Showing papers in "Journal of Applied Physics in 1956"

Thermoelasticity and Irreversible Thermodynamics

Abstract: A unified treatment is presented of thermoelasticity by application and further developments of the methods of irreversible thermodynamics. The concept of generalized free energy introduced in a previous publication plays the role of a ``thermoelastic potential'' and is used along with a new definition of the dissipation function in terms of the time derivative of an entropy displacement. The general laws of thermoelasticity are formulated in a variational form along with a minimum entropy production principle. This leads to equations of the Lagrangian type, and the concept of thermal force is introduced by means of a virtual work definition. Heat conduction problems can then be formulated by the methods of matrix algebra and mechanics. This also leads to the very general property that the entropy density obeys a diffusion‐type law. General solutions of the equations of thermoelasticity are also given using the Papkovitch‐Boussinesq potentials. Examples are presented and it is shown how the generalized coordinate method may be used to calculate the thermoelastic internal damping of elastic bodies.

Theory of Mechanical Damping Due to Dislocations

Abstract: A quantitative theory of damping and modulus changes due to dislocations is developed. It is found that the model used by Koehler of a pinned dislocation loop oscillating under the influence of an applied stress leads to two kinds of loss, one frequency dependent and the other not. The frequency dependent loss is found to have a maximum in the high megacycle range. The second type of loss is a hysteresis loss which proves to be independent of frequency over a wide frequency range which includes the kilocycle range. This loss has a strain‐amplitude dependence of the type observed in the kilocycle range. The theory provides a quantitative interpretation of this loss.

Two‐Dimensional Motion of Idealized Grain Boundaries

Abstract: To represent ideal grain boundary motion in two dimensions, a rule of motion of plane curves is considered whereby any given point of a curve moves toward its center of curvature with a speed that is proportional to the curvature. A general theorem is deduced concerning the change of area enclosed by such a curve. Three families of curves are found that obey the curvature rule of motion while undergoing the shape preserving transformations of uniform magnification, translation, and rotation respectively. Pieces of these curves represent the steady shapes of idealized grain boundaries under certain symmetrical conditions.

Theoretical Considerations Governing the Choice of the Optimum Semiconductor for Photovoltaic Solar Energy Conversion

Abstract: The theory of the photovoltaic effect is used to predict the characteristics of a semiconductor which would operate with an optimum efficiency as a photovoltaic solar energy converter. The existence of such an optimum material results from the interaction between the optical properties of the semiconductor which determine what fraction of the solar spectrum is utilized and its electrical properties which determine the maximum efficiency of conversion into electricity. Considerable attention is devoted to the effect of the forbidden energy gap (EG) of the semiconductor. It is shown that atmospheric absorption causes a shift in the solar spectrum which changes the value of the optimum forbidden energy gap between the limits 1.2 ev

Physical Theory for Capillary Flow Phenomena

Abstract: From the assumption that the microscopic behavior of the liquid in an unsaturated porous medium is controlled by the physical laws of surface tension and viscousflow,differential equations governing the macroscopic flow in such a medium are deduced. No special pore‐shape assumptions are required, but one topological approximation is needed; i.e., that neither isolated drops nor isolated bubbles occur. Several nonessential simplifying assumptions are used; i.e., that the macroscopic properties of the medium, the character of the liquid, and the pressure of the gas are independent of position, time, and direction. The macroscopic equations are obtained in a fully reduced form, permitting comparison between two media— or between two flow systems—that differ only by scaling factors. A novel feature of this calculation is its prediction that the liquid‐transmission and liquid‐capacity properties of an unsaturated medium will exhibit hysteresis in their dependences upon the liquid‐gas pressure differential, p. The properties of the medium depend upon the pressure history but are invariant to monotonic time‐scale distortions of that history. Such time‐invariant functionals have been termed by the authors ``hysteresis functions,'' symbolized by the subscript, H, e.g. FH (p). Although methods for measuring and describing the characteristics of specific ``hysteresis functions'' have not yet been developed, the general validity of this analysis can be studied experimentally by testing predictions that are contained in the reduced variables.

