Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1950"
TL;DR: In this article, it was shown that when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa.
Abstract: It is shown that, when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa. The relationship between the rate of development of the instability and the length of wave-like disturbances, the acceleration and the densities is found, and similar calculations are made for the case when a sheet of liquid of uniform depth is accelerated.
2,839 citations
TL;DR: The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed in this article, where the right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material.
Abstract: The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed. The right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material, but it is necessary to transform to a fixed frame of reference in order to solve the equations of state simultaneously with the equations of continuity and of motion. An illustration is given of the process of formulating equations of state suitable for universal application, based on non-invariant equations obtained from a simple experiment or structural theory. Anisotropic materials, and materials whose properties depend on previous rheological history, are included within the scope of the paper.
1,714 citations
TL;DR: In this paper, a quantitative theory for the shapes of the absorption bands of F -centres is given on the basis of the Franck-Condon principle, and it is shown that the absorption constant as a function of frequency and temperature can be expressed in terms of Bessel functions with imaginary arguments.
Abstract: A quantitative theory for the shapes of the absorption bands of F -centres is given on the basis of the Franck-Condon principle. Underlying the treatment are two simplifying assumptions: namely, ( a ) that the lattice can be approximately treated as a dielectric continuum; ( b ) that in obtaining the vibrational wave functions for the lattice, the effect of the F -centre can be considered as that of a static charge distribution. Under these assumptions, it is shown that the absorption constant as a function of frequency and temperature can be expressed in terms of the Bessel functions with imaginary arguments. The theoretical curves for the absorption constant compare very favourably with the experimental curves for all temperatures. Also considered in the paper are the probabilities of non-radiative transitions, which are important in connexion with the photo-conductivity observed following light absorption by F -centres. The treatment given differs from the qualitative considerations hitherto in one important aspect, namely, the strength of the coupling between the electron and the lattice is taken into account. The adiabatic wave functions for the F -centre electron required for the discussion are obtained by perturbation methods. The probability for an excited F -centre to return to its ground state by non-radiative transitions is shown to be negligible; similar transitions to the conduction band are, however, important if the excited state is separated from the conduction band by not much more than 0·1 eV. The temperature dependence of such transitions is complicated, but, for a wide range of temperatures, resembles e - w/k T . Tentative estimates show that the result is Consistent with the observed steep drop of the photo-conductive current with temperature.
1,314 citations
TL;DR: This paper has now been declassified, and though it has been superseded by more complete calculations, it seems appropriate to publish it as it was first written, without alteration, except for the omission of a few lines, the addition of this summary, and a comparison with some more recent experimental work as discussed by the authors.
Abstract: This paper was written early in 1941 and circulated to the Civil Defence Research Committee of the Ministry of Home Security in June of that year The present writer had been told that it might be possible to produce a bomb in which a very large amount of energy would be released by nuclear fission—the name atomic bomb had not then been used—and the work here described represents his first attempt to form an idea of what mechanical effects might be expected if such an explosion could occur In the then common explosive bomb mechanical effects were produced by the sudden generation of a large amount of gas at a high temperature in a confined space The practical question which required an answer was: Would similar effects be produced if energy could be released in a highly concentrated form unaccompanied by the generation of gas? This paper has now been declassified, and though it has been superseded by more complete calculations, it seems appropriate to publish it as it was first written, without alteration, except for the omission of a few lines, the addition of this summary, and a comparison with some more recent experimental work, so that the writings of later workers in this field may be appreciated An ideal problem is here discussed A finite amount of energy is suddenly released in an infinitely concentrated form The motion and pressure of the surrounding air is calculated It is found that a spherical shock wave is propagated outwards whose radius R is related to the time t since the explosion started by the equation R = S (γ)tt E tρ-t, where ρo is the atmospheric density, E is the energy released and S (γ) a calculated function of of γ, the ratio of the specific heats of air
1,281 citations
TL;DR: In this paper, it was shown that the only obstacle to the evaluation of wave functions of any required degree of accuracy is the labour of computation, and that all necessary integrals can be explicitly evaluated.
Abstract: This communication deals with the general theory of obtaining numerical electronic wave functions for the stationary states of atoms and molecules. It is shown that by taking Gaussian functions, and functions derived from these by differentiation with respect to the parameters, complete systems of functions can be constructed appropriate to any molecular problem, and that all the necessary integrals can be explicitly evaluated. These can be used in connexion with the molecular orbital method, or localized bond method, or the general method of treating linear combinations of many Slater determinants by the variational procedure. This general method of obtaining a sequence of solutions converging to the accurate solution is examined. It is shown that the only obstacle to the evaluation of wave functions of any required degree of accuracy is the labour of computation. A modification of the general method applicable to atoms is discussed and considered to be extremely practicable.
