Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1948"
TL;DR: It is shown that a satisfactory account can be given of open water evaporation at four widely spaced sites in America and Europe, the results for bare soil receive a reasonable check in India, and application of theresults for turf shows good agreement with estimates of evapolation from catchment areas in the British Isles.
Abstract: Two theoretical approaches to evaporation from saturated surfaces are outlined, the first being on an aerodynamic basis in which evaporation is regarded as due to turbulent transport of vapour by a process of eddy diffusion, and the second being on an energy basis in which evaporation is regarded as one of the ways of degrading incoming radiation. Neither approach is new, but a combination is suggested that eliminates the parameter measured with most difficulty—surface temperature—and provides for the first time an opportunity to make theoretical estimates of evaporation rates from standard meteorological data, estimates that can be retrospective. Experimental work to test these theories shows that the aerodynamic approach is not adequate and an empirical expression, previously obtained in America, is a better description of evaporation from open water. The energy balance is found to be quite successful. Evaporation rates from wet bare soil and from turf with an adequate supply of water are obtained as fractions of that from open water, the fraction for turf showing a seasonal change attributed to the annual cycle of length of daylight. Finally, the experimental results are applied to data published elsewhere and it is shown that a satisfactory account can be given of open water evaporation at four widely spaced sites in America and Europe, the results for bare soil receive a reasonable check in India, and application of the results for turf shows good agreement with estimates of evaporation from catchment areas in the British Isles.
6,711 citations
TL;DR: In this article, a theory is suggested which describes the yielding and plastic flow of an anisotropic metal on a macroscopic scale and associated relations are then found between the stress and strain-increment tensors.
Abstract: A theory is suggested which describes, on a macroscopic scale, the yielding and plastic flow of an anisotropic metal. The type of anisotropy considered is that resulting from preferred orientation. A yield criterion is postulated on general grounds which is similar in form to the Huber-Mises criterion for isotropic metals, but which contains six parameters specifying the state of anisotropy. By using von Mises' concept (1928) of a plastic potential, associated relations are then found between the stress and strain-increment tensors. The theory is applied to experiments of Korber & Hoff (1928) on the necking under uniaxial tension of thin strips cut from rolled sheet. It is shown, in full agreement with experimental data, that there are generally two, equally possible, necking directions whose orientation depends on the angle between the strip axis and the rolling direction. As a second example, pure torsion of a thin-walled cylinder is analyzed. With increasing twist anisotropy is developed. In accordance with recent observations by Swift (1947), the theory predicts changes in length of the cylinder. The theory is also applied to determine the earing positions in cups deep-drawn from rolled sheet.
3,426 citations
TL;DR: In this article, it has been shown that the actual yield stress is considerably greater than this minimum, and methods are given for calculating a more probable value, and a minimum possible value can be assigned to the maximum acceleration of the material, and from this a minimum value for the yield stress can be calculated.
Abstract: It has long been known that metals may be subjected momentarily to stresses far exceeding their static yield stress without suffering plastic strain. One of the simplest methods for subjecting a metal to a high stress for a short time is to form it into a cylindrical specimen and fire this at a steel target. The front part of this projectile crumples up, but the rear part is left undeformed. If the target is rigid the distance which this portion travels while it is being brought to rest may be taken as the difference between the initial length and the length of the deformed specimen after impact. Knowing the velocity of impact, a minimum possible value can be assigned to the maximum acceleration of the material, and from this a minimum value for the yield stress can be calculated. The actual yield stress is considerably greater than this minimum, and methods are given for calculating a more probable value.
571 citations
TL;DR: In this article, a simple theory of hardness is given, based on the theoretical work of Hencky and Ishlinsky, which explains a number of empirical relations observed in dynamic hardness measurements, and reproduces the calibration characteristics of the rebound scleroscope.
Abstract: When a hard spherical indenter is pressed into the surface of a softer metal, plastic flow of the metal specimen occurs and an indentation is formed. When the indenter is removed it is found that the permanent indentation is spherical in shape, but that its radius of curvature is greater than that of the indenter. It is generally held that this 'shallowing' effect is due to the release of elastic stresses in the material around the indentation. It is clear that if the recovery is truly elastic it should be reversible and that a second application and removal of the indenter under the original load should not change the size or shape of the indentation. Experiments show that this is the case. This means that when the original load is reapplied, the deformation of the indenter and the recovered indentation is elastic and should conform with Hertz's equations for the elastic deformation of spherical surfaces. Measurements show that there is, in fact, close agreement between the observed deformation and that calculated from Hertz's equations. These results have been applied to the case of indentations formed in a metal surface by an impacting indenter. The energy involved in the elastic recovery of the impacting surfaces is found to account for the energy of rebound of the indenter. This analysis explains a number of empirical relations observed in dynamic hardness measurements, and, in particular, reproduces the calibration characteristics of the rebound scleroscope. The results also show that for very soft metals the dynamic hardness is very much higher than the static hardness, and it is suggested that in rapid deformation of soft metals, forces of a quasi-viscous nature are involved. In the third part of the paper a simple theory of hardness is given, based on the theoretical work of Hencky and Ishlinsky. It is shown experimentally that for a material incapable of appreciable work-hardening, the mean pressure P$\_{m}$ required to produce plastic yielding is related to the elastic limit Y of the material by a relation P$\_{m}$ = cY, where c is a constant having a value between 2$\cdot $6 and 3. An empirical method is described which takes into account the work-hardening produced in metals by the indentation process itself. This results in a general relation between hardness measurements and the stress-strain characteristic of the metal, and there is close agreement between the theory and the observed results. In addition, the theory explains the empirical laws of Meyer.
