Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1935"

Statistical Theory of Turbulence

Abstract: Since the time of Osborne Reynolds it has been known that turbulence produces virtual mean stresses which are proportional to the coefficient of correlation between the components of turbulent velocity at a fixed point in two perpendicular directions. The significance of correlation between the velocity of a particle at one time and that of the same particle at a later time, or between simultaneous velocities at two fixed points was discussed in 1921 by the present writer in a theory of “Diffusion by Continuous Movements.” The recent improvements in the technique of measuring turbulence have made it possible actually to measure some of the quantities envisaged in the theory and thus to verify some of the relationships then put forward. The theory has also been developed in several directions which were not originally contemplated. The theory, as originally put forward, provided a method for defining the scale of turbulence when the motion is defined in the Lagrangian manner, and showed how this scale is related to diffusion. It is now shown that it can be applied either to the Lagrangian or to the Eulerian conceptions of fluid flow.

Statistical Theory of Superlattices

Abstract: In a recent paper, Bragg and Williams have pointed out that the arrangement of the atoms in an alloy depends in a striking way on the temperature. At high temperatures, the atoms are distributed practically at random among the lattice points of the crystal, but at low temperatures a superlattice may be formed such that the atoms of one kind are arranged in a regular lattice of their own and the atoms of the other kind occupy the remaining “sites” in the crystal. The transition from the ordered to the disordered state occurs in a fairly small temperature range, and is accompanied by a large specific heat, an increase in electric resistance, etc. The mathematical method employed by Bragg and Williams is similar to that used in Weiss’s theory of ferromagnetism . Both involve the assumption that the “force” tending to produce order at a given point is uniquely determined by the average state of order throughout the crystal. Actually it will depend on the configuration of the atoms in the immediate neighbourhood of the point under consideration. The order of the crystal as a whole determines this configuration only on the average. In the present paper, the effect of fluctuations in configuration, which was neglected by Bragg and Williams, will be taken into account.

On the Instability of a Cylindrical Thread of a Viscous Liquid Surrounded by Another Viscous Fluid

Abstract: 1—The dynamical theory of the instability of a long cylindrical column of an incompressible perfect liquid under the action of capillary force has been given by Rayleigh, neglecting the effect of the surrounding fluid. According to his results, if the column becomes varicose with wave-length λ , the equilibrium of the column is unstable, provided λ exceed the circumference 2π a of the cylinder, in accordance with the result of Plateau’s statical theory; and the degree of instability, as indicated by the value of q in the exponential eqt to which the motion is assumed to be proportional, depends upon the value of λ reaching a maximum when λ = 4.51 × 2 a . The case of a long cylindrical column of an incompressible viscous liquid has also been discussed by Rayleigh, again leaving out of consideration the effect of the surrounding fluid. Assuming the viscosity to be very great compared with the inertia and neglecting the effect of the latter, he has shown that for a very viscous liquid column the maximum instability occurs when the wave-length of the varicosity is very large in comparison with the radius of the cylinder, i. e ., when λ = ∞ theoretically. Quite recently G. I. Taylor has made interesting experimental researches, together with some theoretical investigations, upon the mode of formation of the cylindrical thread by the disruptive effect of the viscous drag of one fluid on the other, by putting a small drop of a viscous liquid in definable shearing fields of flow of another viscous liquid. He has thus thrown much light upon the mechanism of the formation of emulsions. In the course of his experiments he observed an interesting phenomenon, in one case when the ratio of the viscosity of the liquid forming the thread to that of the surrounding liquid is 0.91, that after the apparatus which was used to produce the field of flow was stopped the final thread gradually broke up into a number of small drops spaced at nearly regular intervals, although it had seemed quite stable while the apparatus was in motion. In connection with this interesting phenomenon, Professor G. I. Taylor kindly suggested to the writer a problem of investigating the character of the equilibrium of a long cylindrical thread of a viscous liquid surrounded by an­other viscous fluid under the action of interfacial surface tension as well as under the effect of viscous forces acting on the liquid inside the column by the surrounding viscous fluid. The effect of the latter is expected to play some important role in the phenomenon under con­sideration, although, as mentioned already, its effect had been neglected by Rayleigh in his investigation.