1,036 citations
TL;DR: In this article, the authors describe measurements of the shape and rate of rise of air bubbles varying in volume from 1·5 to 200 cm. 3 when they rise through nitrobenzene or water.
Abstract: Part I describes measurements of the shape and rate of rise of air bubbles varying in volume from 1·5 to 200 cm. 3 when they rise through nitrobenzene or water. Measurements of photographs of bubbles formed in nitrobenzene show that the greater part of the upper surface is always spherical. A theoretical discussion, based on the assumption that the pressure over the front of the bubble is the same as that in ideal hydrodynamic flow round a sphere, shows that the velocity of rise, U , should be related to the radius of curvature, R , in the region of the vertex, by the equation U = 2/3√( gR ); the agreement between this relationship and the experimental results is excellent. For geometrically similar bubbles of such large diameter that the drag coefficient would be independent of Reynolds’s number, it would be expected that U would be proportional to the sixth root of the volume, V ; measurements of eighty-eight bubbles show considerable scatter in the values of U/V 1/6 , although there is no systematic variation in the value of this ratio with the volume. Part II. Though the characteristics of a large bubble are associated with the observed fact that the hydrodynamic pressure on the front of a spherical cap moving through a fluid is nearly the same as that on a complete sphere, the mechanics of a rising bubble cannot be completely understood till the observed pressure distribution on a spherical cap is understood. Failing this, the case of a large bubble running up a circular tube filled with water and emptying at the bottom is capable of being analyzed completely because the bubble is not then followed by a wake. An approxim ate calculation shows that the velocity U of rise is U = 0·46 √( ga ), where a is the radius of the tube. Experiments with a tube 7·9 cm. diameter gave values of U from 29·1 to 30·6 cm./sec., corresponding with values of U /√( ga ) from 0·466 to 0·490.
999 citations
TL;DR: In this article, a simple statistical theory, based upon the calculation of the probability of occurrence of sequences of pairs of pores of all the possible sizes, and of the contribution to the permeability made by each such pair, leads to an expression of the porosity as the sum of a series of terms.
Abstract: The permeability of a porous material to water is a function of the geometry of the boundary between the solid component and the pore space. Expressions of the Kozeny type purporting to represent this function are based upon the particle size or specific surface of the solids, and whilst, for engineering practice, they have given satisfaction for saturated sands, they may fail badly in other cases. By developing a Kozeny type of expression for the particular structure of a bundle of capillary tubes of assorted radii, we demonstrate the cause of the failure. Such failure may be avoided by relating permeability to pore-size distribution, which is the factor of prime concern and which may be measured directly by even simpler means than are used to determine particle-size distribution. The pore-size distribution is arrived at by an interpretation of the moisture characteristic of the material, i.e. of the curve of moisture content plotted against pressure deficiency. A simple statistical theory, based upon the calculation of the probability of occurrence of sequences of pairs of pores of all the possible sizes, and of the contribution to the permeability made by each such pair, leads to an expression of the permeability as the sum of a series of terms. By stopping the summation at a selected upper limit of pore size one may calculate the permeability at any chosen moisture content and plot it as a function of that content. An example is presented, using a coarse graded sand specified by its moisture characteristic. To check these calculations, experimental determinations of the permeabilities of unsaturated materials are presented, using two different grades of sand and a sample of slate dust, the results being compared with computed values. The agreement seems good, and is certainly better than that provided by the Kozeny formula as developed, with difficulty, for the purpose. The limitations and possible improvements of our concept are very briefly discussed, and finally it is shown how a combined use of the moisture characteristic and the permeability (which is itself derivable from the moisture characteristic) leads to an expression for the coefficient of diffusion of water in the material as a function of moisture content. From this it should be possible, in principle, to calculate in suitable cases the course of water movement down a gradient of moisture content. Such a calculation awaits a satisfactory solution of the problem of non-linear diffusion.
794 citations
TL;DR: In this article, the authors measured the radius of the luminous globe or "ball of fire" which spread out from the centre of the first atomic explosion in New Mexico, and the radius, R, was determined for a large range of values of t, the time measured from the start of the explosion.