Application of Dislocation Theory to Internal Friction Phenomena at High Frequencies

Abstract: A detailed discussion of data obtained over the past 15 years concerning the damping of mechanical vibrations in the kilocycle and megacycle range is given. The dependence of the decrement and modulus change on the variables of frequency and strain‐amplitude and many other parameters is compared with predictions of the dislocation theory developed in an earlier paper. Although general agreement is obtained, and many interesting quantitative results are found, it is not possible to say that the theory agrees everywhere since not all the necessary parameters are known well enough theoretically. A number of new experiments are suggested which may permit stronger conclusions to be made. This part may be read independently of the earlier paper by the reader who does not wish to follow the development of the theory in detail.

Scattering of a Plane Longitudinal Wave by a Spherical Obstacle in an Isotropically Elastic Solid

Abstract: Scattering by a spherical obstacle of a plane longitudinal wave propagating in an isotropically elastic solid is computed. Expressions for the scattered wave and the total scattered energy are given. Three special types of obstacle—an isotropically elastic sphere, a spherical cavity, and a rigid sphere—are discussed in detail, especially for Rayleigh scattering. The result for the isotropically elastic sphere is compared with the well‐known result of scattering of a plane wave propagating in an ideal fluid by a sphere of another ideal fluid.

Polymorphism of Iron at High Pressure

Abstract: Propagation of compressive waves generated by high explosive in Armco iron may, under certain conditions, occur with three stable shocks, which are characterized by abrupt and possibly quasi‐discontinuous change of pressure. The characteristics of these have been investigated experimentally and found to be consistent with the hypothesis of a polymorphic transition in iron at about 0.13 megabar. The same phenomenon has been observed in mild steel.

Tensile Strength of Whiskers

Abstract: Tensile tests have been performed on whiskers of iron, copper, and silver 1.2 to 15 μ in diameter. The strongest whiskers which were less than 4 μ in diameter exhibited resolved elastic shear strengths of from two to six percent of their shear moduli. Stress‐strain determinations on iron have shown that large deviations from Hooke's law occur beyond two percent strain. As the whiskers increase in size, their strengths decrease with considerable scatter.

Paramagnetic Resonance of Lattice Defects in Irradiated Quartz

Abstract: Various kinds of silica glass and natural and synthetic crystals have been irradiated with fast neutrons at temperatures of ∼250°C and above 300°C In the glasses, asymmetric paramagnetic resonance lines resulting from the irradiation have been observed with apparent g values of 20013±00006 and 20090±00007, and half‐widths of ≈17 gauss and ≈40 gauss, respectively Two groups of lines were found in the natural crystals and have been associated with the two lines in the glasses, the two lines in the glasses being the envelope of the lines in the crystals when they are summed over random orientations with respect to the magnetic field On the basis of g values, absence of common impurity and of hyperfine splitting, and the thermal stability of the lines, it is concluded that the observed resonances in the silica glasses and crystals have their origin in defects in the basic SiO4 tetrahedra generated by fast neutrons and primary and secondary knockons From preliminary optical bleaching data in neutron i

Dynamic Method for Measuring the Pyroelectric Effect with Special Reference to Barium Titanate

Abstract: A dynamic method has been developed for the study of the pyroelectric effect. Transient currents produced in single crystals of barium titanate when subjected to flashes of light are shown to be pyroelectric in origin. The illumination results in a small change in the temperature of the crystal which, in turn, causes the polarization to change. This change is recorded as a current in the external circuit. It is shown that from room temperature up to the Curie point, the pyroelectric current is consistent with the polarization as a function of temperature, as determined from hysteresis loop measurements. The technique proves to be a sensitive and nondestructive method for studying the state of polarization of a crystal.The technique is used to study the pyroelectric effect induced by applied static electric fields at temperatures above the Curie point. The results are consistent with Devonshire's theory of the ferroelectricity of BaTiO3, and they confirm that the Curie point transition is of the first order.

Theory of deformation of a porous viscoelastic anisotropic solid

Abstract: Equations are established for the deformation of a viscoelastic porous solid containing a viscous fluid under the most general assumptions of anisotropy. The particular cases of transverse and complete isotropy are discussed. General solutions are also developed for the equations in the isotropic case. As an example the problem of the settlement of a loaded column is treated. The second–order effect of the change of permeability with deformation is also discussed.

