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

Diffusion in regular and disordered lattices

For complex electrical networks, we can take advantage of this result to simply our network. When we raise the resistance of edges, the effective resistance in our modified graph is an upper bound on the effective resistance of – and a lower bound on the effective conductance of – the original graph. When we lower the resistance of edges, the effective resistance in our modified graph is a lower bound on the effective resistance of – and an upper bound of the effective conductance of – the original graph.

Random walks on lattices

Since approximately 1950 an increasing portion of experimental solid state physics research has been concerned with studying defects in crystals. This trend might appear to be a rather belated recognition that most of the materials we come into contact with have a random structure. In fact the theoretical understanding and controlled preparation of compounds with defects or random structure has been very slow in developing. The present paper examines and reviews our knowledge of the lattice vibrations associated with defects. The coverage is extremely broad, as shown by the table of contents. It includes localized and resonant modes of isolated defects as well as the modes in highly disordered mixed crystals and glasses. It is primarily a review of experimental work but theoretical results are included where the latter explain or predict significant features. In order to be self-contained several sections of the paper deal entirely with theoretical matters. There is a chapter on explicit solutions of the linear chain vibration problem and a short chapter on Green's function methods. The review emphasizes the infrared absorption and Raman scattering of defects. This is simply because other techniques have not yielded nearly so much information. Neutron scattering and electron tunneling are referred to only where they have shed light on certain systems. Extensive tables of defects and mode frequencies are included for each type of solid. The major solids which are reviewed include semiconductors, ionic compounds, organic compounds, and amorphous insulators.

Optical studies of the vibrational properties of disordered solids

Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)

Laser calorimetric or thermal rise techniques are useful for the determination of very low absorption coefficients in solids. A number of improvements in this technique are described of which the most important is a means of separating surface and bulk absorption. These techniques have been applied to study alkali halides in the ir but are applicable where laser sources of sufficient power are available.

Measurement of very low absorption coefficients by laser calorimetry.

On the basis of a nearest-neighbor random-walk model for exciton propagation, we calculate the timedependent luminescent emission from an organic phosphor containing exciton traps. For the situation in which the traps themselves also emit a characteristic luminescence, we also calculate the time dependence of that luminescence. The results agree qualitatively with recent measurements of the time dependence of the fluorescence of tetracene-doped anthracene; detailed comparison yields reasonable values for the lifetime of each phosphor and for the time the anthracene singlet exciton spends on one site. There is no need to postulate a long-range energy-transfer mechanism.

Luminescent Emission from an Organic Solid with Traps

Sum Rule for Lattice Vibrations; Application to Forces in Diamond Structures

We consider random walks on simple cubic lattices containing two kinds of sites: ordinary ones and ``traps'' which, when stepped on, absorb the walker. We study two related problems: (a) the probability of returning to the origin and (b) the situation in which the particle can meet its end, not only by absorption at a trap, but also by a process, called spontaneous emission, which has a constant probability per step. In problem (b), we ask for the probability that emission, rather than absorption, occurs. The solution to (a) is known for 1 dimension, and given here for the 3‐, 4‐, … dimensional cases; the 2‐dimensional case remains unsolved. The solution to (b) is known for the 1‐, 3‐, 4‐, … dimensional cases; we give it for 2‐dimensional case.

Random Walks on Lattices with Traps

The concentrations of the various radioactive decay products of radium in the air have been determined by observing the beta activity (1) of pieces of filter paper through which air has been passed, and (2) of chemically separated isotopes obtained from such filter papers. The relative amounts of long-lived and short-lived products found indicate that, besides the radioactive decay of the substances, some other process which removes radioactive particles from the air is active. The mean life of the particles with respect to this removing process is found to be about 10 days. The hypothesis that this process is the capture of the radioactive particles by rain droplets (clouds) is consistent with measurements made of the radioactive content of rain-water. On the other hand, the removal of the ionized radioactive particles by the electric field of the earth leads to a much smaller effect.

Herbert B. Rosenstock

Papers

Measurement of very low absorption coefficients by laser calorimetry.

Luminescent Emission from an Organic Solid with Traps

Sum Rule for Lattice Vibrations; Application to Forces in Diamond Structures

Random Walks on Lattices with Traps

On the natural radioactivity in the air