Abstract: Photographs by J. E. Mack of the first atomic explosion in New Mexico were measured, and the radius, R, of the luminous globe or 'ball of fire' which spread out from the centre was determined for a large range of values of t, the time measured from the start of the explosion. The relationship predicted in part I, namely, that R$^{\frac{5}{2}}$ would be proportional to t, is surprisingly accurately verified over a range from R=20 to 185 m. The value of R$^{\frac{5}{2}}$t$^{-1}$ so found was used in conjunction with the formulae of part I to estimate the energy E which was generated in the explosion. The amount of this estimate depends on what value is assumed for $\gamma $, the ratio of the specific heats of air. Two estimates are given in terms of the number of tons of the chemical explosive T.N.T. which would release the same energy. The first is probably the more accurate and is 16,800 tons. The second, which is 23,700 tons, probably overestimates the energy, but is included to show the amount of error which might be expected if the effect of radiation were neglected and that of high temperature on the specific heat of air were taken into account. Reasons are given for believing that these two effects neutralize one another. After the explosion a hemispherical volume of very hot gas is left behind and Mack's photographs were used to measure the velocity of rise of the glowing centre of the heated volume. This velocity was found to be 35 m./sec. Until the hot air suffers turbulent mixing with the surrounding cold air it may be expected to rise like a large bubble in water. The radius of the 'equivalent bubble' is calculated and found to be 293 m. The vertical velocity of a bubble of this radius is ${\textstyle\frac{2}{3}}\ \surd $(g29300) or 35$\cdot $7 m./sec. The agreement with the measured value, 35 m./sec., is better than the nature of the measurements permits one to expect.
661 citations
TL;DR: In this paper, it was shown that the magnetic energy of the disturbance will increase provided the conductivity is greater than a critical value determined by the viscosity of the fluid, and the rate of growth of magnetic energy is approximately exponential, with a doubling time which can be simply related to the properties of the turbulence.
Abstract: Several recent investigations in geophysics and astrophysics have involved a consideration of the hydrodynamics of a fluid which is a good electrical conductor. In this paper one of the problems which seem likely to arise in such investigations is discussed. The fluid is assumed to be incompressible and in homogeneous turbulent motion, and externally imposed electric and magnetic fields are assumed to be absent. The equations governing the interaction of the electromagnetic field and the turbulent motion are set up with the same assumptions as are used to obtain the Maxwell and current flow equations for a metallic conductor. It is shown that the equation for the magnetic field is identical in form with that for the vorticity in a non-conducting fluid; immediate deductions are that lines of magnetic force move with the fluid when the conductivity is infinite, and that the small-scale components of the turbulence have the more powerful effect on the magnetic field. The first question considered is the stability of a purely hydrodynamical system to small disturbing magnetic fields, and it is shown that the magnetic energy of the disturbance will increase provided the conductivity is greater than a critical value determined by the viscosity of the fluid. The rate of growth of magnetic energy is approximately exponential, with a doubling time which can be simply related to the properties of the turbulence. General mechanical considerations suggest that a steady state is reached when the magnetic field has as much energy as is contained in the small-scale components of the turbulence. Estimates of this amount of energy and of the region of the spectrum in which it will lie are given in terms of observable properties of the turbulence.
409 citations
TL;DR: In this article, an approximation to the heat transfer rate across a laminar incompressible boundary layer, for arbitrary distribution of main stream velocity and of wall temperature, is obtained by using the energy equation in von Mises's form, and approximating the coefficients in a manner which is most closely correct near the surface.