Switching Time in Ferroelectric BaTiO3 and Its Dependence on Crystal Thickness

Abstract: The switching time ts and the switching current imax have been measured as a function of applied field E and of the size of the sample. It has been observed that the ``activation field'' α for the nucleation of new domains is inversely proportional to the thickness of the sample. This behavior can be explained by assuming a surface layer. The thickness of this layer has been calculated to be of the order of 10−4 cm. The same way we can explain the thickness dependence of the 60‐cycle coercive field strength. Furthermore, it has been found that the switching time depends to a first approximation linearly on the thickness of the sample if the field E is kept constant. This can be explained by assuming a domain wall motion primarily in the forward direction or by assuming a nucleation mechanism. The maximum velocity of the domain growth was found to be of the order of the velocity of sound. The switching time does not depend on electrode area.

Copper Precipitation on Dislocations in Silicon

Abstract: Precipitates formed by copper diffused into silicon crystals were studied by optical means using an infrared image tube. Linear aggregates are identified as dislocations by correlation with etch pits. Dislocations in as‐grown crystals are usually curvilinear. Dislocation loops formed during plastic deformation consist of straight‐line composite portions and pure screw portions, all of which lie in 〈110〉 directions. Precipitates frequently are not found on screw dislocations; when present they differ in detail from those on composites. Interactions between adjacent dislocations can be seen. Etch pits associated with either composite or screw dislocations are similar in form, indicating that the pits formed in as‐grown crystals by the technique described identify all the dislocations. Dislocation loops enter from the surface upon deformation of crystals with relatively few grown‐in dislocations. Dislocation loops formed entirely within the crystal in patterns consistent with operation of the Frank‐Read mech...

Damping of Underwater Explosion Bubble Oscillations

Abstract: When an explosive detonates underwater it creates a bubble of gas which performs damped radial oscillations of large amplitude. The usual theory of these oscillations treats the water as incompressible and yields undamped oscillations of constant period. We have modified this theory by taking account of the compressibility of the water. Our theory predicts damped oscillations of diminishing period. Comparison of the predicted and observed radius‐time curves for one particular case shows fairly good agreement. Radius‐time curves for four representative cases have been computed with a large number of periods in each case. These can be used to describe a variety of explosions.

Diffusion of donor and acceptor elements in silicon

Abstract: The diffusion of the Group III (B, Al, Ga, In, and Tl) and Group V (P, As, Sb, and Bi) elements in silicon has been measured in the temperature range 1050–1350°C. A method based on change in conductivity through the penetration layer has been used for B and P. The p‐n junction method has been used for the other elements. Aside from B and P, which have similar diffusional properties, the acceptor elements diffuse more rapidly than the donor elements. Diffusion coefficients are given by DB, P=10.5 exp − (85 000/RT), DA1=8.0 exp − (80 000/RT), DGa=3.6 exp − (81 000/RT), DIn, T1=16.5 exp − (90 000/RT), DAs=0.32 ×exp − (82 000/RT), DSb=5.6 exp − (91 000/RT), DBi=1030 exp − (107 000/RT) with an average estimated error of about ±40%. This corresponds to an error in the activation energies of about ±5 kcal. Sources of error including the effects of impurities in the oxides are discussed. D0 values in most cases conform to the predictions of Zener for substitutional diffusion.

Invariants in Experimental Data on Linear Kinetics and the Formulation of Models

Abstract: A new mathematical approach for the analysis of radioactive tracer experiments on compartmentalized systems in steady states is presented. The experimental measurements of the amounts of tracer in one or more compartments are approximated by a sum of exponential functions of time. The coefficients and exponential factors of these functions are shown to represent a set of invariants of the data. These are then related in a concise matrix equation to the compartmental model parameters, which are defined as the transition probabilities per unit time per unit quantity (turn‐over rates) of molecules from one compartment (or state) into another.When measurements are incomplete, the freedom in choosing a model is expressed in turns of a minimum number of variables equal to the difference between the number of model parameters and the number of invariants found in the data. It is shown how other information regarding the model parameters or the amounts of material in steady state may be combined with the tracer data to reduce the degrees of freedom of a proposed model.Matrix transformations are worked out to allow the mapping of all mathematically consistent models in the configuration space of the minimum variables. The boundaries in this space corresponding to the limits for physically realizable models are found. Thus, a model is expressed by a set of coordinates and the values of all the parameters are obtained by substituting the values of the coordinates. Investigation of the range of variation of each parameter over the bounded region is also possible.