The effect of thermal agitation on atomic arrangement in alloys

Abstract: When two metals are allowed together in various proportions, a series of solid phases is formed. A characteristic phase diagram of a binary alloy system has regions of single phase, throughout which the alloy is homogeneous, alternating with regions in which two neighbouring phases coexist. The composition of a single phase can be varied continuously over a certain range. This feature of an alloy is in contrast to tire constant atomic ratio of a chemical compound, and is explained by the nature of the binding forces in an alloy which are predominantly those between the metal atoms of both kinds on the one hand and the common electronic system on the other hand, as opposed to the binding forces between atom and atom which predominate in other chemical compounds. Not only may the atomic ratio in a given phase be varied, but also an orderly space distribution of one kind of atom relative to the other, as found in topical chemical compounds, does not necessarily exist in an alloy. Although each phase is distinguished by possessing a characteristic crystalline structure which differs from that of other phases in the same alloy system, yet this structure may be merely an orderly arrangement of sites occupied by atoms. The manner in which the atoms are distributed amongst the sites of a given phase is often variable, and is, for instance, affected by the thermal treatment which the alloy has undergone.

The electromagnetic equations of the supraconductor

Abstract: Electric currents are commonly believed to persist in a supra-conductor without being maintained by an electromagnetic field. Thus the relation between the field strength E and the current density J in a supraconductor has sometimes been described by means of an "acceleration equation," of the form ∆J = E; A = m / ne 2. (1) This equation, which might replace Ohm;s law for supraconductors, simply expresses the influence of the electric part of the Lorentz force on freely movable electrons of the mass m and charge e , the number per cm3 being n (we use rational units). By definition the constant A must be positive. As a direct consequence of this equation (1) stationary currents in supraconductors are possible when E = 0. We shall see, however, that actually equation (1), which we will refer to as the "acceleration theory," implies more than is verified by experiment; moreover, presupposing an acceleration without any friction it implies a premature theory, the development of which has presented a hopelessly insoluble problem to mathematical physicist. Apparently a model was wanted which would explain that in its most stable state the supraconductor has always a persistent current. We shall give a formulation which is somewhat more restricted in this respect. On the other had it includes one more important fact, namely, the experiment of Meissner and Ochsenfeld. In this way we get a new description of the electromagnetic field in a supraconductor, which is consistent and, as it eliminates unnecessary statements, is in closer contact with experiment. This new description seems to provide an entirely new point of view for a theoretical explanation.

Progressive Lightning. VI

Abstract: An account has previously been given of some results obtained with the Boys camera in the study of lightning discharges and of some deductions as to the nature of these discharge. The present paper is concerned with further studies in the same field with improved technique and apparatus. The material now available includes 41 photographs comprising 95 separate flashes and over 200 lightning strokes. These have been obtained from 13 different thunderstorms over a period of three years, and may therefore be considered to offer a fair sample of the lightning discharges of Southern Africa. We have derived from them a general picture of the nature of the discharge and the manner of its development which seems to be valid for the great majority of the cases studied.

Quantum theory of the diplon

Abstract: The work of Heisenberg Majorana and Wigner seems to show that the behaviour of protons and neutrons and their interaction in the nucleus may be described by the ordinary methods of quantum mechanics It is of particular interest to study the simplest nuclear system, i e , the diplon, which almost certainly consists of a proton and a neutron In dealing with such a two-body problem, the wave equation can be rigorously solved if the forces are known, and this problem therefore has the same importance for nuclear mechanics as the hydrogen atom has for atomic theory The force acting between a proton and a neutron has been investigated by Wigner ( loc cit ) who showed that in order to understand the high mass defect of He4 compared with H2 one must assume interaction forces with a range much smaller than the radius of H2 Without knowing about these forces more than the binding energy of H2, one can, then, investigate quantitatively the behaviour of H2 against various perturbations

A quantum mechanical investigation of the cohesive forces of metallic copper

Abstract: The calculations of Wigner and Seitz have shown that the Bloch theory is capable of yielding, at any rate roughly, the lattice energy, lattice constant, and compressibility of the metal sodium, and it seems probable that their method could be applied with equal success to the other alkalis. On the other hand, no calculations have as yet been made for the noble metals. Such a calculations is given in this paper. The noble metals differ from the alkalis in having cores which are much larger compared with the atomic volume than those of the alkalis, and the main purpose of this paper is to see whether direct interaction between the pressibility, etc. We have also attempted to find out why the noble metals have the face-centred cubic structure, while the alkalis are body-centred. In 2 we calculate the wave functions and energies of the valence electrons (4 s ) of metallic copper. We chose copper for this calculation, because a self-consistent field due to Hartree is available. Wigner and Seitz did not use the self-consistent field for sodium, but a half empirical potential function for the sodium ion, calculated by Prokofjew from the observed term values. This potential therefore includes implicitly the exchange interaction between the valence electrons and the inner shells, so far as this may be represented by an electrostatic potential. With the self-consistent field the exchange has to be taken into account explicitly; we have done this by solving approximately, instead of the simple Schroedinger equation, Fock’s wave equation, which includes the exchange. Apart from this the calculation in 2 is entirely analogous to that of Wigner and Seitz. It will be shown that for copper this calculations given at least the right order of magnitude for the lattice constant and for the heat of vaporization, but a totally incorrect value for the compressibility.