Abstract: An approximation to the heat transfer rate across a laminar incompressible boundary layer, for arbitrary distribution of main stream velocity and of wall temperature, is obtained by using the energy equation in von Mises's form, and approximating the coefficients in a manner which is most closely correct near the surface. The heat transfer rate to a portion of surface of length l (measured downstream from the start of the boundary layer) and unit breadth is given as -$\frac{\frac{1}{2}k}{(\frac{1}{3})!}\left(\frac{3\sigma \rho}{\mu ^{2}}\right)^{\frac{1}{3}}\int\_{0}^{l}\left(\int\_{x}^{l}\surd \{\tau (z)\}\,dz\right)^{\frac{2}{3}}$ dT$\_{0}$(x), where k is the thermal conductivity of the fluid, $\sigma $ its Prandtl number, $\rho $ its density, $\mu $ its viscosity, $\tau $(x) is the skin friction, and T$\_{0}$(x) the excess of wall temperature over main stream temperature. A critical appraisement of the formula (section 3) indicates that it should be very accurate for large $\sigma $, but that for $\sigma $ of order 0$\cdot $7 (i.e. for most gases) the constant $\frac{1}{2}$3$^{\frac{1}{3}}$/ ($\frac{1}{3}$)! = 0$\cdot 807$ should be replaced by 0$\cdot $73, when the error should not exceed 8% for the laminar layers that occur in practical aerodynamics. This yields a formula Nu = 0$\cdot $52$\sigma ^{\frac{1}{3}}$(R$\overline{\surd C\_{f}}$)$^{\frac{2}{3}}$ for Nusselt number in terms of the Reynolds number R and the mean square root of the skin friction coefficient C$\_{f}$, in the case of uniform wall temperature. However, for the boundary layer with uniform main stream, the original formula is accurate to within 3% even for $\sigma $ = 0$\cdot $7. By known transformations an expression is deduced for heat transfer to a surface, with arbitrary temperature distribution along it, and with a uniform stream outside it at arbitrary Mach number (equation (42)). From this, the temperature distribution along such a surface is deduced (section 4) in the case (of importance at high Mach numbers) when heat transfer to it is balanced entirely by radiation from it. This calculation, which includes the solution of a non-linear integral equation, gives higher temperatures near the nose, and lower ones farther back (figure 2), than are found from a theory which assumes the wall temperature uniform and averages the heat transfer balance. This effect will be considerably mitigated for bodies of high thermal conductivity; the author is not in a position to say whether or not it will be appreciable for metal projectiles. But for stony meteorites at a certain stage of their flight through the atmosphere it indicates that melting at the nose and re-solidification farther back may occur, for which the shape and constitution of a few of them affords evidence. An appendix shows how the method for approximating and solving von Mises's equation could be used to determine the skin friction as well as heat transfer rate, but this line seems to have no advantage over established approximate methods.
346 citations
TL;DR: In this paper, an apparatus for accelerating small quantities of various liquids vertically downwards at accelerations of the order of 50g ( g being 32.2 ft/sec) is described, and the behavior of small wave-like corrugations initially imposed on the upper liquid surface has been observed by means of high-speed shadow photography.
Abstract: An apparatus for accelerating small quantities of various liquids vertically downwards at accelerations of the order of 50g ( g being 32.2 ft./sec. 2 ) is described, and the behaviour of small wave-like corrugations initially imposed on the upper liquid surface has been observed by means of high-speed shadow photography. The instability observed under a wide variety of experimental conditions has been analyzed, and the initial phases have been found to agree well with the first-order theory given in part I. When the disturbance has attained a considerable amplitude the first-order equations cease to apply and it changes from a wave into a form which has the appearance of large round-ended columns of air extending into the liquid and separated by narrow sheets of liquid. The air columns attain a steady velocity relative to the accelerating liquid and continue to penetrate into the liquid until the lower surface of the liquid is reached. In spite of these very large surface disturbances, the main body of liquid below them is accelerated as though they did not exist.
TL;DR: In this paper, the non-orthogonality of atomic orbitals has been considered and generalized definitions of fractional bond order and charge distribution are introduced, and it is shown that under certain conditions, which are satisfied by most conjugated hydro-carbons, the overlap factor is completely irrelevant in calculating bond orders and charges.
Abstract: This part reconsiders the previously neglected non-orthogonality of atomic orbitals. New generalized definitions of fractional bond order and charge distribution are introduced, and it is shown that under certain conditions, which are satisfied by most conjugated hydro-carbons, the overlap factor is completely irrelevant in calculating bond orders and charges. For heteromolecules this is not the case, and substantial changes may sometimes be made by its inclusion. The various polarizability coefficients introduced in earlier papers are always affected, but numerical results suggest that these changes are not very large.
TL;DR: In this paper, simple approximate methods have been developed for evaluating the electrical conductivity of films and wires of a size comparable with the mean free path of the conduction electrons of a thin wire.
Abstract: In the first part of this paper, simple approximate methods have been developed for evaluating the electrical conductivity of films and wires of a size comparable with the mean free path of the conduction electrons. In the second part, a rigorous theory has been given of the electrical conductivity of a thin wire, on the assumptions that the Fermi velocity surface is spherical and that the collisions of the electrons at the surface of the wire are inelastic. In the third part of the paper, this theory has been generalized to cover the case where the scattering is no longer inelastic. In the final part, Andrew’s recent experimental results for a thin mercury wire have been fitted to the theoretical curves obtained, and the mean free path evaluated.
TL;DR: The amplitude of the long sea waves is found to be approximately proportional to the amplitude of ordinary waves and independent of their period as discussed by the authors, and it has been shown that they are due to the varying height of groups of waves breaking on the shore.