Magnetic Granulometry and Super‐Paramagnetism

Abstract: The magnetic technique of particle size measurement based on the work of Elmore consists of a comparison of a modified Langevin curve with the observed magnetization curve for an appropriate system of ferromagnetic particles. The average magnetic moment per particle is obtained from this analysis, and the average particle size is thus determined. The prediction by Neel of thermal relaxation in single‐domain ferromagnetic particles broadens the range of applicability of the granulometry method, with special emphasis on the size range from 20 A to about 100 A. A possible objection to the method, arising from evidence for an abnormal temperature dependence of the saturation magnetization of very small magnetic particles, is re‐examined experimentally and found to be invalid. Several examples are presented that demonstrate the predicted thermal relaxation and that point to a wide utilization of the granulometry technique.

Further Investigation of Laminar Flow in Channels with Porous Walls

Abstract: The problem of two‐dimensional steady‐state laminar flow in channels with porous walls has been extended to the case of moderate to high suction or injection velocity at the walls. An exact solution of the Navier‐Stokes equations, reduced to a third‐order nonlinear differential equation with appropriate boundary conditions, is obtained. The velocity components, the pressure, and the coefficient of wall friction are expressed as functions of velocity through the porous walls, the average axial velocity of Poiseuille's flow, the coordinates and dimensions of the channel, and the physical properties of the fluid.

On the Minimum of Magnetization Reversal Time

Abstract: A modified Landau‐Lifshitz equation is solved for a single‐domain sphere and an infinitely‐wide thin single‐domain sheet of ferromagnetic material neglecting anisotropy. The external magnetic field is switched from one direction to its opposite instantaneously at the initial time and the behavior of the magnetization vector is investigated thereafter. It is shown that there is a critical value of the damping constant corresponding to the minimum value of the (repetitive) magnetization reversal time. If the damping constant is larger than the critical value, the magnetization vector moves slower; if it is smaller, the magnetization vector moves faster but oscillates so that it takes longer time until it comes to a rest at the final position. The critical values of the Landau‐Lifshitz damping constant λ are γM for the sphere and 0.013γM for the thin sheet, where γ and M are the gyromagnetic ratio and the magnetization, respectively. The computed minimum switching time for the thin sheet of 4–79 molybdenum Permalloy is of the order of 10−9 sec.

Missile Guidance by Three‐Dimensional Proportional Navigation

Abstract: Following a brief comparison of three collision‐seeking types of navigation—pure pursuit, constant‐bearing collision, and proportional navigation—the usual definition of planar proportional navigation is extended to three dimensions. Based upon a simple criterion for optimum navigation to the line‐of‐sight motion, a proper formulation is found in terms of the geodesic and normal curvatures of the missile path on the surface generated by the line of sight. By a suitable choice of a reference coordinate system, the missile‐target kinematic relationships are then linearized, assuming relatively small deviations of the missile from a collision course. Upon combining these ``geometry feedback'' equations with the equations of guidance, the missile trajectory is found to be given in terms of two independent linear differential equations of identical form and of one higher order than the missile transfer function. Typical solutions are found for some simple cases. The character of the trajectory is shown to depend on an ``effective navigation constant'' proportional to the missile navigation constant (or gain) and the ratio of missile speed to closing speed; a value of this parameter greater than two is found to be necessary to insure finite terminal missile acceleration. Two example trajectories are calculated from both the exact and the linearized trajectory equations to indicate the accuracy of the linearization.

Stress Relaxation Studies of the Viscoelastic Properties of Polymers

Abstract: Extensive studies of the viscoelastic properties of polymers undertaken in the author's laboratory by means of the method of stress relaxation are here reviewed The discussion is divided into four parts: chemical stress relaxation, stress relaxation in amorphous polymers, stress relaxation in crystalline polymers, and stress relaxation in certain natural polymers and polyelectrolytes Mathematical description of the phenomena are presented in simple form The relation between structure and viscoelastic properties of polymers are discussed and a rather complete over‐all picture of these phenomena seems to be emerging

Resolution of the Atomic Structure of a Metal Surface by the Field Ion Microscope

Abstract: A formula for the resolution of the field ion microscope is derived by considering that most of the ions originate from rebounding molecules after random elastic collision with the surface. Cooling the tip with solid N2 or liquid H2 in order to obtain a large accommodation coefficient drastically reduces the lateral velocity of the ions in a helium operated microscope. As a result the potential resolution for a tip of radius 1000 A is improved to 1.5 A. Photographs show the individual atoms of a tungsten surface. The actual resolution is sufficient to resolve neighboring atoms with 2.74 A separation.

