Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1937"

Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy

Abstract: In a previous paper (Jahn and Teller 1937) the following theorem was established: A configuration of a polyatomic molecule for an electronic state having orbital degeneracy cannot be stable with respect to all displacements of the nuclei unless in the original configuration the nuclei all lie on a straight line. The proof given of this theorem took no account of the electronic spin, and in the present paper the justification of this is investigated. An extension of the theorem to cover additional degeneracy arising from the spin is established, which shows that if the total electronic state of orbital and spin motion is degenerate, then a non-linear configuration of the molecule will be unstable unless the degeneracy is the special twofold one (discussed by Kramers 1930) which can occur only when the molecule contains an odd number of electrons. The additional instability caused by the spin degeneracy alone, however, is shown to be very small and its effect for all practical purposes negligible. The possibility of spin forces stabilizing a non-linear configuration which is unstable owing to orbital degeneracy is also investigated, and it is shown that this is not possible except perhaps for molecules containing heavy atoms for which the spin forces are large. Thus whilst a symmetrical nuclear configuration in a degenerate orbital state might under exceptional circumstances be rendered stable by spin forces, it is not possible for the spin-orbit interaction to cause instability of an orbitally stable state. 1—General theorem for molecules with spin Just as before we must see how the symmetry of the molecular framework determines whether the energy of a degenerate electronic state with spin depends linearly upon nuclear displacements. This is again determined by the existence of non-vanishing perturbation matrix elements which are linear in the nuclear displacements. These matrix elements are integrals involving the electronic wave functions with spin and the nuclear dis­placements, and we deduce as before from their transformation properties whether for a given molecular symmetry they can be different from zero.

Mechanism of the production of small eddies from large ones

Abstract: The connexion between the statistical representation of turbulence and dissipation of energy has been discussed in relation to the decay of the isotropic turbulence which is produced in a wind tunnel by means of regular grids. It was shown that a length λ can be defined which may be taken as a measure of the scale of the small eddies which are responsible for dissipation. This λ can be found by measuring the correlation R y between the indications of two hot wire anemometers set at a distance y apart on a line perpendicular to the axis of the tunnel. Then 1/ λ 2 = Lt y→0 1 - R y / y 2, and the mean rate of dissipation of energy per unit volume is W¯ = 15 μ u 2¯/ λ 2, (1) where u2¯ is the mean of the square of one component of velocity. When turbulence is generated in a wind stream by a grid of regularly spaced bars it may be expected to possess a definite scale proportional to the linear dimensions of the grid. In any complete statistical description of turbulence this scale must be implicitly or explicitly involved. One way in which the scale can be defined is to measure the distance y apart by which the two hot wires must be separated before the correlation between the indications disappears. Another way is to define the scale as l 2 = ∫ y R y d y . (2)

Experiments with Fluid Friction in Roughened Pipes

Abstract: Nikuradse (1933; Prandtl 1933), experimenting with pipes roughened internally by a uniform layer of sand, found that such pipes were indistinguishable from perfectly smooth ones, provided that the pressure gradient was less than that given by pV*k / μ = 4, where V = √( T / p ), T = shear stress at wall, p = density of fluid, μ , —- viscosity of fluid, k = diameter of roughness grains. With lesser flows neither the resistance nor the distribution of velocity was measurably influenced by the size of the roughness grains, and the observed resistance law was satisfactorily of the type required by the Karman-Prandtl theory for smooth surfaces. This law is usually expressed in the following form:

Critical phenomena in gases - I

Abstract: The exact measurements of the isotherms of gases have proved extremely valuable in the determination of interatomic forces. For this purpose it has been found necessary to express the pv values of a gas as a finite power series in the density or in the pressure, and the coefficients so obtained have been compared with theoretical expressions in terms of interatomic fields. Many accounts of the method have been given and it is not necessary to give further details here (cf. Lennard-Jones 1931). While these methods are valid for gases at low densities where binary encounters are predominant, they fail for gases at high densities such as obtain in the neighbourhood of the critical point. Michels and his collaborators (Michels and others 1937) have recently studied the isotherms of gases at pressures as high as 3000 atm., and they find that the usual method of representing isotherms as simple functions of density or pressure ceases to be useful. The equation of state of van der Waals was astonishingly successful in accounting for the critical phenomena of gases and the form of the isotherms for temperatures below the critical temperature. Other empirical equations of state, for example that of Dieterici, were even more successful in reproducing the observed relations between the critical pressure, volume and temperature, and their very success has often obscured the fact that they were not logical theories of critical phenomena in gases, based as they were on arguments which were valid only for gases of low concentration. Thus the van der Waals equation, valuable as it has been and useful as it still is, implies that the internal energy of a vapour and its liquid phase is proportional only to the first power of the density, and this cannot be true for gases or vapours at densities comparable with those of liquids. The problem still remains of explaining why gases exhibit critical properties and of correlating the observed values of the critical temperature with the forces which atoms or molecules exert on each other.