560 citations
Abstract: The problem of metallic conduction at high frequencies and low temperatures, recently discussed by Pippard, is reformulated using the general methods of the theory of metals, and exact solutions are obtained which are valid for all frequencies and temperatures. It is shown that, for large values of the free path of the conduction electrons, the electric field is propagated through the metal as a ‘surface wave’ which differs considerably from the classical exponential solution. The temperature variation of the surface impedance in the microwave region is considered in detail. Pippard’s simplified theory is shown to be qualitatively correct, and a quantitative discussion of his experimental results is given. The frequency variation of the surface impedance at low temperatures is also discussed, and it is shown that relaxation effects are negligible in the microwave region but become important in the infra-red and eventually restore the validity of the classical theory. The theory predicts that, as the frequency is increased, the reflexion coefficient of metals passes through a minimum in the far infra-red.
461 citations
TL;DR: In this article, the authors generalized the Newtonian definition of the mass-centre to the restricted theory of relativity in several ways, leading to fairly simple expressions in terms of instantaneous variables for quite general systems.
Abstract: The Newtonian definition of the mass-centre can be generalized to the restricted theory of relativity in several ways. Three in particular lead to fairly simple expressions in terms of instantaneous variables for quite general systems. Of these only one is independent of the frame in which it is defined. It suffers from the disadvantage that its components do not commute (in classical mechanics, do not have zero Poisson brackets), and are therefore unsuitable as generalized co-ordinates in mechanics. Of the other two, one is particularly simply defined, and the other has commuting co-ordinates. The Poisson brackets can be derived from quite general considerations because the various mass-centres are expressible in terms of integrals of the energy-momentum tensor which are directly connected with the infinitesimal operators of the group of Lorentz transformations. The definitions are readily applicable to a single particle in theories, such as are current for elementary particles, where a co-ordinate observable does not exist, but an energy-momentum tensor does, and furnish the nearest approach possible to such observables. They are applied to electrons, particles of spin 0 and $\hslash $ (scalar- and vector-meson theories), and to photons.
455 citations
TL;DR: In this paper, the authors described direct measurements of all the terms in the equation for the rate of change of mean square vorticity in isotropic turbulence and with the experimental verification of some recent theoretical investigations.
Abstract: In a previous paper the authors described direct measurements of all the terms in the equation for the rate of change of mean square vorticity in isotropic turbulence. The present paper is concerned with developments arising from the earlier work and with the experimental verification of some recent theoretical investigations. The results of measurements of the turbulent intensity u 9 and of λ are presented; these establish that u9 -2 and λ 2 are each proportional to the time of decay provided that the time is not too large. Within this initial period of the decay, the double and triple velocity correlation functions are found to maintain their form, i.e. to be self-preserving, for small values of the distance r between the two points at which the correlations are taken. For larger separations the double velocity correlation function changes its form slightly during decay and direct measurements of λ and of the integral scale L show that λ/ L increases during the decay. Theoretical predictions about the shape of the correlation function, for limited ranges of r , at high and at low Reynolds numbers are compared with measurements. Theory has shown that the above decay law cannot persist indefinitely, and the present experiments confirm that the decay law changes in the expected direction when the time is large. A division of the life-history of the turbulence into initial, transition and final periods is suggested; within the initial period, a classification based on the Reynblds number is also possible. Some speculations on the interpretation of the initial period are presented.
294 citations
TL;DR: In this paper, it was shown that homogeneous turbulence tends to an asymptotic statistical state which is independent of the initial conditions, and that the energy of turbulence is proportional to t$ −\frac{5}{2}}$ and the longitudinal double-velocity correlation coefficient for two points distance r apart, where r is the time of decay.
Abstract: The final period of decay of a turbulent motion occurs when the effects of inertia forces are negligible. Under these conditions the instantaneous velocity distribution in the turbulence field may be solved as an initial value problem. It is shown that homogeneous turbulence tends to an asymptotic statistical state which is independent of the initial conditions. In this asymptotic state the energy of turbulence is proportional to t$^{-\frac{5}{2}}$ and the longitudinal double-velocity correlation coefficient for two points distance r apart is e$^{-r^{2}/8
u t}$, where t is the time of decay. The asymptotic time-interval correlation coefficient is found to be different from unity for very large time intervals only, showing the aperiodic character of the motion. The whole field of motion comes gradually to rest, smaller eddies decaying more rapidly than larger eddies, and the above stable eddy distribution is established when only the largest eddies of the original turbulence remain. Relevant measurements have been made in the field of isotropic turbulence downstream from a grid of small mesh. The above energy decay and space-interval correlation relations are found to be valid at distances from the grid greater than 400-mesh lengths and at a mesh Reynolds number of 650. The duration of the transitional period, in which the energy decay law is changing from that appropriate to the initial period of decay to the above asymptotic law, increases very rapidly with R$_{M}$. There is a brief discussion of the criterion for the existence of final period decay, although clarification must wait until the existence and termination of the initial period of decay are better understood.
