We review the methods which have been developed over the past several years to determine the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys"). We begin with a survey of the various elementary excitations and put the dynamics of electrons, phonons, magnons, and excitons into one common descriptive Hamiltonian; we then review the use of double-time thermodynamic Green's functions to determine the experimental properties of systems. Next we discuss these aspects of the problem which derive from the statistical specification of the microscopic parameters; we examine what information can and cannot be obtained from averaged Green's functions. The central portion of the review concerns methods for calculating the averaged Green's function to successively better approximation, including various self-consistent methods, and higher-order cluster effects. The last part of the review presents a comparison of theory with the experimental results of a variety of properties---optical, electronic, magnetic, and neutron scattering. An epilogue calls attention to the similarity between these problems and those of other fields where random material heterogeneity has played an essential role.

The theory and properties of randomly disordered crystals and related physical systems

Publisher Summary The physical properties given in this compilation are listed in the accompanying tabulation according to the table given in this chapter in which they can be found. Values for two quantities that have frequently been used in alloying theory studies, viz., the metallic radius and electronegativity, are not included here. Other properties, such as magnetic susceptibility, electrical resistivity, and thermal conductivity, which are generally of interest to those studying solids, are not included in this compilation primarily because such properties do not appear to be important in alloying theory. The derived properties and some interrelationships of the physical properties are examined and discussed in this chapter. Those subjects which also appear in tabulated form are listed in the accompanying tabulation. The initial goal in our alloy-theory program was a set of Gruneisen constants and a set of size factors. To calculate these quantities, almost all of the physical properties given herein are needed in the computations. Some of the other derived quantities are not only intermediate steps in the computations of the Gruneisen constant and size factor, but also serve as checks on the consistency of initial data (i.e., the measured physical properties of the elements). Some of the “constants” of the elements—that is, the Griineisen constant, entropy of fusion, etc., which have been derived for a few elements and then generally assumed to apply to all elements—are examined to see if this generalization is valid.

Physical Properties and Interrelationships of Metallic and Semimetallic Elements

It is shown rigorously that the paramagnetic state of an electron gas is never the Hartree-Fock ground state, even in the high-density---or weak-interaction---limit. The paramagnetic state is always unstable with respect to formation of a static spin density wave. The instability occurs for spin-density waves having a wave vector $Q\ensuremath{\approx}2{k}_{F}$, the diameter of the Fermi sphere. It follows that the (Hartree-Fock) spin susceptibility of the paramagnetic state is not a monotonic decreasing function with increasing $Q$, but rather a function with a singularity near $Q=2{k}_{F}$. Rather convincing experimental evidence that the antiferromagnetic ground state of chromium is a large-amplitude spin density wave state is summarized. A number of consequences of such states are discussed, including the problem of detecting them by neutron diffraction.

Spin Density Waves in an Electron Gas

Electrons in transition metals

The three elastic constants of tantalum, tungsten, and molybdenum have been measured in the temperature range between 4.2 and 300\ifmmode^\circ\else\textdegree\fi{}K. The directly measured quantities ${C}_{L}$ and $\frac{1}{2}({C}_{11}\ensuremath{-}{C}_{12})$ for molybdenum and the computed bulk moduli of all three elements show an anomalous temperature dependence. These data along with published results for other metals generally show a difference in temperature dependence of the bulk modulus between $3d$ and $4d\ensuremath{-}5d$ metals. Overlap contributions to the elastic constants are found to favor an antiferromagnetic ordering. The Debye characteristic temperatures calculated from elastic constants are found to be in substantial agreement with results obtained from calorimetry.

Elastic constants of tantalum, tungsten, and molybdenum

Results of measurements of the magnetic susceptibilities of a series of binary alloys are presented. The systems investigated were Pd-Rh, Pt-Ir and Pt-Au. Certain similarities between the alloys of palladium alloyed with silver or rhodium and those of platinum alloyed with gold or iridium are apparent. In addition, the specific heats of palladium-rhodium and platinum-gold alloys have been measured in the liquid helium temperature range and the electronic heats and Debye temperatures computed. The overall picture presented by the results is consistent with the existence of an unfilled d band, having approximately 0.6 holes per atom both for palladium and platinum, which may be filled by alloying with the element of next highest atomic number. Recent work has shown, however, that it is unlikely that the band shape is unchanged on alloying and therefore the calculation by which the number of holes is obtained is probably invalid.