Slippage of Water over Nonwettable Surfaces

Abstract: When water flows over glass which has been treated with the vapor of dimethyldichlorosilane and thus made water repellent, slipping on the boundary between the solid surface and the water takes place. This is is shown in capillary tubes of various diameters. The amount of slipping is small, but measurable. It disappears or becomes extremely small in case of turbulent flow.

Review of Germanium Surface Phenomena

Abstract: Germanium surface behavior has become of great interest recently, chiefly for its importance in the understanding of diode and transistor technology. In general, the surface may be treated as an assemblage of allowed electron states occurring in the normally forbidden energy range. A review of the measurements of the electrical properties suggests that there are two distinct types of state. The ``fast'' state has a hole or electron capture time not greater than a microsecond and is chiefly involved in the recombination process. The ``slow'' state has capture times from a millisecond to several minutes and determines the density and type of carrier at the surface. ``Fast'' states are believed to occur at the interface between the germanium and the oxide layer, and their density of about 1011 cm−2 is determined by the initial surface treatment. ``Slow'' states are associated with the structure of the oxide layer and the gaseous ambient, and have a density greater than 1013 cm−2. Since these states determine the conductivity type at the surface, they contribute to surface ``leakage'' in diodes and transistors and, because of their long equilibrium times, to low‐frequency noise. The adsorption of gases such as water vapor, not only controls the density and energy of the ``slow'' states but also leads to possible electrolytic conduction along the surface, in addition to the normal electron flow in the bulk semiconductor.

Breakdown of Air at Microwave Frequencies

Abstract: A theory for uniform field breakdown in air at microwave frequencies is developed and applied with success to predicting values of breakdown over a wide range of experimental conditions. Three distinct types of breakdown are treated; c.w. (continuous wave) breakdown, single pulse breakdown, and multipulse breakdown. The conditions for breakdown are determined from a solution of the electron continuity equation for an average electron, in which electron ionization, attachment, and diffusion are the dominant mechanisms. Modulation of the electron average energy at twice the frequency of the applied field becomes important at either high pressure or low frequency and modifies the values of the breakdown field. The breakdown field strengths are shown to be determined from a single curve for each type of breakdown power, either c.w. or pulsed. These theoretical curves are in accordance with the experimental results, thus verifying the assumption and the values of the coefficients used in the theory.

Penetration of Electrons and Ions in Aluminum

Abstract: The depth of penetration of 0.5‐ to 11‐kev electrons and 1‐ to 25‐kev H+, H2+, and He+ ions in aluminum has been measured. It has been found that the Thompson‐Whiddington law for electrons does not hold for electrons having energies less than 10 kev. The practical range‐energy relation obtained for electrons in aluminum was found to be R = KE1.3, where K is 0.042 if E is expressed in kev and R in microns. The practical range‐energy relation for light ions in aluminum was found to be R = KE0.83 where K = 0.020 for H+, K = 0.021 for He+, and K = 0.015 for H2+ if E is expressed in kev and R in microns. Results are compared with those obtained by previous observers.

Observations of Dislocation Glide and Climb in Lithium Fluoride Crystals

Abstract: It is shown that reagent CP‐4 plus ferric ions can be used to detect the positions of dislocations in LiF crystals. The technique is highly selective and both edge and screw (110) [110] dislocations can be detected and distinguished. A double etching method is used to observe, for the first time, the glide and climb of individual dislocations in LiF.

Fracture in the Extrusion of Amorphous Polymers through Capillaries

Abstract: In the extrusion of an amorphous, un‐cross‐linked polymer through a capillary, there is a critical stress at and beyond which the emerging stream becomes irregular in shape. This effect appears to originate in the approach to the capillary rather than within the capillary. Tearing or fracture of the molten polymer appears to occur in this region.