282 citations
TL;DR: In this article, it was shown that the sign of the mutual polarizability of two bonds in a chain depends upon whether there is an even or odd number of other bonds between them.
Abstract: This part is concerned with the changes in bond order that take place in a conjugated system when the resonance integral of a bond, or the coulomb term of an atom is altered. It is shown that the sign of the mutual polarizability of two bonds in a chain depends upon whether there is an even or odd number of other bonds between them. Numerical values are given for mutual bond polarizabilities in some important hydrocarbons, and it is found that in the polyenes the effect of a perturbation decreases rapidly with distance along the chain. The effect of hetero-substitution on bond orders is considered, and the results compared with the predictions of the qualitative resonance theory.
201 citations
TL;DR: The positions of the nitrogen atoms in the Fe-iron nitride phase (Fe$\_{2}$N) and in the zeta phase (Zeta $ N) were determined by passing ammonia over iron at temperatures not exceeding 450 degrees C.
Abstract: The $\zeta $-iron nitride phase (Fe$\_{2}$N), the existence of which is confirmed, is prepared by passing ammonia over iron at temperatures not exceeding 450 degrees C and under such conditions that the partial pressure of hydrogen is negligible. The positions of the nitrogen atoms in $\gamma ^{\prime}$(Fe$\_{4}$N) and in $\zeta $ are determined. $\gamma ^{\prime}$ is a 'normal' 12a6 interstitial structure, the cubic unit cell of which contains four iron atomsat 0 0 0, $\frac{1}{2}\frac{1}{2}\,0$, $\frac{1}{2}\, 0\,\frac{1}{2}$, $0\,\frac{1}{2}\frac{1}{2}$, and one nitrogen atom at $\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2}$; at 6$\cdot $1 weight% N, a=3$\cdot $787kX. $\zeta $ has a distorted 12b6 structure. The unit cell, with dimensions a, 5$\cdot $512; b, 4$\cdot $820; c, 4$\cdot $416 kX at 11$\cdot $3 weight% N, contains eight iron atoms at 0 0 0, $\frac{1}{2}\, 0\,0$, $\frac{1}{4}\,\frac{1}{2}\,0$, $\frac{3}{4}\,\frac{1}{2}\,0$, $0\,\frac{1}{3}\,\frac{1}{2}$, $\frac{1}{2}\,\frac{1}{3}\,\frac{1}{2}$, $\frac{1}{4}\,\frac{5}{6}\,\frac{1}{2}$, $\frac{3}{4}\,\frac{5}{6}\,\frac{1}{2}$, and four nitrogen atoms at $\frac{1}{4}\,\frac{1}{6}\,\frac{1}{4}$, $\frac{1}{2}\,\frac{2}{3}\,\frac{1}{4}$, $\frac{3}{4}\,\frac{1}{6}\,\frac{3}{4}$, $0\,\frac{2}{3}\,\frac{3}{4}$. Both $\gamma ^{\prime}$ and $\zeta $ are fully ordered interstitial alloys.
180 citations
TL;DR: In this paper, an extension of the work of Bashforth & Adams (1883) is presented to give the shapes of pendant drops of liquids for values of β from − 0.25 to −0.6 at intervals of 0.001 to 0.025.
Abstract: The calculations presented are an extension of the work of Bashforth & Adams (1883) to give the shapes of pendant drops of liquids for values of β from — 0.25 to — 0.6 at intervals of 0.025. The results have been used to calculate, to an accuracy of 0.001 to 0.01 %, the constants needed for measuring surface and interfacial tensions by the method suggested by Andreas, Hauser & Tucker (1938), which has thus been made independent of calibration.
TL;DR: In this paper, the dynamic compressive yield strength was computed from the density of the specimen, the striking velocity, and from measurements of the dimensions of the test piece before and after test.
Abstract: A description is given of the experimental technique devised to apply the method outlined theoretically in part I to the measurement of the dynamic compressive yield strength of various steels, duralumin, copper, lead, iron and silver. A polished piece of armour steel was employed as a target, and cylindrical specimens were fired at it at various measured velocities from Service weapons. The distance between the weapon and target was made short to ensure normal impact, and apparatus was devised for the precise measurement of striking velocity over this short range. The dynamic compressive yield strength was computed from the density of the specimen, the striking velocity, and from measurements of the dimensions of the test piece before and after test. Details are given of the accuracy of the various measurements, and of their effect on the values of yield strength. The method was found to be inaccurate at low and high velocities. For instance, with mild steel, satisfactory results were only obtainable within the range 400 to 2500 ft. /sec. The range of velocities within which satisfactory results could be obtained varied with the quality of the material tested, soft metals giving results within a much lower range than that necessary for harder materials. Because of its failure at low velocities, the method could not be employed to bridge the gap between static and dynamic tests. The rate of strain employed in the dynamic tests could not be measured, but was estimated to be of the order of 10,000 in. /in. /sec. With the materials tested little change of dynamic strength occurred within the range of striking velocities employed, probably because the rate of strain did not vary to any great extent with the striking velocity. Within the range of weapons available, that is, from a 0·303 in. rifle up to a 13 pdr. gun (calibre 3·12 in.), little change of dynamic strength occurred with alteration of the initial dimensions of the specimens, probably because the corresponding change of rate of strain was not large. In general, the dynamic compressive yield strength S was greater than the static strength Y represented by the compressive stress giving 0·2% permanent strain. For steels of various types, regardless of chemical composition and heat treatment, there was a relation between S / Y and the static strength Y , the ratio decreasing from approximately 3 when Y was 20 tons/sq. in. to 1 when Y was 120 tons/sq. in. A similar relation occurred with duralumin, S / Y varying from 2·5 at Y = 8 tons/sq. in. to 1·4 at Y = 25 tons/sq. in. Dynamic compressive yield values were obtained for soft materials such as pure lead, copper and Armco iron, which, under static conditions, gave no definite yield values. A plot of the unstrained length of the specimen X , expressed as X / L (where L = initial overall length), versus the final overall length L 1 , expressed as L 1 / L , was made for the various materials. Any specified value of X / L was associated with greater values of L 1 / L for the more ductile materials, such as copper and lead, than for the brittle materials, such as armour plate and duralumin.