The crystal structure of ice at low temperatures

Abstract: The use of X-ray in the study of the mechanism of crystal growth showed that if the vapour of a substance came in contact with a cold surface, it condensed to form a solid, the structure of which depended on the temperature of the condensing surface. The authors announced recently that ice formed by condensation at low temperatures and pressures also changed form with temperature. Details of the apparatus and the results follow in this paper. 2-Method and Preliminary Work Analysis was carried out by the Hull-Debye-scherrer method, using the Ka doublet of copper. With a camera of 5 cm radius, photographs were obtained in 4 to 6 hours, using a shearer tube operated at 30 KV (R. M. S.) and 10 mA. The Ks line was removed from the radiation by a filter of nickel foil.

Self-Consistent Field, with Exchange, for Beryllium

Abstract: Except for the lightest atoms, most calculations of approximate wave functions and fields for many-electron atoms have been carried out by the method of the “self-consistent field,” of which the principle is, shortly, the determination of a set of one-electron wave functions such that each represents a stationary state of an electron in the field of the nucleus and the Schrodinger charge distribution of the electrons occupying the other wave functions of the set. This method has been found quite practicable for numerical work, even for the heaviest atoms. As so far applied, it involves three main approximations, namely, ( a ) neglect of relativity and spin effects, ( b ) neglect of exchange effects, and ( c ) treatment of the wave function of the whole atom as built up of functions of the co-ordinates of the individual electrons only, its depen­dence on the mutual distances between every pair of electrons being neglected; or, in other words, each electron is replaced by a statistical average distribution, in calculating its effect on the other electrons on the atom.

The statistical mechanics of regular solutions

Abstract: In a previous paper the methods of statistical mechanics were used to obtain the laws of ideally dilute solutions and of perfect solutions. The methods there used will be extended in the present paper to apply to solutions of a more general type which may be called “regular solutions,” a term due to Hildebrand. These various kinds of solutions may be defined as follows. A solution will be ideally dilute if there are no long range forces between the solute molecules and if the ratio of solute to solvent molecules is so small that of all relevant possible configurations the number of them, in which two solute molecules are within range of each other’s field of force, is negligible. A solution will be perfect if, starting with any given configuration, the interchange of position of any molecule of any one species with any molecule of any other species does not alter the total potential energy of the system. We shall apply the epithet “regular” to solutions whose properties correspond, with fair accuracy, to those of a simplified model which we shall now describe. For the sake of simplicity we shall consider a mixture of two species “A” and “B.” The extension to mixtures of more than two species will be obvious. In our model of a regular solution we postulate first the absence of long-range (electrostatic) forces between the molecules. Our second assumption is that the “A” and “B” molecules may be treated as spheres of at least approximately the same size. Thirdly, we assume that each molecule whether of “A” or “B” is directly surrounded by the same number r of other molecules. If the molecules are close- packed, r will have the value 12, but for our present purpose, there is no need to assign any specified value to r, provided its value is the same for the “A” molecules as for the “B” molecules. Fourthly, we assume that the liquids “A” and “B” mix in all proportions without volume change. Our fifth assumption is that, for varying configurations (all of the same volume) of the system, the total potential energy may be regarded as the sum of contributions of each pair of molecules in direct contact. This assumption is slightly less drastic than assuming that the field of a given molecule does not extend beyond the further side of the next molecule. It is equivalent to ignoring differences between the field of an “A” molecule and that of a “B” molecule at distances exceeding one molecular diameter. Obviously our first assumption is included in our fifth.