Abstract: Long sea waves with periods of 2 to 3 min. and a few inches in amplitude have been measured. It has been shown that they are due to the varying height of groups of waves breaking on the shore. The amplitude of the long waves is found to be approximately proportional to the amplitude of the ordinary waves and independent of their period. The mechanism by which they are produced is discussed.
TL;DR: In this paper, it was shown that the surface tension forces due to thin films of liquid trapped between solid surfaces may be very large and that the absence of adhesion between clean hard surfaces is not due to the non-formation of metallic junctions.
Abstract: Earlier experiments have shown that the friction between metals is due to the shearing of junctions formed by adhesion or welding at the points of intimate contact. This suggests that when the load is removed the junctions should remain and an appreciable normal force should be needed to separate the surfaces. Experiments show that with clean hard surfaces in dry air the adhesion is negligibly small. In moist air appreciable adhesion may be observed, and it is shown that this is due to the surface tension of a thin film of adsorbed water. The surface-tension forces due to thin films of liquid trapped between solid surfaces may be very large. Under certain conditions the viscosity of the liquid may also be important. The absence of adhesion between clean hard surfaces is not due to the non-formation of metallic junctions. Experiments show that it is due to the released elastic stresses which break the junctions one by one as the load is removed. With very soft metals, such as lead or indium, where the effect of released elastic stresses is very much less important, marked adhesion is observed in air, if the surfaces are freed of grosser contaminants. This adhesion provides direct evidence for the formation of metallic junctions by a process of cold welding or pressure welding at the points of Contact. If the surfaces are covered with oxide films of appreciable thickness, the amount of metallic interaction is diminished with a corresponding reduction in the adhesion. Lubricant films have a similar effect, and in general those materials which are most effective in reducing the adhesion are also most effective, as boundary lubricants, in reducing the friction.
TL;DR: In this paper, a new technique of flash photolysis and spectroscopy has been developed, using gas-filled flash discharge tubes of very high power, and the properties of these lamps as spectroscopic and photochemical sources have been studied and details of their construction, spectra, duration of flash, and luminous efficiency in the photochemical useful region.
Abstract: Photochemistry provides us with one of the most generally useful methods of studying the reactions of free radicals and atoms, but the concentration of these intermediates in the usual photochemical systems is too low to allow the use of direct physical methods of investigation such as absorption spectroscopy. To overcome this difficulty a new technique of flash photolysis and spectroscopy has been developed, using gas-filled flash discharge tubes of very high power. The properties of these lamps as spectroscopic and photochemical sources have been studied and details are given of their construction, spectra, duration of flash, and luminous efficiency in the photochemicaliy useful region. An apparatus is described which produces a very great photochemical change, in some cases over 80%, in one-thousandth of a second and in a gas at several cm. pressure contained in an absorption tube 1 m. long, and which photographs the absorption spectrum at high resolution in one twenty-thousandth of a second at short intervals afterwards. Examples of the rapidly changing spectra of substances undergoing reaction, including the spectra of some of the intermediate radicals involved, are shown. These include the recombination of chlorine atoms, the absorption Spectra of S 2 and CS obtained during the photochemical decomposition of carbon disulphide and new spectra attributed to the CIO and CH 3 CO radicals.
TL;DR: In this paper, it was shown that the flow behind the detonation front of an explosive contained in a tube strong enough to confine the motion to one dimension is a progressive wave of finite amplitude of the type studied by Riemann.
Abstract: The flow behind the detonation front of an explosive contained in a tube strong enough to confine the motion to one dimension is shown to be a progressive wave of finite amplitude of the type studied by Riemann. The wave is similar at all stages of its progress if the initiation of the explosion is instantaneous, the linear scale of the whole field of flow increasing at a uniform rate. If the products of combustion obey the law Pρ ¯γ = constant the distribution of gas velocity along the tube is linear. If the initiation end of the tube is closed a fixed proportion of the whole detonating column is at rest. This last case has an analogy in three dimensions. The dynamics of spherical detonation from a point in an explosive is analyzed. As in the one-dimensional case, a fixed proportion of the whole volume of burnt gas is at rest. The radial rate of change of the variables, velocity, pressure and density become infinite at the detonation front, but it is unlikely that this result would be true in a real explosive where the time of reaction is not zero. The results are applied in both linear and spherical cases to the detonation of T. N. T., using data given by Jones & Miller (1948).