TL;DR: The theory of compressible flow in a laminar boundary layer has been developed for the case when the viscosity is proportional to the absolute temperature and the Prandtl number is unity.
Abstract: The theory of compressible flow in a laminar boundary layer has been developed for the case when the viscosity is assumed to be proportional to the absolute temperature and the Prandtl number is unity. (These assumptions may be compared with the empirical relations u∝ oc T® and σ = 0*715 suggested by Cope.) It is shown that a transformation of the ordinate normal to the layer can lead to a simplified form of equation of motion very similar to the ordinary incompressible equation but modified by a multiplicative factor G in the pressure term. This factor is greater than unity at the boundary and tends to one at the outside of the layer. Several particular solutions are considered including accelerated flow with a linearly increasing velocity and retarded flow along a flat plate with a linearly decreasing velocity. The general implications of the theory are discussed and qualitative conclusions are drawn when the mainstream velocity starts from a stagnation point, rises to a maximum and subsequently falls. It is concluded that for such a velocity distribution increasing compressibility will reduce the skin friction, increase the boundary layer thickness and cause earlier separation as compared with the incompressible flow with the same mainstream velocity distribution and the kinematic viscosity corresponding to conditions at the stagnation point.
TL;DR: In this article, the Vegard's law of apparent atomic diameter (A.A.D) was used to measure the lattice spacings of solid solutions of lithium, magnesium, silicon, copper, zinc, germanium and silver in aluminium.
Abstract: Measurements have been made of the lattice spacings of solid solutions of lithium, magnesium, silicon, copper, zinc, germanium and silver in aluminium. The lattice of aluminium is expanded by the solution of magnesium or germanium, and contracted by the solution of lithium, silicon, copper or zinc. No change in lattice spacing can be detected when silver is dissolved in aluminium, although microscopic examination shows that a solid solution is formed, and this is confirmed by the absence of any diffraction lines other than those of the solid solution in aluminium. If the lattice spacing/composition curve for dilute solutions is extrapolated to 100% of solute, the resulting lattice spacing refers to a hypothetical face-centred cubic modification of the solute, and the corresponding closest distance of approach of the atoms is called the apparent atomic diameter (A.A.D.) of the solute when in solution in aluminium. Previous work enables the corresponding A.A.D. values to be deduced for the above solute elements when dissolved in univalent copper, silver or gold, and in divalent magnesium. The differences between the A.A.D. values of a given element when dissolved in various solvents are discussed, and it is suggested that they are controlled by the interplay of four factors: (1) the relative volume per valency electron in crystals of the solvent and solute, (2) the relative radii of the ions of solvent and solute, (3) Brillouin zone effects, and (4) the difference between solvent and solute in the electrochemical series. If this line of approach adopted be correct, it follows that it is only in exceptional circumstances that the so-called Vegard's law will apply to metallic solid solutions.
TL;DR: In this paper, the effect of a magnetic field on the thermoelectric power of a metal containing two overlapping energy bands of normal form was discussed, and exact solutions of the transport equation were obtained for the three limiting cases of high temperatures, low temperatures and very strong magnetic fields, and it was shown that the formulae can be generalized to give approximate expressions for all temperatures and all fields.
Abstract: The methods of a previous paper are used to discuss the effect of a magnetic field on the thermoelectric power of a metal containing two overlapping energy bands of normal form. Exact solutions of the transport equation are obtained for the three limiting cases of high temperatures, low temperatures and very strong magnetic fields, and it is shown that the formulae can be generalized to give approximate expressions for all temperatures and all fields. The magnetic change of the thermoelectric power is found to be small at very low and high temperatures, and to pass through a maximum at intermediate temperatures. The transverse galvano- and thermomagnetic effects are also considered, and the formulae which hold for free electrons are generalized so as to be approximately valid for all temperatures. For free electrons, the Hall coefficient remains constant as the temperature decreases, the Righi-Leduc coefficient increases, and the Ettingshausen and Ettingshausen-Nernst coefficients decrease and change sign at very low temperatures. The corresponding formulae for a metal containing two bands are also obtained, and are used to show that the theoretical predictions for free electrons cannot hold for real metals except in special cases. Finally, the two-band model is used to discuss the effect of the magnitude of the magnetic field on the coefficients of the transverse effects.
TL;DR: The mechanism of deformation of metals at high velocities has been studied by examining the microstructure of cylinders of mild steel, duralumin and standard silver of similar hardness.