The Relativistic Self-Consistent Field

Abstract: 1—The method of the self-consistent field for determining the wave functions and energy levels of an atom with many electrons was developed by Hartree, and later derived from a variation principle and modified to take account of exchange and of Pauli’s exclusion principle by Slater* and Fock. No attempt was made to consider relativity effects, and the use of “ spin ” wave functions was purely formal. Since, in the solution of Dirac’s equation for a hydrogen-like atom of nuclear charge Z, the difference of the radial wave functions from the solutions of Schrodinger’s equation depends on the ratio Z/137, it appears that for heavy atoms the relativity correction will be of importance; in fact, it may in some cases be of more importance as a modification of Hartree’s original self-nsistent field equation than “ exchange ” effects. The relativistic self-consistent field equation neglecting “ exchange ” terms can be formed from Dirac’s equation by a method completely analogous to Hartree’s original derivation of the non-relativistic self-consistent field equation from Schrodinger’s equation. Here we are concerned with including both relativity and “ exchange ” effects and we show how Slater’s varia-tional method may be extended for this purpose. A difficulty arises in considering the relativistic theory of any problem concerning more than one electron since the correct wave equation for such a system is not known. Formulae have been given for the inter-action energy of two electrons, taking account of magnetic interactions and retardation, by Gaunt, Breit, and others. Since, however, none of these is to be regarded as exact, in the present paper the crude electrostatic expression for the potential energy will be used. The neglect of the magnetic interactions is not likely to lead to any great error for an atom consisting mainly of closed groups, since the magnetic field of a closed group vanishes. Also, since the self-consistent field type of approximation is concerned with the interaction of average distributions of electrons in one-electron wave functions, it seems probable that retardation does not play an important part. These effects are in any case likely to be of less importance than the improvement in the grouping of the wave functions which arises from using a wave equation which involves the spins implicitly.

Distribution of velocity and temperature between concentric rotating cylinders

Abstract: It is not possible to distinguish between the Momentum Transport and the Vorticity Transport theories of turbulent flow by measurements of the distribution of velocity in a fluid flowing under pressure through pipes or between parallel planes. Only simultaneous measurements of temperature and velocity distribution are capable of distinguishing between the two theories in these cases. On the other hand, it will be seen later that measurements of the distribution of velocity between concentric rotating cylinders are capable of distinguishing between the two theories; in fact the predictions of the two theories in this case are sharply contrasted and mutually exclusive.

The Function of Hydrogen in Intermolecular Forces

Abstract: We already know the essential nature of the four chief types of forces that can exist between atoms in molecules or crystals, and to a certain extent a quantitative quantum mechanical account can be given of them. The metallic, homopolar, ionic, and residual (van der Waals’s) forces are, however, only idealized types, and in very many substances the nature of the interatomic forces partakes at the same time of two or more of these types. Such intermediate forces are more difficult to study, and consequently present some of the most interesting problems of interpretation.

On Gauss' theorem and the concept of mass in general relativity

Abstract: The present communication is concerned with the extension to general relativity theory of the well-known theorem of Gauss on the Newtonian potential, viz., that the total flux of gravitational force through a simple closed surface is equal to (-4π) x the total gravitating mass contained within the surface: and to various questions which arise in connection with this. In the extended theorem, which is found in 2, the Newtonian concept of "gravitating mass" is naturally replaced by that of the energy-tensor, which does not in general consist solely of the "material" energy-tensor, and need not involve any "matter" at all. This new feature is illustrated in 3 by an example in which the "gravitating mass" is simply an electrostatic field. In 4 a theorem of "energy" is obtained which is required later, and which enables us to make precise the concept of the "potential energy" of an infinitesimal particle in a statical field in general relativity; this "potential energy" is shown to be the product of two factors, one depending on the particle alone (which may be called its "potential mass") and the other depending solely on its position. It is shown in 5 that the definition of "potential mass" introduced in 4 enables us to express the generalized Gauss' theorem of 2, in the case when the energy-tensor is due to actual masses, by a simple statement practically identical with the original Gauss' theorem of Newtonian theory. Finally in 6 it is shown that the electrostatical form of Gauss' theorem in Newtonian physics, viz., that the total strength of the tubes of force issuing from a closed surface is equal to the total electric charge within the surface, can also be extended to General Relativity, but that this extension is different in character from the gravitational theorem of 2.

