Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1970"

The Singularities of gravitational collapse and cosmology

Abstract: A new theorem on space-time singularities is presented which largely incorporates and generalizes the previously known results. The theorem implies that space-time singularities are to be expected if either the universe is spatially closed or there is an ‘object’ undergoing relativistic gravitational collapse (existence of a trapped surface) or there is a point p whose past null cone encounters sufficient matter that the divergence of the null rays through p changes sign somewhere to the past of p (i. e. there is a minimum apparent solid angle, as viewed from p for small objects of given size). The theorem applies if the following four physical assumptions are made: (i) Einstein’s equations hold (with zero or negative cosmological constant), (ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’), (iii) there are no closed timelike curves, (iv) every timelike or null geodesic enters a region where the curvature is not specially alined with the geodesic. (This last condition would hold in any sufficiently general physically realistic model.) In common with earlier results, timelike or null geodesic incompleteness is used here as the indication of the presence of space-time singularities. No assumption concerning existence of a global Cauchy hypersurface is required for the present theorem.

Random packings and the structure of simple liquids. I. The geometry of random close packing

Abstract: In his Bakerian Lecture, Bernal (1964) discussed those ideas of restricted irregularity which are physically realized in random packings of equal hard spheres, with particular reference to the structure of simple liquids. He stressed the need for a science of ‘statistical geometry ’, and took the first steps himself by proposing possible ways of describing such arrays. In this paper, these and other associated ideas are briefly described and extended by deriving an equivalent set of polyhedral subunits essentially inverse to the packing in real space. Examination of two independent high density arrays demonstrates the repro-ducibility of certain metrical and topological properties of these polyhedra, and their correlations over larger elements of volume. As a result, several possible ‘descriptive parameters’ are proposed. Although these essentially ‘numerical’ characteristics facilitate sensitive structural descriptions of any assembly of micro- and macroscopic subunits, we are still unable to characterize an irregular array in formal mathematical terms. Such a formulation of statistical geometry could be a powerful tool for tackling important problems in many branches of science and engineering.

The morphology and mechanism of formation of porous anodic films on aluminium

Abstract: The morphology of porous anodic oxide films formed on aluminium in phosphoric acid electrolytes at constant current density or voltage, and under changing electrical or electrolytic conditions, has been studied quantitatively by electron microscopy. Replicas from film sections and from both film interfaces have been prepared, as well as transmission micrographs of thin films, produced under accurately defined conditions. During formation at constant current density, pore initiation occurs by the merging of locally thickening oxide regions, which seem related to the substructure of the substrate, and the consequent concentration of current into the residual thin areas. The pores grow in diameter and change in number until the steady-state morphology is established. The film barrier layer thickness has been measured directly for the first time. The steady-state barrier-layer thickness, cell diameter and pore diameter are all observed to be directly proportional to the formation voltage. It becomes evident that the barrier-layer thickness, decided largely by an equilibrium established between oxide formation in the barrier-layer and field-assisted dissolution (probably thermally enhanced) at the pore bases, determines the cell and pore sizes by a simple geometrical mechanism. Anion incorporation into the film and its hydrogen-bonded structure play secondary roles to these factors in determining the actual film morphology, although not its subsequent properties. A consequence of the mechanism is that, at constant current density, relatively non-aggressive electrolytes give thicker barrier layers, larger cells and larger pores next to the barrier layer than aggressive media, although subsequent pore widening at the outer surface of the film by simple chemical dissolution is more severe in aggressive electrolytes.

The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy

Abstract: A transmission electron micrograph is essentially a projection of the specimen in the direction of view. In order to reconstruct a three-dimensional image of the specimen, it is necessary to be able to combine data from a number of different views. A formal solution of this problem is given in terms of Fourier transforms. Its realization requires data reduction and interpolation. The final solution is given by a least squares approach, which also indicates how many views must be included to give a valid reconstruction of a given particle to a given degree of resolu­tion. Interpolation procedures of varying power are given, to be employed according to the economy with which the available data must be used. An alternative procedure is described for direct reconstruction without the use of Fourier transforms, but it is shown to be in general less practicable than the Fourier approach.