Abstract: The mechanism of deformation of metals at high velocities has been studied by examining the microstructure of cylinders of mild steel, duralumin and standard silver (of composition Ag 92$\cdot $5%, Cu 7$\cdot $5%) of similar hardness, and of steel balls after impact on steel plates at velocities from 300 to 3830 ft./sec. The means by which the stress on impact was relieved depended on the material, and was first the formation of twins or 'compression bands', i.e. by block movement of wedges of material within individual grains, or by cracking. When the applied stress could no longer be relieved in this way, plastic deformation occurred. The amount of residual strain in deformed mild steel and duralumin has been examined by observing changes of microstructure after annealing. Hardness surveys were made on longitudinal sections of the cylinders and the results correlated with the microstructures and with observations of the limits of strain due to impact.
TL;DR: The starting field strength of an electrodeless discharge in a uniform high-frequency electric field at pressures of order 10 -3 mm was investigated in this paper, where it was shown that the starting field is independent of the gas and only slightly dependent on its pressure.
Abstract: The starting field strength of an electrodeless discharge in a uniform high-frequency electric field at pressures of order 10 -3 mm. Hg in helium, mercury, hydrogen, and air, is found to be independent of the gas and only slightly dependent on its pressure. By increasing the wavelength (> 4 m.) the starting field varies first inversely with the wave-length, then becomes constant, and at a critical value rises discontinuously probably to infinity. This cut-off wavelength is proportional to the diameter of spherical and the length of cylindrical vessels respectively, the axis of the cylinder being parallel to the field. The facts that the results are independent of the gas and that at these pressures the electrons collide very rarely with gas molecules both indicate that directionization by collision with the gas cannot account for the multiplication of the few electrons originally present. A theory is worked out on the basis that secondary emission from the walls is the operative factor. The initial electrons assumed to move singly in vacuum come to life at various phase angles of the alternating field. However, only those contribute effectively to the multiplication process whose transit time between the walls corresponds to one or an odd number of half-cycles, and whose velocity is large enough to liberate from the wall more than one secondary electron. For each phase angle an appropriate transit time, that is, an appropriate wave-length, can be found to give this requisite final velocity so that discharges are possible over a limited range of wave-lengths corresponding to the allowable phase angles. From similarity follows the proportionality between cut-off wave-length and the size of the vessel. This theory agrees well with the experimental results. The dependence on secondary emission was confirmed because by coating the inner glass wall with a poor secondary emitter the starting field is increased, while the cut-off wave-length decreases; the reverse is observed when clear Pyrex glass is exchanged for soda glass, which agrees with the larger value of the coefficient of secondary emission of the latter.
TL;DR: In this article, the authors considered the problem of determining the distribution of electric charge in crystals, metals, and large molecules in a regular lattice, each particle being acted upon by its nearest neighbours and perhaps by a fixed boundary.
Abstract: In many problems in physics and chemistry certain determinants of large order and of a special type require to be evaluated. The determinants in question usually arise from some kind of secular equation, and in most cases they owe their origin to a system of particles of one kind or another which, in their equilibrium positions, form a regular lattice, each particle being acted upon by its nearest neighbours and perhaps by a fixed boundary. These forces may be assumed to be elastic in character, to the degree of approximation required. For simplicity we shall refer to these particles as atoms although they may be electrons, molecules or even material particles in the Newtonian sense, according to the nature of the problem under investigation. In particular we may mention the occurrence of these determinants in problems involving the solution of Schrodinger's Wave Equation for permitted energy levels and in determining the distribution of electric charge in crystals, metals and large molecules. They have also been used by Born in his investigations of crystal structure by means of X-rays. It is hoped therefore that the results achieved in this paper will be of interest to the physicist and chemist as well as to the mathematician.
TL;DR: The detailed mechanism of the reaction between carbon dioxide and coconut shell charcoal has been studied by both flow and static methods as mentioned in this paper, and accurate measurements of the rate of reaction have been made by the flow method.
Abstract: The detailed mechanism of the reaction between carbon dioxide and coconut shell charcoal has been studied by both flow and static methods. The temperature has been varied in the range 700 to 830 degrees C and the pressures of the gases from 10 to 760 mm. The static method has been used to investigate the adsorption of the gases on the carbon surface during the course of the reaction, and thus to illustrate in a very direct way the general nature of the mechanism. Accurate measurements of the rate of reaction have been made by the flow method. At a given temperature the rate of reaction can be represented in terms of the partial pressures of the gases by an expression of the form rate = $\frac{k\_{1}p\_{\text{CO}\_{2}}}{1+k\_{2}p\_{\text{CO}}+k\_{3}p\_{\text{CO}\_{2}}}$. The three separate constants have been evaluated and each has been found to vary exponentially with temperature. From these results the rates of the individual stages of the mechanism have been calculated. The first stage is the decomposition of the carbon dioxide molecule into an atom of oxygen which is adsorbed by the carbon and a molecule of carbon monoxide which passes into the gas phase. Only certain sites on the charcoal surface take part in the reaction; they represent about 0$\cdot $5% of the total area and probably consist of some of the less firmly bound carbon atoms situated at lattice discontinuities. The rate of the first stage can be accounted for by assuming that reaction occurs in those collisions in which the combined energy of the active carbon atom and the incident carbon dioxide molecule exceeds 68 kcal. The second stage is the evaporation of the adsorbed oxygen atom, together with an atom of carbon from the solid, to form gaseous carbon monoxide; the activation energy is thought to be 38 kcal., and the low value of 10$^{7}$ sec.$^{-1}$ obtained for the non-exponential factor is discussed. The retarding effect of carbon monoxide is due to the adsorption of the gas on the reaction sites, the heat given out in the change being 46 kcal. On the basis of this value, which implies that the molecule is adsorbed chemically, it is possible to calculate theoretically the order of magnitude of the retardation constant, k$_{2}$.