Theory of Electrical Breakdown in Ionic Crystals

Abstract: Recently a theory of electrical breakdown in solids has been developed (Frohlich 1937). This theory is based on the idea that electrical breakdown is a phenomenon due to the acceleration of electrons, as has been suggested by von Hippel (1935) and others. The critical field strength at which the breakdown occurs has been calculated in the following way: In strong external electrical fields, there are always some electrons in the conduction levels of an ionic crystal. These electrons, which are not in thermal equilibrium with the lattice, may be brought into these levels by cold emission or by some similar “pulling out” mechanism. Such an electron will make collisions with the lattice vibrations and thus lose per second a certain energy B ( E ), which depends upon its kinetic energy E . On the other hand, it will gain per second an energy A ( E, F ) from the external field F . Now it has been shown in I that B decreases but that A increases with increasing energy E . Thus there exists always an energy E' for which A = B . An electron with energy E less than E' will, on the average over several collisions, lose energy, whereas an electron with E greater than E' will gain more and more energy.

An X-Ray Analysis of the Nickel-Aluminium System

Abstract: An X-ray analysis of the nickel-aluminium system has been made. A comprehensive series of powder photographs were taken of slowly cooled alloys, and in addition some quenching experiments were made on the nickel-rich alloys. In several important particulars the results differ from the phase-equilibrium diagram published by Gwyer (1908) (fig. 1 a ). This diagram has therefore been modified in the direction indicated by our work (fig. 1 b ). The solid solution of aluminium in nickel shown by Gwyer has been found to contain a miscibility gap at room temperatures. The α -phase, which is a face-centred cubic solid solution of aluminium in nickel, extends to 89 atomic % of nickel at room temperatures. The α' -phase, which is a facecentred cubic solid solution closely corresponding to the composition Ni3Al, has a superlattice (Westgren and Ekman 1930), nickel atoms occupying centres of cube faces and aluminium atoms cube corners. Quenching extends the range of the α -phase until in alloys quenched from 1100° there is no break between the α and α' structures.

A Theoretical Formula for the Solubility of Hydrogen in Palladium

Abstract: It is known that the amount of hydrogen dissolved by Pt, Mo, Cu, Co, Fe, and Ni is directly proportional to the square root of the hydrogen pressure p . Furthermore, these metals take up hydrogen without changing their lattice structure. It has been shown by Fowler and Smithells (1937) that one can get a theoretical interpretation of the observed solubility, if one assumes that the hydrogen dissolves as protons which are free to move throughout practically the whole volume of the metal. The theory has been extended, using slightly different assumptions, to include the metals V, Ta, Ti, and Zr. In these cases definite metallic hydrides are formed having lattice structures differing from that of the pure metal, and at higher pressures the amount of absorbed hydrogen saturates, being no longer proportional to √ p . In the present paper we shall give a statistical interpretation of the solubility of hydrogen in palladium. Palladium seems to offer an advan­tageous starting-point for an attempt at a more profound extension of the theory outside the region where the √ p law is valid. It has been thoroughly investigated experimentally and is capable of taking up large volumes of hydrogen without changing its lattice structure (Linde and Borelius 1927; Kruger and Gehm 1933)

The Distribution of Electricity in Thunderclouds

Abstract: The mechanism of thunderstorms has been the subject of much controversy in recent years, and at the present time there are at least two theories which find considerable support. To arrive at a theory which would satisfy all criticism would be a great advance towards a solution of the more fundamental problem of the circulation of electricity through the atmosphere. The evidence put forward in support of the present theories of the mechanism of thunderstorms is derived almost wholly from observations made at the surface of the earth; in many cases this evidence is ambiguous, and the same observations have, in fact, sometimes been used to support opposing theories. It seemed highly desirable that fresh evidence should be sought, and the most promising line of attack appeared to lie in attempting measurements in and above the thunderclouds instead of confining the observations to ground-level. In 1934, after some preliminary experiments, a simple method of recording the distribution of electricity in thunderclouds by means of instruments attached to sounding balloons was devised at Kew Observatory, and a sufficient number of soundings has now been made to enable some important generalizations to be made. The method of observation and the results obtained are described and discussed in this paper.