Elastic-plastic behaviour of polycrystalline metals and composites

Abstract: Stress-strain curves of polycrystalline metals and composites using single crystal elastoplastic model

The Properties of Random Surfaces of Significance in their Contact

Abstract: In recent work it has been shown that many types of surfaces used in engineering practice have a random structure. The paper takes, as a representation of the profile of such a surface, the waveform of a random signal; this is completely defined by two parameters, a height distribution and an auto-correlation function. It is shown how such a representation can be transformed into a model, appropriate for the study of surface contact, consisting of an array of asperities having a statistical distribution of both heights and curvatures. This theory is compared with the results of an analysis of surface profiles presented in digital form. The significance of these findings for the theory of surface contact and for the measurement and characterization of surface finish is discussed.

Wavetrains in Inhomogeneous Moving Media

Abstract: When a slowly varying wavetrain of small amplitude propagates in a general medium, changes of frequency and wavenumber are determined along definite paths known as rays. It is shown that, for a wide class of conservative systems in fluid dymamics changes in amplitude along the rays may be computed from conservation of wave action, which is defined as the wave energy divided by the intrinsic frequency. The intrinsic frequency is the frequency which would be measured by an observer moving with the local mean velocity of the medium. This result is the analogue for continuous systems of the adiabatic invariant for a classical simple harmonic oscillator.

Dynamics of extended bodies in general relativity. I. Momentum and angular momentum

Abstract: Definitions are proposed for the total momentum vector p$^\alpha$ and spin tensor S$^{\alpha\beta}$ of an extended body in arbitrary gravitational and electromagnetic fields. These are based on the requirement that a symmetry of the external fields should imply conservation of a corresponding component of momentum and spin. The particular case of a test body in a de Sitter universe is considered in detail, and used to support the definition p$\_\beta$S$^{\alpha\beta}$ = 0 for the centre of mass. The total rest energy M is defined as the length of the momentum vector. Using equations of motion to be derived in subsequent papers on the basis of these definitions, the time dependence of M is studied, and shown to be expressible as the sum of two contributions, the change in a potential energy function $\Phi$ and a term representing energy inductively absorbed, as in Bondi's illustration of Tweedledum and Tweedledee. For a body satisfying certain conditions described as 'dynamical rigidity', there exists, for motion in arbitrary external fields, a mass constant m such that $M = m + \frac{1}{2}S^\kappa\Omega\_\kappa + \Phi$, where $\Omega_\kappa$ is the angular velocity of the body and S$^\kappa$ its spin vector.

Exact analysis of unsteady convective diffusion

Abstract: An exact solution to the unsteady convective diffusion equation for miscible displacement in fully developed laminar flow in tubes is obtained. The most interesting result of this work is that it shows the generalized one-dimensional dispersion model describes the average concentration distribution exactly for all values of the independent variables if the dispersion coefficients are defined properly as functions of time. The analysis given reveals the precise structure of the dispersion coefficients necessary for the dispersion model to be exact. Also, it is shown that an exact solution for the local distribution can be constructed in which the dispersion coefficients play the role of eigenvalues which are functions of $\tau $.

Interface effects and the work of fracture of a fibrous composite

Abstract: The stresses near the tip of a crack which lies normal to a set of alined fibres is discussed when the elastic properties of the composite are appropriate to those of a carbon reinforced epoxy resin. Splitting parallel to the fibres is expected to occur before fibre fracture only if an interface parallel to the fibres of one fiftieth the strength of the composite parallel to the fibres is present. The two mechanisms of energy dissipation which have been suggested to occur in a fibre composite of brittle fibres in a brittle matrix, namely pull out and debonding, are discussed. Experiments to measure the latter are described. The work of debonding ( 4 J m -1 ) is usually less than the work of pull out. The theory of pull out is described and experiments to support it noted. An important result is that the work of pull out increases linearly with fibre diameter, and is likely to be inconveniently small for fibres of diameter ≾ 10 μ m. The relative advantages and disadvantages of fibres of various diameters are discussed. Fibres of diameter ≿ 25 μ m lead to large works of pull out and are stable and easily handled. Thinner fibres ( μ m diameter) give rise to constraint effects which are important in metallic matrices.