TL;DR: A set of classical relativistic equations of motion of an electron in an electromagnetic field is postulated, which give the same results as the Maxwell-Lorentz theory for non-relativistic motions when the external electromagnetic field does not vary too rapidly.
Abstract: A set of classical relativistic equations of motion of an electron in an electromagnetic field is postulated. These equations are free from ‘run-away’ solutions, and give the same results as the Maxwell-Lorentz theory for non-relativistic motions when the external electromagnetic field does not vary too rapidly. For the scattering of light by an electron, the scattering crosssection is independent of the frequency and is a universal constant. This brings out a point of difference from the Lorentz-Dirac equations according to which the scattering cross-section varies inversely as the square of the frequency of the incident light, for large frequencies. For the motion of an electron towards a fixed proton, the equations allow a collision, unlike the Lorentz-Dirac equations according to which the electron is brought to rest before it reaches the proton.
TL;DR: In this article, the authors examined the mechanism whereby grain boundaries are delineated and striations formed on polished surfaces of heated metal specimens and suggested that the chief means whereby the boundary grooves are formed is that of surface migration of ions: both surface migration and evaporation of silver oxide molecules may play major parts in the formation of the striations.
Abstract: The mechanism whereby grain boundaries are delineated and striations formed on polished surfaces of heated metal specimens has been examined. Experiments on electrolytically polished silver show that grooves form at grain boundaries at temperatures as low as 300° C and striations at 500° C in air. Striations only appear in the presence of oxygen and may be removed by heating in nitrogen. A furnace for high-temperature photomicrography, suitable for specimen temperatures up to about 950° C, is described. Previous theories are found inadequate to explain the effects observed in silver, and a theory which regards the surface etching as an approach to equilibrium by the reduction of surface free energy is suggested. Thus the equilibrium condition of the boundary is a groove whose shape is determined by the relative magnitudes of the free energy per unit area of the boundary and the surface energies of the crystalline faces meeting the boundary in the surface of the specimen. The striations are caused by the development of those crystalline planes having the lowest free energy; the relative surface energies of different planes being modified by the presence of oxygen. It is suggested that the chief means whereby the boundary grooves are formed is that of surface migration of ions: both surface migration of ions and evaporation of silver oxide molecules may be expected to play major parts in the formation of the striations.
TL;DR: In this article, it is shown that at a high loading density only negligibly small amounts of hydrogen and carbon monoxide are present in the detonation wavefront, and this is due to the dominant role of the repulsive forces between the molecules of the tightly compressed gases during the early stages of the expansion.
Abstract: Assuming chemical and thermal equilibrium to be maintained in the detonation wave-front, and using the equation of state in the form of the virial expansion, the velocity of detonation has been determined as a function of the loading density. In the absence of data at sufficiently high pressures and temperatures for the products of detonation of T.N.T., it has been assumed that the virial coefficients are constant and their values have been determined to give agreement with the measured values of the detonation velocity for loading densities less than 1$\cdot $5 g.cm.$^{-3}$. The pressure-volume-temperature relation in the detonation wave-front can then be determined. The pressure in the detonation wave-front is found to be of the order of 2 $\times $ 10$^{11}$ dyne cm.$^{-2}$ for a loading density of 1$\cdot $5 g.cm.$^{-3}$, compared with the value of 9$\cdot $4 $\times $ 10$^{10}$ dyne cm.$^{-2}$ given in the earlier work of other authors using the co-volume method. With the equation of state adopted in this paper it is found that at a high loading density only negligibly small amounts of hydrogen and carbon monoxide are present in the detonation wave-front, a fact which facilitates the calculation of the adiabatic relations in this case. It is shown (part B) that these gases do, however, develop rapidly during the initial stages of the adiabatic expansion. The calculation of the equilibrium conditions in the detonation wave-front with the adopted equation of state (part A) determines the initial conditions for the calculation of the adiabatic relations for a high loading density. The chemical composition of the gases during the adiabatic expansion and the external work done during it have been calculated for a loading density of 1$\cdot $5 g.cm.$^{-3}$ (part B). It is shown that the large amount of chemical energy released in the early stages of the expansion is to be correlated with the high effective value of the exponent in the adiabatic in this region, and this is due to the dominant role of the repulsive forces between the molecules of the tightly compressed gases during the early stages of the expansion. The same effect is also observed in the case of a low loading density (part C). The difference with regard to the amounts of hydrogen and carbon monoxide present in the detonation wave-front for a low loading density complicates the solution of the equations in this case. In part C it is shown how this can be done for a loading density of 1$\cdot $0 g.cm.$^{-3}$, and the detonation velocity and the pressure, density and temperature in the wave-front have been determined using the same equation of state as in parts A and B. The adiabatic pressure-volume relation for the expansion of the products of detonation, and the chemical composition during, and up to the end of, the adiabatic expansion have also been determined. Compared with the results for a high loading density, there is considerably more carbon monoxide and less carbon dioxide and a substantial rise in the total number of moles of gas produced per mole of explosive. The ratio is found to be sensitive to the pressure in the detonation wave-front, from which, by comparison with the observed values of this ratio, independent evidence is obtained for the detonation pressure calculated in part A. The chemical energy released per gram of explosive is less for a loading density of 1$\cdot $0 g.cm.$^{-3}$ than for a loading density of 1$\cdot $5 g.cm.$^{-3}$, and the external work done is also less in the former than in the latter case. The amounts of ammonia and of hydrocyanic acid in chemical equilibrium with the other gases are determined and they are found to be negligibly small. It is concluded that these gases, observed in experiments, are probably formed by catalytic action with the bomb fragments during the cooling period after the adiabatic expansion has been completed. The calculations have been compared with the available experimental data and are in reasonable agreement with it. An explanation is suggested for the observed difference in the composition of the gaseous products of the detonation of T.N.T., initiated at a given loading density, with detonators of different power.
