raises many unanswered questions, so the reader is cautioned to be careful in using the information from Fig. 1 823-27 (1976).
Reports the changes in the distribution and density of dislocations in single crystals of silver (purities: series 1, 99.99 %; series 2, 99.999 %) subjected to stress reversal at small strains. The dislocation density was reduced from 10 6 /cm 2 to the order of 10 4 to 10 5 /cm 2 by thermal cyclic annealing in vacuum for about ten days. Pictures of etch-pit configurations are shown. One plot of shear stress (in compression, then in tension, then again in compression) versus cumulative shear strain is given (Spec.  203-211 (1957). †33 The fatigue of Ag, Al, Cu, and Au has been mea- †34 sured at 4 K, 20 K, 90 K and 293 K as a function †44 of number of cycles to fracture. The fatigue strength improves considerably as the temperature is reduced. The ultimate tensile strength is given as a function of temperature, and is also shown normalized by the fatigue strength.
Yield strength, tensile strength, percent elongation, true stress, true strain, flow stress and strain-rate sensitivity are reported for annealed silver of three grain sizes and "0.9997 purity," over the range 20 to 1173 K. Correlations are also given with grain size.
[143]. Brandenberg, W. M., The Reflectivity of Solids at Grazing Angles, Measurement of Thermal Radiation Properties of Solids, NASA SP-31, Joseph C. Richmond, ed., NASA, Wash., DC (1963) pp. 75-82. †20 An integrating-sphere reflectometer is described for measurement of reflectivity of imperfectly diffuse specimens. Correct operation was checked by measurements on black glass and a platinum mirror. Reflectivities of Ag, Al, Au, and stainless steel were measured and graphed for angles from 0Њ to 90 o .
The total hemispherical absorptivity of electroplated silver at 75.8 K was measured as a function of blackbody-radiation (BBR) temperatures from 268 K to 367 K. Over this range the total hemispherical absorptivity decreases with increasing BBR temperature. The normal absorptivity increases with increasing mean wavelength.
In an introduction to a symposium session, the thermal radiative properties of solids at 0 K < T < 200 K, together with the experimental methods used to obtain such properties, are briefly reviewed. A short but wide-ranging bibliography is included.
[ Classical electromagnetic theory is reviewed, along with the optical properties of noble, alkali and divalent metals, Al, and liquid metals. For silver, dispersion relations: conductivity (nk ), (1-⑀ ) = (n-k 2 ) and the optical "constants" n and k are plotted as functions of wavelength for 0.2 m < < 1.5 m.
Data are tabulated for emittances (hemispherical total; normal total; normal spectral), reflectances (normal spectral; angular spectral, absorptances (hemispherical integrated; normal spectral; angular spectral; normal solar), and transmittance (normal spectral). For most of the quantities indicated above the data are also plotted. Because of the paucity of data for silver, no recommended values are given.
The theory of the optical properties of metals in the infrared agrees much better with experimental data when the anomalous skin effect is considered, and diffuse, rather than specular, reflection of electrons at the metallic surface is assumed. Data for absorptivity at normal incidence at 17 ЊC and -188 ЊC are given for silver foil 14 m thick.
Absorption of EM radiation by silver at LHe temperatures is treated.
A layer of superconductivity has been induced in multiple-layer thin-film specimens of silver on a superconducting lead-bismuth alloy by a proximity effect. The alloy was made with a mole ratio of 5 parts Bi to 100 parts Pb+Bi. A layer of Pb-Bi alloy, 100 nm thick, was deposited on specimens of Ag ranging in thickness from 53 nm to 200 nm. For triple-layer specimens, 2 layers of 200 nm Pb-Bi alloy were sandwiched with specimens of Ag ranging from 80 nm to 500 nm thick. The energy gap at 0 K is e 0 = 1.76 k B иT c ; T c is not explicitly defined but seems to be the superconducting transition temperature for the substrate.
By a variational method, for two different pseudopotentials (Moriarty; Nand et al.), the electronic contributions to the low-temperature thermal conductivities of Ag and Cu were calculated. The temperature dependences of thermal resistivity are plotted for Ag and Cu for temperatures from 1.5 K to 15 K.
[156]. Cezairliyan, A., and Touloukian, Y. S., Correlation and Prediction of Thermal Conductivity of Metals Through the Application of the Principle of Corresponding States, Advances in Thermophysical Properties at Extreme Temperatures and Pressures, Serge Gratch, ed., ASME, NY (1965) pp. 301-313. Values for thermal conductivities of 22 metals were shown to be correlated by universal relations based on reduced values k * and T* of thermal conductivity and temperature. For cryogenic temperatures the reduced values are defined by k * = k/k m and T * = T/T m , where k m is the maximum value of conductivity and T m is the corresponding value of temperature. For moderate and high temperatures, the reduced values are defined by T * = T/T ⌰ and k * = k/k ⌰ , where T ⌰ is the Debye characteristic temperature and k ⌰ is the thermal conductivity at the Debye temperature. Recommended values for thermal conductivities of Ag, Au, and Cu are plotted as functions of temperature and of an impurity parameter ␤ . This parameter is defined by ␤ = 0 /L , where 0 is the residual electrical resistivity and L is the Lorenz number, 2.45ϫ10 -8 V 2 /K 2 . [158]. Ehrlich, A. C., and Schriempf, J. T., The Temperature Dependent Thermal and Electrical Resistivity of High Purity Silver From 2 to 20 K, Solid State Commun. 14, 469-473 (1974) [159]. Gerritsen, A. N., and Linde, J. O., Thermal Conductivity of Some Dilute Silver Alloys; Physica 22, 821-831 (1956).