Selective Adsorption at Graphite/Hydrocarbon Interfaces

Abstract: A study was made of adsorption of n -dotriacontane and n -butanol dissolved in n -heptane onto graphitized carbon black and various ground graphites having surface areas ranging from 5 to 700 m 2 g -1 . It was established that the adsorption of n -dotriacontane by the graphites is confined entirely to the basal planes of graphite crystals. This is attributed to a remarkable fit between the hydrogen atoms attached to one side of the zig-zag carbon chain in the normal paraffins and the centres of hexagons formed by the carbon atoms in the basal planes of the substrate. The longer the chain the more contacts it can form with the graphite surface and the more strongly it is adsorbed. More detailed studies of the adsorption of n -paraffins on ground graphites have shown that they form close-packed monolayers of horizontally disposed molecules on the basal planes. There is little further adsorption after the monolayers are complete. The heat of adsorption per molecule increases uniformly with the chain length reaching very high values for the normal paraffins having more than 30 carbon atoms. The formation of the close-packed layers has been used for the measurement of the proportion of basal plane surface in different types of graphites. n -Butyl alcohol also forms closely packed monolayers, but on the polar sites of graphites, which can be used for the estimation of their area. The basal plane and polar sites act independently in adsorption and their relative proportions characterize the adsorptive properties of graphites. Examination of graphite ground in n -heptane, which consists of plates of average area of several square micrometres and average thickness of 5 nm, shows that its surface consists predominantly of basal planes having a high adsorptive capacity for n -paraffins.

A model for inhomogeneous turbulent flow

Abstract: A set of model equations is given to describe the gross features of a statistically steady or 9slowly varying’ inhomogeneous field of turbulence and the mean velocity distribution. The equations are based on the idea that turbulence can be characterized by ‘densities’ which obey nonlinear diffusion equations. The diffusion equations contain terms to describe the convection by the mean flow, the amplification due to interaction with a mean velocity gradient, the dissipation due to the interaction of the turbulence with itself, and the dif­fusion also due to the self interaction. The equations are similar to a set proposed by Kolmo­gorov (1942). It is assumed that both an ‘energy density’ and a ‘vorticity density’ satisfy diffusion equations, and that the self diffusion is described by an eddy viscosity which is a function of the energy and vorticity densities; the eddy viscosity is also assumed to describe the diffu­sion of mean momentum by the turbulent fluctuations. It is shown that with simple and plausible assumptions about the nature of the interaction terms, the equations form a closed set. The appropriate boundary conditions at a solid wall and a turbulent interface, with and without entrainment, are discussed. It is shown that the dimensionless constants which appear in the equations can all be estimated by general arguments. The equations are then found to predict the von Karman constant in the law of the wall with reasonable accuracy. An analytical solution is given for Couette flow, and the result of a numerical study of plane Poiseuille flow is described. The equations are also applied to free turbulent flows. It is shown that the model equations completely determine the structure of the similarity solutions, with the rate of spread, for instance, determined by the solution of a nonlinear eigenvalue problem. Numerical solutions have been obtained for the two-dimensional wake and jet. The agreement with experiment is good. The solutions have a sharp interface between turbulent and non-turbulent regions and the mean velocity in the turbulent part varies linearly with distance from the interface. The equations are applied qualitatively to the accelerating boundary layer in flow towards a line sink, and the decelerating boundary layer with zero skin friction. In the latter case, the equations predict that the mean velocity should vary near the wall like the 5/3 power of the distance. It is shown that viscosity can be incorporated formally into the model equations and that a structure can be given to the interface between turbulent and non-turbulent parts of the flow.