TL;DR: In this article, two methods are described for calculating the distribution of mass at a given depth that will produce a specified gravitational field, and the limit to the possible depth of the masses producing an anomaly is also discussed.
Abstract: Two methods are described for calculating the distribution of mass at a given depth that will produce a specified gravitational field. In one the observed field is extrapolated downwards by a finite difference procedure; in the other a solution is obtained in the form of an integral which can be evaluated numerically. Tables of the quantities necessary for the practical use of the method are provided, and real and artificial examples are given. The limit to the possible depth of the masses producing an anomaly is also discussed.
TL;DR: In this article, the wave functions for the normal configurations of neutral nitrogen and N$ √ n−1/n−2 were calculated by the method of the self-consistent field with exchange (Fock's equations).
Abstract: Wave functions for the normal configurations of neutral nitrogen and N$^{-}$ have been calculated by the method of the self-consistent field with exchange (Fock's equations). To the accuracy of the approximation represented by these equations, the N$^{-}$ ion would be unstable and liable to auto-ionization, but it is estimated that a better approximation of the treatment of a many-electron atom would give a small positive ionization potential for N$^{-}$. Revised wave functions for Na$^{+}$ and the normal state of neutral Na have also been calculated. Tables of results are given.
TL;DR: In this paper, a series of ternary interstitial alloys containing iron, carbon, and nitrogen is described, with a range of homogeneity extending from Fe$\{8}$N$\_{4}$ to Fe${1}$C$\_3}N. The latter is isomorphous with Fe{1]$N.
Abstract: Chemical and X-ray investigation of the reaction of carbon monoxide with iron nitrides and of the reaction of ammonia with iron carbides discloses the existence of iron carbonitrides- a series of new ternary interstitial alloys containing iron, carbon and nitrogen. $\zeta $-phase carbonitrides, with structures similar to those of $\zeta $-iron nitrides, have a range of homogeneity extending approximately from Fe$\_{8}$N$\_{4}$ to Fe$\_{8}$C$\_{3}$N. The latter is isomorphous with Fe$\_{2}$N. $\epsilon $-phase carbonitrides, which are isomorphous with $\epsilon $-iron nitrides, have a composition range of approximately 25 to 33 atomic% nitrogen plus carbon, i.e. from Fe$\_{3}$X to Fe$\_{2}$X, in which the higher carbon concentration limit is not less than 16 atomic%. Probable phase fields for part of the iron-carbon-nitrogen system are given on a ternary diagram. Prolonged reaction of carbon monoxide with iron nitrides results in complete elimination of nitrogen. Below 500 degrees C the product is a carbide of iron, now called iron percarbide, the narrow composition range of which includes Fe$\_{20}$C$_{9}$. Above 500 degrees C the product of the same reaction is cementite.
TL;DR: In this paper, the authors discussed the interaction of two conjugated systems across a single bond and showed that the conjugation energy across the bond and the mobile order of the bond are both positive, and that both are closely related to the self-polarizabilities of the atoms forming the bond.
Abstract: In this paper there is discussed the interaction of two conjugated systems across a conj ugated single bond. It is shown that the conjugation energy across the bond, and the mobile order of the bond, are both positive, and that both are closely related to the self-polarizabilities of the atoms forming the bond. This relation forms the basis for a definition of the conjugating power of a position as β times its self-polarizability. The conjugating power of various hydrocarbon residues is shown to increase in the order: phenyl, β-naphthyl, 2-butadienyl, α-naphthyl, vinyl, 1-butadienyl.
TL;DR: The detailed mechanism of the reaction between steam and coconut shell charcoal has been studied by the method described in the preceding paper as discussed by the authors, and the activation energy for the conversion of an adsorbed oxygen atom into gaseous carbon monoxide was found to be 55 kcal.