The Contribution to the Electrical Resistance of Metals from Collisions between Electrons

Abstract: The theory of electrical resistance developed by Bloch and others reats the conduction electrons as moving independently of one another but interacting with the lattice vibrations. The theory gives for the resistance of a metal, subject to certain simplifying assumptions, the formula R = const. G (ʘ/T) G ( x ) = 5/x5 ∫x o ξ4dξ/eξ-1 - 1/ex-1, (1) where ʘ is the Debye characteristic temperature. At low temperatures formula (1) leads to the conclusion that R varies as T5. The function G ( x ) was first proposed by Gruneisen.

The Electronic Structure of Some Polyenes and Aromatic Molecules. I. The Nature of the Links by the Method of Molecular Orbitals

Abstract: The characteristic property which some molecules have of trans­mitting influences from one part to another is an indication that the electrons constituting the bonds cannot be segregated into closed localized pairs. This feature is represented in the Pauling method by the super­ position of a number of canonical structures, each of which corresponds to a chemical picture of localized bonds. The state of the molecule has properties which are different from those of the individual canonical structures, but can be defined or interpreted in terms of a set of them. The energy of the lowest or normal state is usually lower than that of any one canonical structure, even than that which would appear from the orthodox method of drawing bonds to be the most stable. The amount by which the energy sinks as a result of the superposition has been called resonance energy by Pauling, and numerical estimates of its value have been made for a large number of organic molecules. In calculating this energy change, however, no account seems to have been taken of the change in the lengths of the links which may be caused by the interaction of the electrons. Links are regarded as single or double, and the appropriate energy content calculated as though they were isolated single or double links, whereas the interaction between adjoining links may result in a length and an energy of link, to which the description single or double is no longer appropriate. Pauling has recently discussed the relationship between the nature and the lengths of links in typical organic molecules. Taking the known lengths of links in ethylene, benzene, graphite, and ethane, and the frac­tional order of their links (as two, three-halves, four-thirds, and unity) derived from the coefficients which occur in the linear sum of canonical structures, Pauling has derived an empirical relation between them. From it he deduces in other cases the extent to which the double-bonded character enters into other links. Pauling’s method will be made more precise in a paper which follows this.

The Passage of Fast Electrons and the Theory of Cosmic Showers

Abstract: It is well known that according to relativistic quantum mechanics, electrons and positrons with energy large compared with their rest mass have a very large probability when passing through the field of a nucleus of losing a large fraction of their energy in one process by emitting radiation. Hard quanta have a correspondingly large probability of creating electron pairs. Until recently it was believed that the direct measurements of Anderson and Neddermeyer on the energy loss of fast electrons showed that though this energy loss by radiation existed, it was much smaller for energies greater than about 108 e-volts than that theoretically predicted, and it was therefore assumed that the present quantum mechanics began to fail for energies greater than about this value. More recent experiments by Anderson and Neddermeyer (1936) have, however, led them to revise their former conclusions, and their new and more accurate experiments show that up to energies of 300 million e-volts (the highest energies measured in their experiments) and probably higher, the experimentally measured energy loss of fast electrons is in agreement with that predicted theoretically. In fact, one may say that at the moment there are no direct measurements of energy loss by fast electrons which conclusively prove a breakdown of the theory. This is particularly satisfactory, inasmuch as the theoretical reasons for expecting a breakdown of the theoretical formulae at energies greater than about 137 mc 2, namely the neglect of the classical “radius” of the electron, have been shown by v. Weizsacker (1934) and Williams (1934) to be unfounded. Under these circumstances, and in view of the experimental evidence mentioned above, it is reasonable as a working hypothesis to assume the theoretical formulae for energy loss and pair creation to be valid for all energies, however high, and to work out the consequences which result from them. It is our aim to deduce results which can be compared directly with cosmic ray experiments and which will then allow one to decide whether or not the theory fails for extremely high energies, and in the latter case, at what point the failure begins.