Constitutive inequalities for isotropic elastic solids under finite strain

Abstract: Possible restrictions on isotropic constitutive laws for finitely deformed elastic solids are examined from the standpoint of Hill (1968). This introduced the notion of conjugate pairs of stress and strain measures, whereby families of contending inequalities can be generated. A typical member inequality stipulates that the scalar product of the rates of change of certain conjugate variables is positive in all circumstances. Interrelations between the various inequalities are explored, and some statical implications are established. The discussion depends on several ancillary theorems which are apparently new; these have, in addition, an intrinsic interest in the broad field of basic stress—strain analysis.

The strength and stiffness of polymers

Abstract: Fully alined long chain polymers would have a Young modulus in the alinement direction similar to that of steel. Practically available polymeric materials have moduli less than one-tenth, and usually less than one-fiftieth of this, even with a high degree of molecular alinement. The paradox is of course resolved in principle by recognizing the predominance of chain folding in polymer crystallization, which allows strong alinement to occur with little or no extended chain continuity in the alinement direction. Questions which arise from this are: (1) what are the actual mechanisms of compliance in the material of relatively low modulus? and (2) by what means may it be possible to achieve full extension and alinement of a high proportion of the chains? Some tentative answers can be given in the light of current researches. Fully extended alinement for a small proportion is obtainable. Extension of a substantial proportion, rather than all, is the desideratum, since the folded chains con­tribute toughness. It can be useful to think of polymers as intrinsically composite materials, even when chemically homogeneous.

Glacier Sliding Without Cavitation in a Linear Viscous Approximation

Abstract: A model (Nye 1969) of ice sliding over a smooth wavy surface is applied to a temperate glacier sliding over its rock bed. The linear viscous approximation enables one to take full account of the continuous spectrum of obstacle sizes and to see the part played by the 9controlling obstacle size9 (Weertman 1957). The form of the power density spectrum of the bed is inferred from qualitative observations; it depends essentially on a single parameter that is connected with the roughness of the bed, roughness transverse to the direction of flow being included. The spectrum is used to calculate the drag, the dependence of the drag on scale, and the overburden pressure necessary to suppress cavitation. The thickness of the regelation layer calculated by the theory agrees well with the observations of Kamb & LaChapelle (1964).

Dynamics of Extended Bodies in General Relativity. II. Moments of the Charge-Current Vector

Abstract: The problem is considered of defining multipole moments for a tensor field given on a curved spacetime, with the aim of applying this to the energy-momentum tensor and charge-current vector of an extended body. Consequently, it is assumed that the support of the tensor field is bounded in spacelike directions. A definition is proposed for 'a set of multipole moments' of such a tensor field relative to an arbitrary bitensor propagator. This definition is not fully determinate, but any such set of moments completely determines the original tensor field. By imposing additional conditions on the moments in two different ways, two uniquely determined sets of moments are obtained for a vector field J$^{\alpha}$. The first set, the complete moments, always exists and agrees with moments defined less explicitly by Mathisson. If $ abla $$\_{\alpha}$J $^{\alpha}$ = 0, as is the case for the charge-current vector, these moments are interrelated by an infinite set of corresponding restrictions. The second set, the reduced moments, exists if and only if $ abla $$\_{\alpha}$J $^{\alpha}$ = 0. These avoid such an infinite set of interrelations, there being instead only one such restriction, the constancy of the total charge of the body. The energy-momentum tensor will be treated in a subsequent paper.

Random Packings and the Structure of Simple Liquids. II. The Molecular Geometry of Simple Liquids

Abstract: Machine calculations by Monte Carlo methods enable us to simulate real liquids in real space, and to reproduce the thermodynamic properties of a real liquid assembly. On the basis of the geometrical descriptions of part I, the structures of these simulated arrays are compared with the random packing. The results demonstrate the essential validity of Bernal's concept of the liquid state; moreover, by artificially hardening the interaction potential, we can throw light upon the structural differences between real and idealized systems.