Abstract: The detailed mechanism of the reaction between steam and coconut shell charcoal has been studied by the method described in the preceding paper. The temperature has been varied in the range 680 to 800 degrees C and the pressures of the gases from 10 to 760 mm. Steam first reacts with the carbon to give oxygen and hydrogen atoms separately adsorbed on neighbouring sites. An initial dissociation into an adsorbed hydrogen atom and an adsorbed hydroxyl radical is probably followed by the more rapid transfer of the second hydrogen atom to the carbon. Only about 2% of the total surface takes part in the reaction; these sites are distinct from the smaller group which reacts with carbon dioxide, but they are also thought to be atoms at the edges of lattice planes. The rate of the first stage can be accounted for by assuming that reaction occurs in those collisions in which the combined energy of the incident steam molecule and the two active carbon atoms exceeds 75 kcal. Adsorbed hydrogen evaporates rapidly, but in the steady state much remains on the surface. A close correlation has been observed between the fraction of the active sites occupied by hydrogen and the extent to which the reaction is retarded by that gas. Adsorbed oxygen reacts much more slowly to form gaseous carbon monoxide; the latter, which has no retarding effect, is not appreciably adsorbed by the sites accessible to steam. The activation energy for the conversion of an adsorbed oxygen atom into gaseous carbon monoxide is found to be 55 kcal., and the non-exponential factor to be 10$^{11\pm 1\cdot 7}$ sec.$^{-1}$ which may be compared with the value of 10$^{13}$ sec.$^{-1}$ predicted by simple theory. As the active carbon atoms are thought to be exerting less than their maximum valency, it is suggested that the two types differ in the number of extra bonds which they can form. Energetic considerations show that whereas those which can form a single bond should react with steam, only the relatively few capable of forming a double bond should react with carbon dioxide. This theory also explains why hydrogen is strongly adsorbed by both the steam and the carbon dioxide sites, but carbon monoxide only by the latter type. The relation of these views to outstanding problems of the oxygen-carbon and nitrous oxide-carbon reactions is discussed, and an explanation of the main kinetic features of those processes is given.
TL;DR: In this paper, the variation of plastic deformation in aluminium specimens consisting of large crystals has been determined by measuring elongation and hardness at various points after tensile deformation, and also within each grain the deformation near the boundary was greater or smaller than at the centre according to whether the neighbour was more or less deformed.
Abstract: The variation of plastic deformation in aluminium specimens consisting of large crystals has been determined by measuring elongation and hardness at various points after tensile deformation. The deformation varied from grain to grain, and also within each grain the deformation near the boundary was greater or smaller than at the centre according to whether the neighbour was more or less deformed, i. e. there is not necessarily inhibition of slip near grain boundaries. These results were supported by metallographic and X-ray observations. Their importance with respect to the calculation of the stress-strain curve of aggregates from those of single crystals is discussed. It is suggested that a mechanism other than slip operates near the grain boundaries during deformation, and even within the crystals during large extensions.
TL;DR: In this paper, the magnetization curves for iron single crystals above the knee are derived on the basis of domain theory for the case where the specimen is finite and the field is applied in an arbitrary direction with respect to the crystal axes.
Abstract: Magnetization curves for iron single crystals above the ‘knee’ are derived on the basis of domain theory for the case where the specimen is finite and the field is applied in an arbitrary direction with respect to the crystal axes. The shape of the specimen is important for the magnetization process and it is shown that in many cases the demagnetizing field must be such as to make the field actually acting in the crystal have a direction of symmetry (e. g. [111] or [110]) whatever the direction of the applied field. The cases of an oblate spheroid with its equatorial plane a (100) plane of the crystal and of a long rod with arbitrary orientation are considered in detail. In these cases simple expressions for the magnetization curves are obtained. There is good agreement with the experimental results of various authors for both parallel and normal components of magnetization. A method for correcting for the effect of internal strains is indicated.
TL;DR: In this article, a uniform gas of polyatomic molecules is treated as an assembly of classical vibrating systems, which dissociate when one internal coordinate q reaches a critically high value q$\_{0}$ which is related to the dissociation energy E$\_0}$.
Abstract: A uniform gas of polyatomic molecules is treated as an assembly of classical vibrating systems, which dissociate when one internal co-ordinate q reaches a critically high value q$\_{0}$ which is related to the dissociation energy E$\_{0}$. The unimolecular velocity constant at temperature T is found to be K = $
u $ exp (-E$\_{0}$/kT), where the 'frequency factor' $
u $ lies in the range of molecular vibration frequencies. The factor $
u $ may be interpreted (i) as a weighted average of the normal vibration frequencies, or (ii) as the ratio of the product of the normal frequencies to the product of the frequencies with q fixed, or (iii) in the case where dissociation is due to the rupture of an isolated bond, as the vibration frequency of an imaginary diatomic molecule consisting merely of the two bonded atoms connected by the original bond-force. The dissociation rate is formulated in several ways, which represent different aspects of the physical picture. The main contrast is between (a) the rate as the average frequency with which the normal-mode vibrations come sufficiently into phase to carry q to the critical value q$\_{0}$, and (b) 'transition state' formulations, giving the rate as the product of the relative concentration of activated complexes (molecules with q near q$_{0}$) and the mean transition frequency. The mathematical equivalence of these methods is shown by a study of the asymptotic distribution of values of sums of harmonic vibrations. The present model is used to illustrate some concepts of transition state or activated complex theory, such as the 'effective mass' in the reaction co-ordinate, and the partition function of the activated complex. The relation of the model to Kassel's theory is shown by calculating the dissociation rates of molecules of specified total energy.