Magnetic Anisotropy and Electronic Structure of Aromatic Molecules

Abstract: In a recent paper Pauling (1936) calculates the diamagnetic anisotropy of benzene and other aromatic hydrocarbon molecules on “the assumption that the 2 p z electrons are free to move under the influence of the impressed fields from carbon atom to adjacent carbon atom”. He points out that this is a quantitative extension of previous and generally accepted ex­planations of magnetic anisotropy given by Ehrenfest (1925, 1929) for Bi and Sb, and by Raman and Krishnan (1927) and Raman (1929 a , b ) for benzene and its derivatives. The idea that certain electrons can move in this way is implicit in the theory of molecular orbitals, as applied by Huckel (1931 a , b , 1932, 1933) to aromatic and unsaturated molecules. Huckel (1934), in fact, pointed out that the application of the theory to such problems as magnetism and light absorption would be a task for further research. With the object of applying the theory of molecular orbitals to the quantitative explanation of diamagnetic anisotropy, I have for some time been engaged on a determination of the diamagnetic constants of substances which can be used for this purpose, including some which had been pre­viously examined and which then gave the results used by Pauling (Krishnan, Guha and Banerjee 1933; Krishnan and Banerjee 1935). It was clear, almost from the first, that the original measurements on naphthalene and anthracene were out of harmony with the rest; these measurements were repeated independently by Krishnan and myself, and corrected results were published in a joint paper (Lonsdale and Krishnan 1936). These new results remove a difficulty commented upon by Pauling. I have also determined the dia­magnetic constants of certain new compounds, especially cyanuric triazide, cyanuric trichloride and metal-free phthalocyanine. This work was begun and completed and the following paper written, in ignorance that Pauling was making the calculations described in his recent paper, and it therefore adds independent evidence in favour of the essential principle.

The Isotherms of CO 2 in the Neighbourhood of the Critical Point and Round the Coexistence Line

Abstract: The isotherms of CO2 between 0 and 150°C. and up to 3000 atm. have been previously published by two of the authors (Michels, A. and C. 1935). The method used for these measurements was not suitable, however, for determinations in the neighbourhood of the critical point and the coexistence line. A second method has therefore been developed by which both the critical data and the coexistence line can be determined. This method and the results obtained are described in the present paper. The Method and Apparatus The method was based on the one developed by Michels and Nederbragt (1934) for the determination of the condensation points of a binary mixture. While, however, for the measurements of condensation points, it was not necessary to know the quantity of gas in the apparatus, this knowledge is essential for the determination of isotherms. A new apparatus was therefore constructed in which this quantity could be determined. A diagrammatic sketch showing the principle employed is given in fig. 1. In a steel vessel A , a glass bell B is suspended which is connected through the steel valve H and the capillary J to a cylinder containing a supply of the gas to be examined. A steel capillary C connects A with a second steel vessel D , placed on one scale pan of a balance. Inside D a steel tube E , which is coupled to C , reaches to the bottom. The capillary F is connected to the top of D and leads to a cylinder of pure nitrogen and to an apparatus for measuring the gas pressure. The capillaries C and F are flexible, and are supported at G at such a distance from the scale pan that the variations in the forces acting on the latter during the swinging can be neglected. Before starting the measurements, the vessels A , the glass bell B and the tube C are completely, and the vessel D is partly filled with mercury. The valve H is then opened and CO2 gas admitted to the glass bell, driving mercury out of A into D . The pressure in D is balanced by nitrogen introduced through F . When sufficient CO2 has entered the glass bell, the valve H is shut. As the filling operation is carried out at a temperature and pressure at which the isotherms of CO2 are known, the amount of gas in B can be calculated from a knowledge of the volume.

The Quantum Theory of Atomic Polarization. I. Polarization by a Uniform Field

Abstract: Two problems of atomic energy, the energy of polarization of an atom in a plane electrostatic field and the energy of interaction, or van der Waals energy, of two distant atoms, are particularly suited to attack by approximate methods. In each, the disturbing field, whether that of the static field or of the distant atom, is small in comparison with the internal fields acting on the electrons of the atomic system, and so the standard methods by which quantum mechanics deal with small disturbances, namely the perturbation and variation methods, can be and have been successfully applied to these problems. Though the perturbation method, strictly applied, is the more accurate, since it takes into account the possible excited states of the system, its usefulness is restricted to atom s of simple structure, and having relatively simple wave functions. Generally speaking the variation methods, which require a knowledge of the unperturbed state of the system only, are more suitable for larger atom s; ignorance of the excited states is largely compensated for by expressing the perturbed wave functions in term s of parameters, which are then chosen to make the total energy a minimum. As one would expect, the atom s of hydrogen and helium have been studied most fully, the perturbation theory being used by Wang, Eisenschitz and London, and Lennard-Jones, and variation methods by Atanasoff, Hasse, Slater and Kirkwood, Pauling and Beach. Owing to the first order Stark effect, the calculated polarizability of the hydrogen atom cannot be compared with experiment, but the calculated value for helium agrees well in most cases with that observed. The van der Waals energy of two hydrogen atoms given by Pauling and Beach includes accurate values not only of the usual dipole-dipole term, which varies as 1/ R 6 but also of the dipole-quadripole and quadripole-quadripole terms, varying as 1/ R s, 1/ R 10 respectively.