An Investigation of Trapped Holes and Trapped Excitons in Alkaline Earth Fluorides

Abstract: The e.p.r. and optical absorption spectra of the self-trapped hole (V$\_{K}$) centres in CaF$\_{2}$, SrF$\_{2}$ and BaF$\_{2}$ have been measured and it is shown that the molecular axes of these centres are alined along $\langle $100$\rangle $. Preferential alinement of the V$\_{K}$ centres along a particular cube edge is readily produced by bleaching into the optical absorption bands of the V$\_{K}$ centre at 77 K with linearly polarized light. The anisotropy created in the crystals by preferential alinement is detected in both the e.p.r. and optical spectra. The thermally induced decay of the optically alined V$\_{K}$ centres is monitored by e.p.r. methods and it is shown that, in CaF$\_{2}$, linear (0 degrees) motion of the V$\_{K}$ centres predominates over the entire decay range (85 to 138 K). In SrF$\_{2}$ linear motion predominates in the early stages of decay (ca. 110 K) but random (90 degrees) motion sets in at higher temperatures (ca. 120 K). In BaF$\_{2}$ only random motion is observed. The kinetics of reorientation and decay of the V$\_{K}$ centre are investigated. A band observed in the X-ray fluorescence of CaF$\_{2}$ (at 279 nm), SrF$\_{2}$ (at 300 nm) and BaF$\_{2}$ (at 310 nm) is assigned to emission of the self-trapped exciton by comparison with a similar band in the recombination luminescence of the V$\_{K}$ centre (due to e$^{-}$ + V$\_{K}$). The temperature dependence of the X-ray fluorescence intensity is described. The polarization properties of the recombination luminescence are investigated on crystals containing optically alined V$\_{K}$ centres. The e.p.r. and optical absorption spectra of trapped hole centres involving other lattice defects are also described. One such centre, with its molecular axis close to $\langle $111$\rangle $, involves an interstitial fluorine ion and is named the V$\_{H}$ centre. Other centres with molecular axes close to $\langle $100$\rangle $ appear to be stabilized by impurity ions and are labelled V$\_{KA}$ and V$\_{KA^{\prime}}$ centres. The V$\_{H}$ centre occurs in CaF$\_{2}$, SrF$\_{2}$ and BaF$\_{2}$, the V$\_{KA}$ centre in SrF$\_{2}$ and BaF$\_{2}$, and the V$\_{KA^{\prime}}$, centre only in BaF$\_{2}$. The kinetics of reorientation and decay of the V$_{KA}$ centre have been investigated. We give a general discussion of the properties of hole centres, including the H centre, in alkaline earth fluorides.

Absorption of Oxygen in Silicon in the Near and the Far Infrared

Abstract: Previous work on oxygen in silicon has shown that oxygen dissolves interstitially in silicon forming a complex which may be approximately described as Si$\_{2}$O. Absorption bands of Si: O occur at 517, 1136 and 1203 cm$^{-1}$ and these have been assigned by earlier authors to the $\upsilon \_{2}$ (symmetric bending), $\upsilon \_{3}$ (antisymmetric stretch) and $\upsilon \_{1}$ (symmetric stretch) normal modes of vibration of Si$\_{2}$O. The present investigation confirms the $\upsilon \_{3}$ origin of the 1136 cm$^{-1}$ band (the well known 9$\mu $m band) but we disagree with the earlier assignments of the 517 and 1136 cm$^{-1}$ bands. The results reported here are relevant to organic siloxanes. We have extended the investigation of Si: O into the far infrared and we find sharp absorption lines at 29.3, 37.8, 43.3 and 49.0 cm$^{-1}$ which we have assigned to the $\upsilon \_{2}$ mode of Si$\_{2}$O. The isotope shift due to $^{18}$O has been observed in the far infrared spectrum. Effects of uniaxial stress on the 29.3 cm$^{-1}$ line have been investigated and are found to be consistent with the assignment to the $\upsilon \_{2}$ mode. The main features of the far infrared spectrum are accounted for with a simple anharmonic potential which ignores coupling of the Si$\_{2}$O to the crystal lattice. We have investigated effects of uniaxial stress on the 517, 1136 and 1203 cm$^{-1}$ bands of Si$\_{2}$O. Our stress results for the 1136 cm$^{-1}$ band are consistent with the earlier $\upsilon \_{3}$ assignment. Using our normal mode description, we conclude that the 1203 cm$^{-1}$ band is a combination band involving $\upsilon \_{3}$ and $\upsilon \_{2}$ excitations. We have not been able to give a clear cut assignment to the 517 cm$^{-1}$ band, but we suggest that $\upsilon _{1}$ type excitation may be involved. The appendix describes the stress splitting of the 836 cm$^{-1}$ band of the silicon A centre in electron irradiated Si: O and our results confirm an earlier model for this centre. In all cases investigated here, the stress splittings arise from raising the orientational degeneracy of the oxygen complex.