The Thermal and Magnetic Properties of Some Transition Element Alloys

Details are given of the construction of a calorimeter to operate in the range 1-2 to 4.2 'K. Analyses of the results obtained for several pure metals (Cu, Ag, Pt, Al, Ni, Fe, Co) are given for various weighting schemes in reducing the results. The effects of errors due to the possible presence of a Kapitza boundary layer affecting the temperatures deduced from vapour pressure bulb measurements are considered. The results are analysed with a view to detecting further lattice contributions to the specific heat (T5 terms) and (in the case of ferromagnetics) spin wave contributions. The nuclear contribution and the effective field is evaluated for cobalt. Over the past ten years a considerable amount of work has been carried out at Leeds on the low temperature specific heat and related properties of metals and alloys. The last report on specific heat measurements from these laboratories (Budworth, Hoare & Preston i960) was concerned with work carried out in a combined liquefier-calorimeter. The present report is concerned with a later calorimeter which has enabled the experimental precision to be improved by an order of magnitude; this has been achieved by refinements in technique, particularly thermometry, following the implementation of the 1958 4He temperature scale. It might be mentioned that this improvement in precision has come at an opportune time, coinciding as it does with an upsurge of interest in dilute alloys. In investigations of the electronic properties of metals and alloys, measurements of specific heat in the liquid helium temperature range are of particular importance. In this range the temperature variation of each contribution to the specific heat can usually be expressed by a simple power law, and the magnitude of these contributions can be determined by least squares analysis of the specific heat measurements.

The low temperature specific heats of some pure metals (Cu, Ag, Pt, Al, Ni, Fe, Co)

The low-temperature, 2 to 4$\cdot $2 degrees K specific heats of a series of nine palladium-silver alloys and the pure metals have been measured. From the results the variation with concentration of $\theta _{D}$, the Debye temperature, and $\gamma $, the electronic heat coefficient, have been determined. According to the collective electron approach the height of the band at the Fermi level is proportional to $\gamma $. This enables an approximate band shape to be deduced on the assumption that alloying leaves the shape substantially unchanged. The band obtained in this way is considered in relation to earlier work on the magnetic susceptibilities of this alloy series.

The Low-Temperature (2 to 4$\cdot $2 degrees K) Specific Heats of Palladium-Silver Alloys

The magnetic susceptibilities of a series of palladium-silver alloys from pure palladium to approximately 50 atomic% silver, have been measured over a range of temperature from 20 to 290 degrees K. The specific heats of a similar series of alloys have been measured in the temperature range 11 to 20 degrees K, and the results used to deduce the electronic specific heats. Owing to the variation of the Debye characteristic temperature with temperature these values are not considered to be of high precision beyond 15% silver content. Making a reasonable correction, however, it has been possible to deduce the band form for palladium. By extrapolation of the susceptibility results to 0 degrees K the variation of Stoner's exchange interaction parameter with concentration has been investigated over a limited range.

The Thermal and Magnetic Properties of Palladium-Silver Alloys

An apparatus is described which makes possible the rapid determination of the temperature variation of the magnetic susceptibilities of small solid specimens. The deflexions of a beam of light passing through a Sucksmith balance are recorded photographically. The temperature of the specimen, measured by a Au-Co/Ag-Au thermocouple in contact with it, is initially reduced to 20° K and deflexions are recorded at small intervals of temperature as the temperature of the specimen rises to 290° K. The whole temperature range can be covered in 4 h. The mean deviation of single measurements from the best smooth curve passing through them is less than 0.2 % of the mass susceptibility of the specimens. The apparatus has been used to measure the temperature variation of the susceptibilities of spectrographically standardized specimens of platinum, rhodium and palladium. Measure­ments on three pure palladium specimens show a maximum in the susceptibility temperature curve at 80° K. The experimental results for platinum show a change in the curvature of the susceptibility temperature curve below 90° K, the susceptibility increasing more rapidly as the temperature decreases. The experimental results for rhodium show a regular increase of susceptibility with increase of temperature. The results for palladium are discussed in relation to the collective electron treatment of metals, and assuming a parabolic distribution of energy states in the d band, estimates are made of the effective Fermi-Dirac degeneracy temperature θ F , the number of holes per atom, q , and the exchange interaction energy θ9 . The estimated values are θ F = 1020° K, q = 0.29, θ9 = 554.0° K. Similar estimates are made in the case of platinum, the values being θ F =1750° K, q = 0.25, θ9 = 770° K.

F. E. Hoare

Papers

The Thermal and Magnetic Properties of Some Transition Element Alloys

The low temperature specific heats of some pure metals (Cu, Ag, Pt, Al, Ni, Fe, Co)

The Low-Temperature (2 to 4$\cdot $2 degrees K) Specific Heats of Palladium-Silver Alloys

The Thermal and Magnetic Properties of Palladium-Silver Alloys

The Magnetic Susceptibilities of Platinum, Rhodium and Palladium from 20 to 290 degrees K