X-Ray Analysis of the Dibenzyl Series. IV. Detailed Structure of Stilbene

Abstract: A preliminary account of the crystal structure of stilbene in relation to other compounds in the dibenzyl series has already been given in Part III of this work (Robertson, Prasad and Woodward 1936). The general arrangement of the four molecules in the unit cell was deduced from certain absolute intensity measurements, the physical properties of the crystal, and a comparison with the accurately known structure of dibenzyl (Robertson 1934, 1935 a ). Although the approximate structure was established beyond any doubt, the precise determination of the atomic positions has proved to be a matter of very great difficulty. This is due to the presence of four molecules in the unit cell and to the fact that it is impossible to formulate a definite model of the molecule from the chemical structure owing to the possibility of free rotation about the single bands. For the accurate analysis it was necessary to determine the phase constants of all the weaker reflexions, and in the course of this work over twenty trial structures were calculated and compared, In this connexion the mechanical sorting method of structure factor calculation previously described (Robertson 1936) proved of great assistance. The repeated application of a double Fourier synthesis to the ( h 0 l ) set of structure factors then gradually refined the values for the atomic co-ordinates, and we believe that the final results given below are now comparable in accuracy to those obtained for the other structures in this series. It is unnecessary to describe in detail the various stages by which this structure has been reaches, and the results of the final Fourier synthesis only are given in the following pages. The relative orientations of the molecules in the crystal now recorded should various physical properties of the crystal to be expressed as functions of the individual molecules. In particular, the results now obtained will justify more accurate measurements being made on the magnetic susceptibilities and the refractive indices of the crystal than are available at present.

The Interaction of Atoms and Molecules with Solid Surfaces. VIII. The Exchange of Energy between a Gas and a Solid

Abstract: In previous papers of this series the problem of energy interchange between a gas atom and a solid has been discussed for the case when the gas atom makes a transition between two adsorbed states or between an adsorbed state and a free state. In this paper we shall discuss the case of a transition between two free states and apply the results to the determination of the thermal accommodation coefficient. In recent years a number of theoretical papers on this subject have appeared, following the new and accurate experimental work of Roberts, who worked with helium and neon on tungsten. The authors, however, neglect, or only roughly take into account, the attractive field which is known to exist between the solid and the gas; the fact that atoms become adsorbed on the surface is clear evidence of the existence of such a field. In this paper we shall suppose that the interaction potentials between solid and gas atom can be represented by a Morse potential function, for it has the right characteristics; in that it is attractive at large distances and repulsive at small ones, and has a minimum in between. The formulae of this paper are accordingly more general than previous ones and contain them as special cases. They are applicable to experimental results such as those of neon on tungsten for which earlier theories would not be adequate.

Progressive lightning III―The fine structure of return lightning strokes

Abstract: It has been established in two previous papers by Schonland, and the authors (Schonland and Collens 1934; Schonland, Malan and Collens 1935) that the lightning flash involves two consecutive processes, a downward moving leader and an upward moving main or return process. These two processes are repeated for each separate of the series which may make up a complete flash. In these papers it has been shown that the return stroke travels with a velocity of the order of 2 x 109 cm./sec., and that the velocity decreases as the stroke travels upwards. Variations in luminosity and in velocity have been found to occur after the stroke has passed points where the original leader channel has branched.

A method for the numerical or mechanical solution of certain types of partial differential equations

Abstract: The development of mechanical means of evaluating solutions of ordinary differential equations, in the form of the differential analyser of Dr. Bush (Bush 1931; Hartree 1935), has made it feasible to undertake the investigation of many problems of scientific and technical interest leading to differential equations which have no convenient formal solution, and which are too elaborate, or for which the range of solutions required is too extensive, for calculation of the solutions by numerical methods to be practicable. The practical success of this machine, and the wide range of equations to which it can be applied, have led to the hope that it may be found possible to apply it to partial differential equations, which are usually regarded as less amenable to numerical methods than ordinary equations. The present paper gives one way of applying it to such equations in two independent variables with certain types of boundary conditions. As will appear, the possibility of applying this method depends more on the form of the boundary conditions than on the exact form of the equations. The method is particularly suited to the differential analyser, though it is also practicable for numerical work.