On the Instability of a Fluid when Heated from below

Abstract: It has been known for some times that when a horizontal layer of fluid is heated from below the fluid stationary, if initially at rest, until a certain temperature difference, depending on the physical constant of the fluid and the depth of the fluid layer, is reached. The equilibrium, which becomes less and less stable as the temperature rises, then becomes unstable for infinitely small disturbances, so that the fluid begins to move. Rayleigh,* in 1916, put forward a theory which gave the temperature at which motion first occurs when the top and bottom layers are free surfaces. The results agreed qualitatively with previous experiments due to Bernard. Later Jeffreys calculated a critical temperature for the stability of a fluid between two rigid horizontal conducting planes. The investigation has since been revised by Jeffreys§ and also by Low.¶

A Study of the Effect of Curvature on Fully Developed Turbulent Flow

Abstract: In aeronautics we are especially interested in the flow of air adjacent to surfaces, such as airfoils. There are two main types of flow of real fluids, laminar and turbulent and it is turbulent flow which is of practical importance in aeronautics. We should like to be able to predict the skin friction and flow conditions for any surface of any shape. There has recently been much success with the problem of predicting flow along a flat plate parallel to the direction of flow, and the problem was attacked by investigation of fully developed turbulent flow in straight channels, and direct application of the semi-empirical laws obtained, to the flow along a flat plate. However, surfaces met with in practice are, in general, curved, so that it would be important to be able to predict the effect of curvature on turbulent flow. Most of the previous work in curved flow, however, has been with curved pipes and channels where the behavior of the flow was complicated by secondary vortices. The present work had the purposes of isolating as far as possible the effect of curvature on a fully developed turbulent flow, with two dimensional mean motion. The curved channels used were 5 cm. in breadth and 90 cm. in depth, and had straight entrance sections over 60 x breadth in length to produce a fully developed straight flow before subjecting it to the effect of curvature. Channel I had inner radius 45 cm. and outer radius 50 cm., while channel II had inner radius 20 cm. and outer 25 cm. In addition, measurements were made in an appratus consisting of two concentric cylinders, the inner one of radius 20 cm. and rotating, the outer of radius 25.4 cm. and fixed. The curvature was made of the same order as channel II for purpose of comparison. Measurements on the channels consisted of pressure drop along the channel walls at several speeds, velocity distribution at 30° intervals around the curved portion, velocity distributions at several speeds, and for channel II, determination of the shearing stress at the walls of one of the curved sections. Measurements on the cylinders consisted of velocity distributions at two speeds and determination of shearing stress at the outer wall. Evaluation of results included: calculation of resistance law, calculation of the shearing stress, distribution in radial direction across the curved portion, determination of the exponential law for the velocity distribution near the walls in the various cases, calculation of the "mixing length" 1, from turbulent exchange theory, and several dimensionless methods of plotting velocity distributions to show similarity between measurements in the channels and in the concentric cylinders. Also included are calculations of the laminar flow distribution in a curved channel, and a discussion of Rayleigh's stability criterion. It appears that the distribution of centrifugal force has a strong influence on the stability of the flow, and affects materially the velocity distribution. The fact that similarity can be obtained for several cases by proper dimensionless reduction based on the effective breadth of the mixing region looks hopeful, and it remains for future investigations to determine more facts about the effective breadth of the mixing region.

Contributions to Born's New Theory of the Electromagnetic Field

Abstract: Born’s theory starts from describing the field by two vectors (or a “six-vector”), B, E, the magnetic induction and electric field-strength respectively. A second pair of vectors (or a second six-vector) H, D, is introduced, merely an abbreviation, if you please, for the partial derivatives of the Lagrange function with respect to the components of B and E respectively (though with the negative sign for E). H is called magnetic field and D dielectric displacement. It was pointed out by Born that it is possible to choose the independent vectors in different ways. Four different and, to a certain extent, equivalent and symmetrical representations of the theory can be given by combining each of the two “magnetic” vectors with each of the two “electric” vectors to form the set of six independent variables. Every one of these four representations can be derived from a variation principle, using, of course, entirely different Lagrange functions—physically different, that is, though their analytic expressions by the respective variables are either identical or very similar to each other. In studying Born’s theory I came across a further representation, which is so entirely different from all the aforementioned, and presents such curious analytical aspects, that I desired to have it communicated. The idea is to use two complex combinations of B, E, H, D as independent variables, but in such a way that their “conjugates,” i. e. , the partial derivatives of L , equal their complex conjugates.

Statistical Theory of Turbulence. IV. Diffusion in a Turbulent Air Stream

Abstract: It was pointed out in Part I that experiments on the spread of heat from a line source (e.g., an electrically heated wire) in a turbulent air stream may be expected to give two elements of the statistical specification of turbulence. If the spread is measured near the source the value of the mean transverse component of velocity V/V2, or v' in the notation of Part I, can be found. If the spread is examined further down-stream it should be possible to analyse the results to find the correlation function R,,, which is the principal element of the representation of turbulence in the Lagrangian system.