Thermal conductivities were measured for pure silver and dilute alloys of silver with Mn, In, and Pb. The thermal resistivity was found to increase faster than linearly with small concentrations of impurity. A plot is given for thermal conductivity from 1 K to 90 K for one specimen of pure silver at three stages of annealing and for one specimen of very pure silver from G. K. White (Ref. 171).
The thermal conductivity was determined by a calorimetric method. The data are given as mean conductivities for temperature differences over the range from 10 ЊC to 97 ЊC or over the range from 15 ЊC to 98 ЊC. [162]. Kannuluik, W. G., The Thermal and Electrical Conductivities of Several Metals Between -183 and 100 ЊC, Proc. Roy. Soc. (London) A141, 159-168 (1931).
[163]. Klemens, P. G., Thermal Conductivity of Solids at Low Temperatures, Encyclopedia of Physics, Vol. XIV: Low Temperature Physics I, Springer, Berlin (1956) pp. 198-279. This work reviews the theory of thermal conductivity of dielectric solids, of metals and alloys (both electronic and lattice components), and of superconductors.
Precise measurements of the thermal conductivity and electrical resistivity of high-purity (RRR=1050) silver are reported for the range of temperatures from 80 K to 350 K. Values reported here agree closely with measurements by White and Woods (Ref. 76). A dip in the ideal Wiedemann-Franz ratio for silver has not been found for Au or Cu; no explanation for the dip is offered by the authors.
The thermal conductivity of silver (99.99 % pure) was measured in the range 2 K to 40 K. An equation is given for the thermal resistance R = ␣T 2 +␤ /T , with the electron-phonon scattering term ␣ = 9.0ϫ10 -5 and the impurity-scattering term ␤ = 1.6 (conductivity units of W/(cmиK)). A conductivity maximum of approximately 9 W/(cmиK) is observed at 20 K. The maximum is rather broad, possibly indicating the presence of a large impurity scattering term.
Article I, pure silver, covers transport properties of 99.9999 % pure silver. Deviations from Matthiessen's Rule are discussed. The ideal thermal resistivity of silver is plotted for 4 K < T < 29 K. The thermoelectric power of 6 specimens of pure silver is plotted for temperatures from 2 K to 10 K.
The thermal conductivity and "ideal" thermal resistance of silver are plotted for temperatures from 1 K to 140 K. The ideal resistance R i is given by R i = 2.86ϫ10 -5 T 2и3 (unannealed) and R i = 1.06ϫ10 -5 иT 2.5 (annealed).
White, G. K., and Woods, S. B., Electrical and Thermal Resistivity of the Transition Elements at Low Temperatures, Phil. Trans. Roy. Soc. London A251(995), 273-302 (1959) 190-205 (1953). †9 The thermal resistances of 27 pure metals were measured in magnetic fields up to 0.4 T. Ag (polycrystalline; purities of 99.99 % and 99.999 %), Cd, Ga, In, Pb, Sn, Tl and Zn showed appreciable magnetothermal resistance (MTR). The effect was always greater for transverse applied fields. The MTR increased quadratically with temperature at very small fields, and then rose linearly in larger fields. The ratio of W H (thermal resistance in magnetic field, H ) versus W 0 (thermal resistance in zero field) is plotted versus applied magnetic field.
Electrical and thermal magnetoresistances were determined at a maximum field of 4.46 T, over the range 80 K < T < 130 K. Results were interpreted in terms of "normal" and "magnetic" Wiedemann-Franz ratios for Ag, Au, and Cu. These "normal" and "magnetic" ratios were plotted. Parameters for the phonon conductivity are tabulated. Results were applied to data for conductivity of tungsten.
Data for contact resistance of an electron-beam weld, as measured by electrical resistance, of silver to silver are given for measurements at 4.2 K; information on electrical resistance of a threaded contact between Ag and Pt is also given. Actual thermal contact resistance, in terms of temperature difference, is not given.
Thermal expansion coefficients of twelve materials, including Ag, measured from 103 K to 283 K, are tabulated and graphed.
The temperature dependences of dilatation ␦ and coefficient of linear thermal expansion, ␣ , are tabulated over the range 17 K to 100 K for Ag, Al, Au, Cu, Fe, and Ni; some data are plotted (but not for Ag). The Grüneisen constant ␥ is plotted as a function of reduced temperature T /⌰ D .
The Nernst-Ettinghausen (NE) effect is the appearance of a transverse electric field E y in a metallic conductor subjected to mutually perpendicular magnetic induction B z and temperature gradient ѨT /Ѩx: E y = QиB z и ѨT/Ѩx. The isothermal NE coefficient Q i , measured over the range 4 K < T < 200 K, shows a strong peak at about 16 K. Thermal conductivity in zero magnetic field and in B=0.885 T , and absolute thermopower of silver, are plotted for temperatures from 4 K to 78 K. Sondheimer's two-band model is found to predict a phonon-drag contribution to the thermopower of the wrong sign. The solubility of oxygen in silver is said to be proportional to the square root of pressure, and is a minimum at 400 ЊC. There is no simple relation that models the dependence on temperature. It is suggested that above 400 ЊC the oxygen is present as atoms.
Some low-temperature properties (density, electrical resistivity, thermal conductivity and specific heat) of coin metal (90.3 % Ag, 9.7 % Cu, by mass) are tabulated. Direct measurements of magnetothermal conductivity for T < 20 K are included.