On the Vibrational Spectrum of a Three-Dimensional Lattice

Abstract: All those properties of a crystal which depend on the heat motion of the constituent particles, require for their detailed explanation a knowledge of the actual form of the vibrational spectrum; about this very little is known, even qualitative features being difficult to obtain. It was supposed at first (Born and v. Karman 1913) that the v 2 law proposed by Debye formed a good approximation to the truth, but the experimental evidence, which gradually accumulated, tended to show that the specific heat did not vary exactly as T 3 at temperatures where this law was expected to hold. Theoretical investigations (Blackman 1935) of the properties of a two-dimensional lattice showed that the spectrum could have markedly different features from those of a continuum distribution. It was furthermore found that large variations of θD with temperature could occur and that spurious T 2 regions were possible. On account of the similarity of the two- and three-dimensional cases as regards the frequency equation and in certain particular features, it was assumed that the spectrum would not be very different in the three-dimensional case; with this assumption it was possible to explain the rise in the θ D value of substances like KCl at low temperatures, and the discrepancies between elastic and thermal data. The theoretical predictions have been confirmed to some extent by the recent experimental work of Keesom and Clark (1935). They find that the rise of the θ D curve stops at helium temperatures, as had been expected, but that the values decrease at still lower temperatures. Whether this last effect is real or not does not appear to be definitely settled. A possible theoretical explanation will be considered below.

The electronic structure of some polyenes and aromatic molecules III—Bonds of fractional order by the pair method

Abstract: Lennard-Jones and Turkevich in the preceding paper have investigated by the orbital method the nature of the binding in various aromatic molecules due to the mobile electrons. The purpose of the present paper is to make somewhat similar calculations by the pair method. The author has had the benefit of several discussions with Professor Lennard-Jones, and he is glad to take this opportunity of acknowledging them.

The Size-Grading of Sand by Wind

Abstract: In the course of experiments on the changes which take place in the size-grading of the grains when sands are picked up, transported and redeposited by a wind, it was found that by plotting the proportional weights of the constituent sieve-separated grades on a log scale, important changes can be observed in the character of the grading which are imperceptible when the usual grading diagram is used. Let the abscissa of the usual diagram be R = log diameter, so that δR is the log ratio between successive sieve apertures. If δp is the weight of a grade (total weight of sample unity), then ordinate ϕ = δp / δR . With an infinite number of sieves ϕ = dp / dR . The whole area included under the diagram is ∫ ϕ dR = 1. 1—The Grading Diagram Fig. 1 a shows the grading curve of a sample of dune sand, together with a suitably chosen “normal probability curve” ϕ' = a /√ π e - a 2( R - R )2 where R is log (most frequent diameter). Though comparison is difficult owing to the low values of the ordinates representing the extreme grades, it is some­times assumed that the grain-size distribution may be a random effect.

The interaction of atoms and molecules with solid surfaces VII—The diffraction of atoms by a surface

Abstract: In the hands of Stern and his collaborators, an accurate technique has been evolved for the measurement of the intensities of reflected and diffracted beams of atoms from the surfaces of crystals, but surprisingly little use has been made of these results to deduce information of theoretical interest except to verify the validity of the de Broglie relation for atomic beams. In this paper we attempt a theory of the diffraction of atoms at surfaces with a view to finding explicit formulae for the intensities of diffracted beams in terms of the constants of the surface field in order that a comparison of the theory and experiment might yield information as to the magnitudes of these constants. A precise knowledge of surface fields would be of value in that it would make possible a calculation of other properties of atoms on surfaces, such as the adsorption and migration of adsorbed atoms, as has already been done in the preceding paper (Part VI) for helium on lithium fluoride.

The Conical Shock Wave Formed by a Cone Moving at a High Speed

Abstract: This paper gives an account of investigations which have been carried out in continuation of word already described by G. I. Taylor and the present writer (1933). In our former paper we developed a method for calculating the air flow and pressure in the neighbourhood of a cone and gave the results of numerical calculations which had been made for cones having semi-vertical angles of 10, 20 and 30°. These calculations were checked by obtaining surface pressure measurements on cones in a high-speed wind channel and by obtaining photographs of conically headed bullets in flight. The results of both experimental investigations were in satisfactory agree­ment with our calculations. In addition the photographs showed that, at speeds for which no mathematical solution was found possible, the shock wave was no longer conical and was detached from the apex of the cone. Part I of the present paper deals with the results of further photographic investigations which were arranged for when our earlier programme was nearing completion. At that time the Research Department, Woolwich, were developing a method of taking photographs of medium-sized projectiles in Flight, and it appeared that such photographs would give a more satisfactory means of checking the calculations than had been possible with the photographs of small bullets. With this object in view the Ordnance Committee requested that photographs should be obtained showing two-pounder projectiles in flight. The results obtained from these new photo­graphs are reviewed below, and a brief description is given also of the experimental methods used at the Research Department.