The creation of electron pairs by fast charged particles

Abstract: We shall discuss in this paper the creation of electron-pairs by the collision of fast charged particles. This calculation goes farther than other calculations on this subject in considering the effect of screening, and in investigating the probability of the creation of a pair as a function of impact parameter, i . e ., the least distance of approach between the two colliding particles. We shall also treat certain other cases which have not been considered before, among them the creation of very slow pairs such that the kinetic energies of the electron and positron of the pair are small compared to their rest energy. When the energy of one of the colliding particles is large compared with its rest mass, we shall also show that to a certain approximation most of the formulae given by the direct calculation can be obtained quite simply by a method similar to that given by Weizsacker for calculating the emission of radiation by fast electrons on colliding with nuclei. The procedure consists in calculating the probability of the transition of an electron from its initial state of negative energy to a final state of positive energy under the perturbing influence of the two colliding particles, the electron and resulting hole then appearing as the electron and positron of the created air. We shall throughout use the Born approximation, in which the interaction between the particles is treated as a perturbation. The transition from the initial to the final state of the system can then happen in two ways. The electron in the negative energy state may either interact with one of the colliding particles and jump at once to its final state, the colliding particle going over into an inter-mediate state. This particle can then interact with the other colliding particle and both jump to their final states. Or, the electron in the negative energy state may interact with one of the colliding particles and jump to an intermediate state, after which its interaction with the other colliding particle causes it to jump to its final state. Both processes are strictly of the second order, but for brevity we shall call the former process a “first-order process,” only in the sense that it involves just one matrix element of the interaction of the electron of the created pair with the colliding particles. The second process involves two matrix elements of the interaction of the electron of the created pair with the colliding particles, and we shall call it a “second-order process.”

The Adsorption of Hydrogen on Tungsten

Abstract: A method for studying the adsorption of hydrogen on a clean tungsten surface has been developed which depends on the fact that the accommodation coefficient of neon is different for a bare surface and for a surface with an adsorbed film on it.It has been shown that on a bare tungsten surface, saturation occurs at a partial pressure of hydrogen of 4.10−4 mm. of mercury and probably much lower, so that the adsorption is of the nature of chemi-sorption.Important deductions can be drawn concerning the process of activation if any for this chemi-sorption of hydrogen on a bare tungsten surface.The accommodation coefficient of neon with a tungsten surface covered with a hydrogen film of the type investigated is 0·17 at 295°K, and 0·32 at 79°K.The experiments described were carried out during my tenure of a Moseley Studentship at the Cavendish Laboratory, and I should like to thank Lord Rutherford for his interest in the work and the Council of the Royal Society for their continued support. I should also like to thank Professor Rideal for his interest in the results.

Isotherms of COFormula between 0 degrees and 150 degrees and Pressures from 16 to 250 Atm (Amagat Densities 18-206)

Abstract: In a previous publication* results of measurements were given of the dielectric constant of CO2 under pressure. In the calculation of the Clausius-Mosotti function, e-1/e+2. 1/ d , from these results and the isotherm data as published by Amagat doubt arose as to the reliability of the latter, and isotherrn measurements have therefore been carried out up to 3000 atm at the same temperature at which the dielectric constant was measured. The results will be given in series of papers, of which the present one describes the measurements up to 250 atm. In this range the method published in previous papers could be used. The apparatus is shown diagrammatically in fig. 1. A is the glass piezometer containing the gas under investigation. It is enclosed in the steel vessel C, which contains some mercury, the remainder being filled with oil. The piezometer consists of a large reservoir B and several small reservoirs connected by narrow capillaries. Through each of these capillaries is sealed a platinum wire, and these all make contact with another platinum wire wound round the outside of the capillary and connected to an insulated lead E through the top of the steel vessel. The volumes from each of the contact to the top of the tube are calibrated by weighing with mercury, using the method described by Michels and Gibson. The volume of gas with which the piezometer is filled is determined separately at 25° and a pressure of about 1 atm, as described by Michels, Wouters, and de Boer. The steel vessel is placed in a thermostat, the temperature of which is electrically regulated in the way previously described by the authors. The temperature is read with thermometers divided in hundredths of a degree and compared with thermometers calibrated at the P. T. R. (Berlin). They were totally immersed in the thermostat and read with the help of a periscope. For the isotherm measurements, pressure is applied to the oil through D, forcing the mercury up inside the piezometer and compressing the gas till the mercury surface makes contact with the platinum wires. This contact is indicated by a drop in the electrical resistance of the platinum wire wound round the capillary. The equilibrium pressure, at which contact is made, is measured with a pressure balance. Corrections must be applied for the height of the mercury column inside the glass piezometer and the hydrostatic head of the oil. From the data thus obtained for p, v , and T, the isotherm can be calculated if the total amount of gas contained in the piezometer is known in standard units.

