Westinghouse Electric

Abstract: In this chapter, we will restrict our attention to the ferrites and a few other closely related materials. The great interest in ferrites stems from their unique combination of a spontaneous magnetization and a high electrical resistivity. The observed magnetization results from the difference in the magnetizations of two non-equivalent sub-lattices of the magnetic ions in the crystal structure. Materials of this type should strictly be designated as “ferrimagnetic” and in some respects are more closely related to anti-ferromagnetic substances than they are to ferromagnetics in which the magnetization results from the parallel alignment of all the magnetic moments present. We shall not adhere to this special nomenclature except to emphasize effects, which are due to the existence of the sub-lattices.

Abstract: In this paper is described a model for polaron motion which incorporates, in simplified form, the principal physical features of the problem. The (crystalline) medium, within which a single excess electron (or hole) is contained, is taken to be a one-dimensional molecular crystal, consisting of diatomic molecular sites; each site possesses a single vibrational degree of freedom, represented by the deviation, x n , of its internuclear separation from equilibrium. The motion of the electron in this medium is treated by a tight-binding approach, in which the wave function is represented as a superposition of local “molecular” functions, φ (r − n a, x n ). In line with the x n dependence of the δ's, it is also assumed that the “local” electronic energy, E n , (which, in the conventional tight-binding theory, has one and the same value for all sites) depends (linearly) on x n . This dependence gives rise to electron-lattice interaction; alternatively, it may be regarded as removing the electronic translational degeneracy, characteristic of the undistorted crystal, and thereby providing the possibility for electron trapping. On the basis of the above-described model, the zeroth order adiabatic treatment of the polaron problem is developed. For values of the parameters such that the linear dimension of the polaron is large compared to a lattice spacing (“large” polaron), an exact solution is obtained; the correspondence between it and Pekar's zeroth-order solution is established. The conditions under which the size of the polaron becomes comparable to a lattice spacing (“small” polaron) are discussed. Finally, by way of exhibiting the relationship of the molecular-crystal concept to the real situation, a description is given of an alternate molecular-crystal model which, in the case of the large polaron, is completely equivalent to the continuum-polarization model of conventional polaron theory.

Abstract: A phenomenological model is developed to facilitate calculation of lattice thermal conductivities at low temperatures. It is assumed that the phonon scattering processes can be represented by frequency-dependent relaxation times. Isotropy and absence of dispersion in the crystal vibration spectrum are assumed. No distinction is made between longitudinal and transverse phonons. The assumed scattering mechanisms are (1) point impurities (isotopes), (2) normal three-phonon processes, (3) umklapp processes, and (4) boundary scattering. A special investigation is made of the role of the normal processes which conserve the total crystal momentum and a formula is derived from the Boltzmann equation which gives their contribution to the conductivity. The relaxation time for the normal three-phonon processes is taken to be that calculated by Herring for longitudinal modes in cubic materials. The model predicts for germanium a thermal conductivity roughly proportional to ${T}^{\ensuremath{-}\frac{3}{2}}$ in normal material, but proportional to ${T}^{\ensuremath{-}2}$ in single-isotope material in the temperature range 50\ifmmode^\circ\else\textdegree\fi{}-100\ifmmode^\circ\else\textdegree\fi{}K. Magnitudes of the relaxation times are estimated from the experimental data. The thermal conductivity of germanium is calculated by numerical integration for the temperature range 2-100\ifmmode^\circ\else\textdegree\fi{}K. The results are in reasonably good agreement with the experimental results for normal and for single-isotope material.

Abstract: The one-dimensional molecular-crystal model of polaron motion, described in the preceding paper, is here analyzed for the case in which the electronic-overlap term of the total Hamiltonian is a small perturbation. In zeroth order—i.e., in the absence of this term—the electron is localized at a given site, p. The vibrational state of the system is specified by a set of quantum-numbers, Nk, giving the degree of excitation of each vibration-mode; the latter differ from the conventional modes in that in each of them, the equilibrium displacement, about which the system oscillates, depends upon the location of the electron. The presence of a nonvanishing electronic-overlap term gives rise to transitions in which the electron jumps to a neighboring site (p→p±1), and in which either all of the Nk remain unaltered (“diagonal” transitions) or in which some of them change by ±1 (“nondiagonal” transitions). The two types of transitions play fundamentally different roles. At sufficiently low temperatures, the diagonal transitions are dominant. They give rise to the formation of Bloch-type bands whose widths (see Eq. 37) are each given by the product of the electronic-overlap integral, and a vibrational overlap-integral, the latter being an exponentially falling function of the Nk (and, hence, of temperature). In this low-temperature domain, the role of the non- diagonal transitions is essentially one of scattering. In the absence of other scattering mechanisms, such as impurity scattering, they determine the lifetimes of the polaron-band states and, hence, the mean free path for typical transport quantities, such as electron diffusivity. With rising temperature, the probability of the off-diagonal transitions goes up exponentially. This feature, together with the above-mentioned drop in bandwidth, results, e.g., in an exponentially diminishing diffusivity. Eventually, a temperature, Tt∼ 1 2 the Debye Θ, is reached at which the energy uncertainty, ℏ/τ, associated with the finite lifetime of the states, is equal to the bandwidth. At this point, the Bloch states lose their individual characteristics (in particular, those which depend upon electronic wave number); the bands may then be considered as “washed out.” For temperatures >Tt, electron motion is predominantly a diffusion process. The elementary steps of this process consist of the random-jumps between neighboring sites associated with the nondiagonal transitions. In conformance with this picture, the electron diffusivity is, apart from a numerical factor, the product of the square of the lattice distance and the total non-diagonal transition probability, and is therefore an exponentially rising function of temperature. The limit, Jmax, of the magnitude of the electronic overlap term, beyond which the perturbation treatment of the present paper becomes inapplicable, is investigated. For representative values of the parameters entering into the theory, Jmax∼0.12 ev and 0.035 ev for the extreme cases of (a) width of the ground-state polaron-band and (b) high-temperature site-jump probabilities (these numbers correspond to electronic bandwidths of 0.24 ev and 0.07 ev, respectively). For electronic bandwidths in excess of these limits, a treatment based on the adiabatic approach is required; preliminary results of such a treatment are given for the above two cases.

Abstract: The following investigation contains a general theory of bending of a bi-metal strip submitted to a uniform heating. This theory is applied in analysis of operation of a bi-metal strip thermostat. The equations are obtained for calculating the temperature of buckling, the complete travel during buckling, and the temperature of buckling in a backward direction.. By using these equations the dimensions of the thermostat for a given temperature of operation and a given complete range of temperature can be calculated. The results obtained are based on certain ideal conditions. For example, it was assumed that the differ­ ence in the coefficients of expansion remained constant during heating, that the friction at the supports could be neglected and that the width