Investigations of Infra-Red Spectra-Absorption of Some Hydroxy Compounds in the Region of $3\mu $

Abstract: It has been realized by several investigators that with some exceptions compounds containing the hydroxyl group have a sharp absorption band close to 2.75μ, and frequently another much wider band about 3μ. Erreta and Mollet (1936, 1937) and Erreta (1937) have shown that the latter is an \"association\" band which diminishes on dilution in a non-polar solvent, or by raising the temperature, when the association complexes split up and the hydroxyl band at 2.75μ becomes more pronounced. (Compare also Barchewitz (1937) and Freymann (1937). In the present investigations of the infra-red absorption spectra of certain hydroxyl compounds in the 3μ region we have made observation of the effect of the structure of the molecule on both the above-mentioned bands, as well as on the CH vibration bands. The OH bands in the region about 1.4-1.6μ have been investigated by Wulf and his co-workers (1935, 1936) for the most part with solutions of about 0.01 mol./l/ in which the association band had practically disappeared.

Kinematics, Dynamics, and the Scale of Time

Abstract: 1—In two previous papers the laws of dynamics and the Newtonian approximation to the law of gravitation have been derived on a purely kinematic basis. That is, they have been deduced rationally, starting from the individual observer’s awareness of a temporal sequence for events at himself, and from his assigning of measures of distance and epoch by means of light-signals and appropriately rated and synchronized clocks; explicit procedure for this rating and synchronization having been stated in terms of the observer’s own experiences. No appeal was made in the derivations to any empirical laws of dynamics or gravitation, or even to the principle of relativity or to the principle of the constancy of the velocity of light. Recourse to these last two was avoided by careful statement of exactly what is meant by “uniform velocity” ,by intro­duction of the notion of kinematic equivalence . Laws of dynamics and gravitation emerged by embodying in analysis the programme out­lined by Mach, according to which all the matter in the universe is relevant to the description of dynamical laws of nature and must necessarily be taken into account in describing the motion even of a free particle; this programme was carried out by introducing the notion of statistical equivalence , which amounts to a definition of the systems of moving particles proposed for consideration (as affording a representation of the universe) as those whose motion and distribution satisfy what has been called the cosmological principle . It is particularly to be noticed that the cosmological principle is not a concealed law of nature but is simply a definition defining the subject of study, just as in any gravitational problem we must define what we are discussing— one-body problem, three-body problem, etc. The reason that, in systems defined by means of the cosmological principle, not only no recourse to empirical laws of nature is needed but regularities playing the part of laws of nature can be derived, is that essentially the cosmological principle is a mode of specifying an extrapolation of certain kinds of phenomena . If laws of nature are supposed to be derived as inductive generalizations from observed phenomena, then it is preferable to extrapolate phenomena and derive the correspond­ing laws rather than to extrapolate laws and predict phenomena. We have no more warrant for extrapolating laws than for extrapolating the phenomena from which they have been learned. We have no justification for believing in world-wide principles as a sort of super-authority.

Physical Properties of Surfaces. IV. Polishing, Surface Flow and the Formation of the Beilby Layer

Abstract: The usual method of polishing surfaces is to rub them together with a fine powder between them. By this process a rough surface, having visible surface irregularities, is changed into one where the irregularities are invisible. If the surface gives specular reflexion the height of these irregularities will be less than half a wave-length of visible light. Beilby has shown that the top layer of the polished solid is “vitreous” in character, it has lost its obvious crystalline properties and has apparently flowed over the surface, bridging and filling up the irregularities in it. The mechanism of the process has been a subject of discussion for many years. Newton (1730, p. 265), Herschel (1830, p. 447) and Rayleigh (1901 a ) considered that polishing was essentially due to abrasion. Beilby (1921, p. 114) considered that the polisher tore off the surface atoms, and the layer below this “retains its mobility for an instant, and before solidification is smoothed over by the action of surface tension”. Adam (1927) has suggested that molecules or small particles are abraded from the surface and subsequently adhere to a different area, so building up an amorphous layer. Hamburger (1932) considered that the abraded units were not molecules but small crystals perhaps 30A in diameter. The view that the temperature is sufficiently high to cause surface melting has been held for some time by Macaulay (1926, 1927, 1931), who was able to detect the products of thermal decomposition of the powder used for polishing glass plates. This evidence has been questioned because many reactions may occur at freshly exposed surfaces due to effects other than temperature. J. W. French (1927) carried out experiments with a thermometer embedded in the polisher, and concluded that the rise in surface temperature was negligibly small. Experiments described in earlier papers (Bowden and Ridler 1935, 1936) provide direct evidence that the surface temperature at the points of contact may, under many conditions of sliding, be sufficiently high to cause a real melting of the metal. The surface temperature was determined by using the rubbing contact of two different metals as a thermo-couple and measuring the e. m. f. developed on sliding. The mass of the metal remained quite cool, the high temperature was localized at the surface of the points of contact. It is just at these points, however, that wear, surface flow and polish occur. The method can, of course, only be applied to metallic conductors, but experiment showed that there was a simple relationship between surface temperature and thermal conductivity. A metal of low thermal conductivity, such as bismuth, showed a much higher surface temperature than copper under the same conditions of sliding. In the case of non-metals such as glass, silk, alumina, etc., which possess a very low thermal conductivity, we should expect the local surface temperature to be very much higher still. If these high local temperatures are reached so easily when one surface is rubbed on another, we should expect it to play an important part in the mechanism of wear, surface flow and polish of solids. The melted or softened solid would be wiped over the surface and would quickly solidify to form the Beilby layer.