Phase-equilibrium of supraconductors in a magnetic field

Abstract: At present it seems to be doubtful whether in the theory of supraconductivity the conception of two phases, a supraconducting and a normal phase, can describe the transformation between the two states and how far the equations which were first proposed for the pure supraconducting state in small fields remain true, when both phases are present. Especially it seems to be impossible to describe a phasetransition when an electric field is present, as probably occurs when one cools a wire carrying a constant current. However, where there is only a magnetic field the problem can be treated consistently, and the results are in agreement with experiment, and so it may be worth while to try the two-phase hypothesis, excluding equilibrium states in an electric field. The disturbance of supraconductivity by a magnetic field HT at a temperature t below the transition point has been treated thermodynamically by Gorter as a phase transition, the supraconducting phase being characterized by a zero value of the magnetic induction B

The Zeros of the Riemann Zeta-Function

Abstract: In my previous paper I described calculations which show that all the zeros of ζ ( s ), where s = σ + it , between t = 0 and t = 390 lie on the line σ = ½. With the help of a Government Grant, these calculations have now been carried as far as t =1468. The number of zeros up to this point is 1041, and they all lie on the line σ = ½. I have to thank Dr. L. J. Comrie for planning and supervising the calculations, which were carried out with Brunsviga, National, and Hollerith machines.

The Accommodation Coefficient and the Evaporation Coefficient of Water

Abstract: Previous experiments have already indicated that the maximum rate of evaporation of water into a vacuum is not so great as would be expected theoretically. The ratio of the experimental to the theoretical rate is defined as the evaporation coefficient f and has been found to have a value of about 0⋅04 for pure water at temperatures about 0o C. This result would indicate that, of the vapour molecules striking the liquid surface, about 96% must return to the vapour without entering the liquid. It is therefore of interest to enquire whether these vapour molecules attain temperature equilibrium with the surface or rebound at once before this equilibrium can be established. In the present paper experiments are described in which vapour molecules are incident on a liquid surface which is at a temperature lower than that of the vapour itself and the itself and the energy transferred to the surface by the vapour molecules is measured. If α, the accommodation coefficient, is defined as usual as the ratio of the energy actually transferred to the maximum possible transfer, it is found that for water at 10o C— α= 1⋅0 f = 0⋅036 so that, while only a very small fraction of the vapour molecules enter the liquid, all of them reach temperature equilibrium with the surface before re-evaporating into the vapour. Method If a drop of water is allowed to form on a glass tip in a vessel maintained at a pressure ( p ) which is lower than the saturated vapour pressure corresponding to the temperature of the drop, steady evaporation takes place from the surface of the latter throughout the period of its formation. This evaporation cools the surface. When the drop is fully formed it falls from the tip and may be collected and the drop weight determined. The surface tension can be deduced therefrom and hence the surface temperature may be obtained. This data makes possible the direct calculation of f as follows.

The Interaction of Atoms and Molecules with Solid Surfaces. I. The Activation of Adsorbed Atoms to Higher Vibrational States

Abstract: A first step towards an understanding of the many important chemical phenomena which are known to take place at solid surfaces is to discover how atoms behave when adsorbed on a surface. They may continue in one place vibrating about a mean position or, in certain circumstances, they may migrate from one part of the surface to another or they may be ejected from the surface altogether by the thermal agitation of the solid below them. It is of some importance to know how often they change from one such state to another and what are the controlling influences involved. The processes of adsorption and evaporation have often been considered by thermodynamic methods, but these, though powerful, deal only with statistical effects and are independent of any picture of the mechanism involved. They are able to give the fraction of atoms in each energy state of the assembly at a given moment, but they are not capable of providing any information as to the probabilities of transfer from one state to another.