TL;DR: In this paper, the steady velocity of translation U may be written in the alternative forms U = ''underset 1\to{\overset \infty \to{\sum}}$c$\_{
u}$Q$^{n}$ = ''understand 1/to-overseet \INfty ∞ \to-sum'' d$\_zeta ^{n]$, where Qe denotes the charge on a sphere and n denotes the zeta-potential; the coefficients c$
u $ and d$
Abstract: A theoretical analysis is given of the motion of solid non-conducting charged spheres through symmetrical electrolytes, under the action of an applied electric field. It is shown that the steady velocity of translation U may be written in the alternative forms U = $\underset 1\to{\overset \infty \to{\sum}}$c$\_{
u}$Q$^{
u}$ = $\underset 1\to{\overset \infty \to{\sum}}$ d$\_{
u}\zeta ^{
u}$, where Qe denotes the charge on a sphere and $\zeta $ the zeta-potential; the coefficients c$
u $ and d$
u $ are proportional to the applied field strength and depend upon the radius of the particle and the concentrations, valencies and mobilities of the ions in the electrolyte. A general method is given for calculating the c$\_{
u}$ and d$\_{
u}$ and the first four coefficients of each series found explicitly. Some quite general properties of the c$_{
u}$ are also deduced. It is shown that under certain conditions, the terms of the series for U, apart from the first, are of considerable importance and must be taken into account in estimating $\zeta $ from U.
TL;DR: In this article, the Boltzmann equation was re-examined from a kinetic theory standpoint, and the same expressions are obtained by this method, usually rather more simply, while the physical picture is considerably clarified.
Abstract: A thin film or wire of metal has a lower electrical conductivity than the bulk material if the thickness is comparable with or smaller than the electronic mean free path. Previous workers have obtained expressions for the magnitude of the effect by integrating the Boltzmann equation and imposing the appropriate boundary conditions. The problem is re-examined from a kinetic theory standpoint, and it is shown that the same expressions are obtained by this method, usually rather more simply, while the physical picture is considerably clarified. The method is applied to an evaluation of the conductivity of a thin wire with a magnetic field along the axis, and it is found that the resistivity should decrease as the magnetic field is increased; it should be possible to derive the mean free path and velocity of the conduction electrons by comparison of theory and experiment. The theory has been confirmed by experimental measurements on sodium; estimates of electronic velocity and mean free path are obtained which are in fair agreement with the values given by the free-electron theory.
TL;DR: In this article, the radial gradient of the potential problem with fixed boundaries was shown to be the same as that of the corresponding potential problem in the case of a phase growth controlled by the diffusion of heat, solute, or both together.
Abstract: Particular solutions of the diffusion equation, with radial symmetry, in three and in two dimensions, found originally by Rieck, represent the diffusion field around a spherical or cylindrical new phase, growing from a negligible initial radius in an initially uniform medium, maintaining equilibrium conditions at the growing surface. The resulting diffusion field is most simply described by saying that the radial gradient is the same as that of the corresponding potential problem with fixed boundaries, multiplied by a factor exp (- r 2 /4 Dt ), where r is radius, D diffusivity, t time. The result is applied to phase growth controlled by the diffusion of heat, solute, or both together. It differs appreciably from the static approximation unless the supercooling, or the supersaturation and the solubility, are small.
TL;DR: In this article, it was shown that both the normal and tangential stresses play a part in the deformation of the metallic junctions formed at the interface, and the detailed behaviour of the junctions during the early stages of the sliding process may be expressed quantitatively in terms of von Mises's criterion for plastic deformation under combined normal and Tangential stresses, and there is good agreement between the theoretical relation and the experimental observations.
Abstract: Simultaneous measurements have been made of the friction and adhesion of steel sliding on indium in air. The results show that both the normal and tangential stresses play a part in the deformation of the metallic junctions formed at the interface. When the surfaces are first placed in contact, a minute tangential force is required to initiate relative motion between the slider and the indium surface, since the junctions are already plastic under the applied load. As relative motion proceeds, the region of contact grows with a corresponding increase in the tangential force and the adhesive force. An upper steady state is reached where the tangential force increases more rapidly than the rate of growth of the region of contact and sliding on a macroscopic scale occurs. The detailed behaviour of the junctions during the early stages of the sliding process may be expressed quantitatively in terms of von Mises's criterion for plastic deformation under combined normal and tangential stresses, and there is good agreement between the theoretical relation and the experimental observations. The results emphasize the reality of the cold welding process which occurs at the points of intimate contact when metal surfaces are placed together. The metallic junctions so formed are responsible both for the friction and the adhesion observed. Lubricant films diminish the amount of metallic contact and so lead to a reduction in the friction and adhesion.