Quaternion Treatment of the Relativistic Wave Equation

Abstract: A set of matrices can be found which is isomorphic with any linear associative algebra. For the case of quaternions this was first shown by Cayley (1858), but the first formal representation was made by Peirce (1875, 1881). These were two-matrices, and the introduction of the four-row matrices of Dirac and Eddington necessitated the treatment of a wave function as a matrix of one row (as columns). Quaternions have been used by Lanczos (1929) to discuss a different form of wave equation, but here the Dirac form is discussed, the wave function being taken as a quaternion and the four-row matrices being linear functions of a quaternion. Certain advantages are claimed for quaternion methods. The absence of the distinction between outer and scalar products in the matrix notation necessitates special expedients (Eddington 1936). Every matrix is a very simple function of the fundamental Hamiltonian vectors α, β, γ , so that the result of combination is at once evident and depends only on the rules of combination of these vectors. At all stages the relationship of the different quantities to four-space is at once visible. The Dirac-Eddington matrices, the wave equation and its exact solution by Darwin, angular momentum operators, the general and Lorentz transformation, spinors and six-vectors, the current-density four-vector are treated in order to exhibit the working of this method. S and V for scalar and vector products are used. Quaternions are denoted by Clarendon type, and all vectors are in Greek letters.

The Relationship between Viscosity, Elasticity and Plastic Strength of a Soft Material as Illustrated by Some Mechanical Properties of Flour Dough. IV. The Separate Contributions of Gluten and Starch

Abstract: In the earlier papers of this series (Schofield and Scott Blair 1932, 1933 a, 1933 b) the endeavour has been to give a quantitative description of the behaviour of flour dough under stress. Use was made of the equation de/dt=(1/n·dS/dt-dx/dt)+1/ƞ·S, which is the expression originally put forward by Maxwell with the addition of — dx/dt to take account of elastic after-effect. In this equation de/dt represents the rate of elongation of a cylinder of dough, and S the shearing stress which is one-third the longitudinal stress per unit area. The equation serves to define n, the modulus of rigidity, and ƞ the viscosity, and enables these to be evaluated from experimental observations of e and S RESP-865,885,910

On the Relations of the Tensor-calculus to the Spinor-Calculus

Abstract: In this paper it is first shown (2) that a six-vector which has the property of being identical with its own dual six-vector, and which, moreover, has its invariant null, has properties equivalent to those of a spinor. It is then shown (3) that the correspondence thus set up between tensor-analysis and spinor-analysis enables us to replace some complicated tensor-operations by simple spinor-operations. In 4 the correspondence is applied to the tensorization of Dirac’s relativistic equation of the electron, in connexion with its generalization to the space-time of general relativity. It is shown that Dirac’s equations are equivalent to the vanishing of an ordinary vector.

An Analysis by Adsorption of the Surface Structure of Graphite

Abstract: The work to be described had as objectives: to measure accurately sets of isothermals of non-polar gases on an inert adsorbent as a function of temperature and of quantity adsorbed; then to employ these isotherms and heats to obtain a modified Langmuir isotherm capable of a general and quantitative application to surfaces of variable adsorption potential. Owing to a quantization of the energy levels of the interacting molecules, in the van der Waals potential energy hollow, the liquid hydrogens H2 and D2 (Urey and Teal 1935) or the hydrogens adsorbed on charcoal (Barrer and Rideal 1935) have different vapour pressures; The quanta are larger for H2 than for D2 and therefore it is easier to evaporate H2. A counteracting influence must be considered, however (Lennard-Jones and Devonshire 1936), which is the effect of the mass on the wave functions whose product determines the probability of evaporation. Calculation shows for a simple case a separation factor which diminishes with temperature at a rate less than the original zero-point energy theory requires, though the zero-point energy effect is the greater. One might expect similar considerations to apply to other adsorbed molecules held by van der Waals forces only, and by lowering the temperature to cause transitions to lower vibrational levels of the adsorbed molecules against the solid, making it more difficult to desorb them. No investigation of the magnitude of these effects has been made, and so it was regarded as important to look for effects of temperature on pure van der Waals heats of sorption.