Stress-Calculation in Frameworks by the Method of "Systematic Relaxation of Constraints". I and II

Abstract: An earlier paper bearing this title dealt in detail (Part I) with the application of relaxation methods to frameworks having frictionless joints, and in general terms (Part II) with their relation to the Moment Distribution Method of Professor Hardy Cross. Given any framework and a (self-equilibrating) system of applied forces, we may imagine constraints to be provided at every joint whereby the three components of displacement and the three components of rotation can be severally controlled. Then initially, If all constraints are fixed before the forces are applied, extension of members is prevented, and hence the forces (of known magnitude) are taken wholly by the constraints. Subsequently, by permitting a suitable relaxation, we can transfer any component of force or moment from a constraint to the framework, and in so doing we shall store strain-energy in the latter. If the successive relaxations are systematically chosen, we can bring about a continuous approximation to the required conditions, in which every constraint is relieved of load and all forces have been transferred to the framework. When all the joints are frictionless (as in the problems of Part I) the constraints at any one joint have only to control its three component displacements, because no stresses are entailed by joint rotations; but If any joint is rigid it can by rotating transmit couples to adjacent joints through the agency of shearing forces and the consequent bending moments. When the members have uniform flexural rigidity, it can be shown that any one which shares in the rotation of a joint, and which in consequence transfers a couple M away from that joint, transmits it necessary to the strict validity of a method which depends (as the relaxation method evidently does) upon the principle of superposition, although it appears that a solution for "P. y effects" could be obtained without difficulty by a slight modification of the standard procedure. some remarks on this aspect are given in Appendix C.

The Crystal Structure of the Alums

Abstract: The literature on the structure of the alums is fairly extensive, for though the formula is complex, the problem is greatly simplified by the high symmetry of the crystal. Nevertheless, none of the proposed structures has been fully supported by X-ray measurements. The first published work was that of Vegard and Schjelderup, but, though they arrived at the correct unit cell, their structure involved an improbable arrangement of atoms, in which the identity of even the SO4 group was lost. On this ground it was strongly criticized by Schaefer and Schubert, and Niggli showed that it was also incompatible with space-group theory. From Vegard and Schjelderup's measurements he assigned the alums to the space-group Th2 (Pn3), but Wyckoff, by means of Laue and rotation photographs, showed that this was incorrect, and that the space-group was Th2 (Pa3).

The Relation between the Alum Structures

Abstract: That the crystals which comprise the alums form an isomorphous series has generally been accepted almost as a self-evident fact. This is due to similarity of their chemical formulae and of their crystal classes, and the evidence was rendered almost complete when Cork showed that a series ranging from ammonium alum to thallium alum was based on the one space-group, Pa3. There were, indeed, some discrepancies in his measurements, but these could be ascribed to small parameter differences. The first indication of polymorphism among the alums was given by methyl ammonium alum, of which the structure was found to be different from that of potassium alum. §

Experiments on Heavy Hydrogen. V. The Elementary Reactions of Light and Heavy Hydrogen. The Thermal Conversion of Ortho-Deuterium and the Interaction of Hydrogen and Deuterium

Abstract: Since the discovery and preparation of deuterium, many of the chemical reactions of this heavy hydrogen isotope have been invalidated and compared with the corresponding reactions of ordinary hydrogen. It was soon recognized that in certain reactions the differences observed between the rate of a reaction involving hydrogen molecules and of the corresponding reaction involving deuterium molecules (D2) could be explained by the assumption that the different zero-point energies of the molecules H2 and D2 contribute a different amount of energy for the activation process. This was, in fact, predicted by Cremer and Polanyi, and by Eyring. This difference in the zero-point energies of the initial states will cause a lower reactivity of heavy hydrogen as compared with light hydrogen. On the other hand, it was pointed out by Polanyi that the activated stats also contains a certain amount of zero-point energy which is smaller for the deuterium complex than for the hydrogen complex and that therefore the energy of activation may appear to be lowered for reactions involving heavy hydrogen when compared with the corresponding reaction of light hydrogen. The subject of the present paper is the investigation of the effect of replacing the hydrogen atoms and molecules by deuterium atoms and molecules on the rate of the interaction of hydrogen atoms and molecules according to the reaction H+H2⇌H2+H. (1) This particular reaction involve only one reactant and a single process, and is Ike simplest chemical reaction we know. It appeared, therefore, to be an especially suitab1e example for such a study, being also well understood experimentally, and, owing to its ideal simplicity, amenable to theoretical treatment.