TL;DR: In this paper, a simple relation is derived between the compressibility and absorption frequency of ionic crystals, taking into account the change which a displacement induces in the electronic configuration, and is independent of the uncertainty concerning the Lorentz force.
Abstract: A simple relation is derived between the compressibility and absorption frequency of ionic crystals. This relation takes into account the change which a displacement induces in the electronic configuration, and is independent of the uncertainty concerning the Lorentz force. Comparison with experimental data shows very good agreement in the case of the alkali halides. The changes in the electronic configuration of each ion are described in terms of a dipole moment and in terms of 'local distortions'. The latter are caused by forces not included in the Lorentz force, and it is assumed that they are localized in small regions in the immediate neighbourhood of the neighbouring ions. The local distortions are responsible for the deviation of the polarization from the value given by the Lorentz force; from the sign of this deviation it is concluded that they are due mainly to the non-electrostatic repulsive forces. Although this model allows deviations from spherical charge distribution around the ions, it nevertheless permits the splitting of the energy of certain types of displacements into two-body interactions and leads to the Cauchy relations between the elastic constants.
TL;DR: In this paper, the theory of the transformation from molecular orbitals to sets of equivalent orbitals is discussed for the general case when there is more than one occupied molecular orbital of given symmetry and more than 1 equivalent set.
Abstract: The theory of the transformation from molecular orbitals to sets of equivalent orbitals is discussed for the general case when there is more than one occupied molecular orbital of given symmetry and more than one equivalent set. The general transformation is worked out for molecules whose component atoms possess inner shells and lone pairs of electrons. The theory is illustrated by reference to some simple molecules such as water and ammonia. Finally, it is shown how the expression for the total energy of a molecule can be divided up in such a way that the interactions between its localized parts are dealt with separately. The significance of lone pairs of electrons in determining the shape of molecules is pointed out.
TL;DR: A systematic survey has been made of chain models which conform to established bond lengths and angles, and which are held in a folded form by N—H—O bonds, indicating that a further intensive study of these proteins, and in particular of myoglobin which has promising features of simplicity, may lead to a determination of the chain structure.
Abstract: Astbury’s studies of α-keratin, and X-ray studies of crystalline haemoglobin and myoglobin by Perutz and Kendrew, agree in indicating some form of folded polypeptide chain which has a repeat distance of about 5·1 A, with three amino-acid residues per repeat. In this paper a systematic survey has been made of chain models which conform to established bond lengths and angles, and which are held in a folded form by N—H—O bonds. After excluding the models which depart widely from the observed repeat distance and number of residues per repeat, an attempt is made to reduce the number of possibilities still further by comparing vector diagrams of the models with Patterson projections based on the X-ray data. When this comparison is made for two-dimensional Patterson projections on a plane at right angles to the chain, the evidence favours chains of the general type proposed for a-keratin by Astbury. These chains have a dyad axis with six residues in a repeat distance of 10·2 A, and are composed of approximately coplanar folds. As a further test, these chains are placed in the myoglobin structure, and a comparison is made between calculated and observed F values for a zone parallel to the chains; the agreement is remarkably close taking into account the omission from the calculations of the unknown effect of the side-chains. On the other hand, a study of the three-dimensional Patterson of haemoglobin shows how cautious one must be in accepting this agreement as significant. Successive portions of the rod of high vector density which has been supposed to represent the chains give widely different projections and show no evidence of a dyad axis. The evidence is still too slender for definite conclusions to be drawn, but it indicates that a further intensive study of these proteins, and in particular of myoglobin which has promising features of simplicity, may lead to a determination of the chain structure.
TL;DR: In this paper, the temperature-pressure-concentration relationship of the hydrogen-titanium system has been studied and its more important thermodynamic aspects considered, and a part of the equilibrium diagram has been established, and free-energy curves for the α- and β -solutions have been plotted for a number of temperatures.
Abstract: The temperature-pressure-concentration relationships of the hydrogen-titanium system have been studied, and its more important thermodynamic aspects considered. Part of the equilibrium diagram has been established, and free-energy curves for the α- and β -solutions have been plotted for a number of temperatures. The experimental results have been used to obtain values for the latent heat of the α ⇌ β transformation and the transformation temperature.
TL;DR: In this paper, the authors examined the horizontal and vertical movements of short-period disturbances in the ionosphere and found that quasi-periodic travelling disturbances with periods of from 10 to 60 min. are of frequent occurrence in the F region by day.
Abstract: Methods have been developed for the examination of the horizontal and vertical movements of short-period disturbances in the ionosphere. It has been found that quasi-periodic travelling disturbances with periods of from 10 to 60 min. are of frequent occurrence in the F region by day. They appear as temporary variations in the vertical distribution of ionization which show a horizontal progression and a vertical progression downwards. The horizontal directions of travel have a well-defined mean direction on most days. The mean direction shows a marked seasonal variation with a sudden change at each equinox. The horizontal rate of travel is usually between 5 and 10 km./min., and the rate of vertical progression downwards is approximately half the horizontal rate. The disturbances are considered to be variations of a compressional type in the atmosphere resulting in changes in the distribution of ionization.
TL;DR: In this paper, a perturbation theory is developed to deal with the changes produced in molecular systems when disturbed from their normal states, and it is shown that certain molecular properties can be evaluated using either equivalent or molecular orbitals.
Abstract: The discussion of molecular orbitals and equivalent orbitals, given in previous papers, is carried a stage further. It is shown that certain molecular properties can be evaluated using either equivalent or molecular orbitals. On the other hand, a study of the changes produced by ionization demonstrates that molecular orbitals have a special significance and that certain energy parameters associated with them are closely related to ionization potentials. For the purpose of this discussion a perturbation theory is developed to deal with the changes produced in molecular systems when disturbed from their normal states.
TL;DR: In this paper, an approximate wave function expressed in terms of exponential functions, spherical harmonics, etc., with numerical coefficients has been calculated for the ground state of the beryllium atom.
Abstract: An approximate wave function expressed in terms of exponential functions, spherical harmonics, etc., with numerical coefficients has been calculated for the ground state of the beryllium atom . Judged by the energy criterion this gives a more accurate result than the Hartree result which was the best previously known. This has been calculated as a trial of a fresh method of calculating atomic wave functions. A linear combination of Slater determinants is treated by the variational method. The results suggest that this will provide a more powerful and convenient method than has previously been available for atoms with more than two electrons.
TL;DR: In this paper, experiments and theoretical work concerned with the variation of conductivity with size in metals are described. But the results for the conductivity in thin wires of pure sodium of varying diameter in the absence of a magnetic field and also in the presence of longitudinal and transverse magnetic fields are given.
Abstract: This paper records experiments and theoretical work concerned with the variation of conductivity with size in metals. Experimental results for the conductivity in thin wires of pure sodium of varying diameter in the absence of a magnetic field and also in the presence of longitudinal and transverse magnetic fields are given. Using the general statistical theory of metals the variation of resistance with size in the case of conductivity wires of square cross-section is calculated for comparison with the first set of experiments. A theoretical investigation follows of the alteration in conductivity produced in metallic films by the application of transverse magnetic fields, and this is compared with the corresponding experimental results obtained on the sodium cylinders.
TL;DR: In this paper, it was shown that only five of the stresses in an arbitrarily chosen reference configuration can be explicitly represented, namely, all the anisotropic stresses, and the expressions for the elastic constants can be obtained subject to the condition that the remaining stress, namely an isotropic pressure, vanishes.
Abstract: A scheme has recently been developed to present the general theories in lattice dynamics without specific assumptions about the atomic interactions. The present note aims at clarifying some basic points in using this scheme, as well as giving the correct expressions for the elastic constants (the available results given in previous works are shown to be correct only for central forces in spite of the use of the general scheme). The fact is emphasized that neither the potential energy of a homogeneous deformation from a reference configuration, nor the complete stresses in the configuration can in general be represented in the general scheme. Hence, for instance, the elastic constants cannot be deduced by straightforward means, nor the equilibrium conditions (vanishing stresses) imposed; the latter itself being necessary for the definition of the usual elastic constants. A different technique is shown to be necessary for discussing such problems; one result obtained is that only five of the stresses in an arbitrarily chosen reference configuration can be explicitly represented, namely, all the anisotropic stresses. After introducing the condition that these stresses should vanish, the expressions for the elastic constants can be obtained, which are, however, to be used subject to the condition that the remaining stress, namely, an isotropic pressure, vanishes. Only when the general theory is applied to a concrete case, can the latter condition be explicitly introduced by the use of the given interaction. The results are illustrated by the special example of central forces in the last section. The Cauchy relations follow as an incidental result; the assumptions upon which the relations rest are clearly exhibited in the simple derivation.