On steady vortex rings of small cross-section in an ideal fluid

Abstract: This paper is concerned with vortex rings, in an unbounded inviscid fluid of uniform density, that move without change of form and with constant velocity when the fluid at infinity is at rest The work is restricted to rings whose cross-sectional area is small relative to the square of a mean ring radius An existence theorem is proved for distributions of vorticity in the core that are arbitrary, apart from the condition imposed by the equation of motion and certain smoothness requirements The method of proof relies on the nearly plane, or two-dimensional, nature of the flow in the neighbourhood of a small cross-section, and leads to approximate but explicit formulae for the propagation speed and shape of the vortex rings in question

I.—A class of symmetric polynomials with a parameter*

Abstract: In an attempt to evaluate the integral (5) below, using a decomposition of an orthogonal matrix (Jack 1968), the author is led to define a set of polynomials, one for each partition of an integer k, which are invariant under the orthogonal group and which depend on a real parameter α. An explicit representation of these polynomials is given in an operational form. When α = − 1, these polynomials coincide with the augmented monomial symmetric functions. When α = 1, a systematic way of taking linear combinations of these polynomials is explained and it is shown that the resulting polynomials coincide with the Schur functions from the representation theory of the symmetric group. A similar procedure in the case α = 2 then appears to give the zonal polynomials as defined by James (1964, p. 478).

Nanosecond Flash Photolysis

Abstract: A flash photolysis system, using a pulsed laser as source, has been designed and used to study events having a duration of a few nanoseconds; an improvement over conventional flash techniques by a factor of a thousand. The apparatus incorporates both spectrographic and photoelectric monitoring techniques which are easily interchangeable and, apart from the laser itself, it is readily constructed from standard components. Its applications to the observation of the absorption spectra of excited singlet states, short-lived excited triplet states and chemical events in the nanosecond time region are described.

Quantum electrodynamics of spinless particles between conducting plates

Abstract: The energy of any system containing charged particles arises partly from its coupling to the quantized electromagnetic field; it changes on inserting the system between conducting plates, because these alter the normal modes of the field relative to free space. We calculate to order e2 such energy shifts for a free electron and for a hydrogen atom, neglecting spin, but allowing in full for electrostatic and retardation corrections as well as for changes in the Lamb shift proper. The potential between a particle and a single plate follows as a limiting case. An appendix details the distinction between classical and quantum effects.

Conservation of action and modal wave action

Abstract: Multiple wave propagation in acoustic duct with winds, using perturbation Lagrangian for ideal fluid flow

On the Global Isometric Embedding of Pseudo-Riemannian Manifolds

Abstract: It is shown that any pseudo-Riemannian manifold has (in Nash's sense) a proper isometric embedding into a pseudo-Euclidean space, which can be made to be of arbitrarily high differentiability. The application of this to the positive definite case treated by Nash gives a new proof using a Euclidean space of substantially lower dimension. The general result is applied to the space-time of relativity, and the dimensions and signatures of the spaces needed to embed various cases are evaluated.

Surface energy and surface tension in solids and liquids

Abstract: Rayleigh's theory of surface tension, based on the quadrant theorem, is extended to solids and presented in a simple general form. The quadrant theorem is generalized to embrace the traction of a. half-space upon a wedge in contact with it, and Young's equation for the angle of contact is derived as a condition of equilibrium between the surface tension of the liquid and the tangential traction of the solid half-space upon the liquid wedge. The historical theories of capillarity are correlated with the general theory. 1. THEORY OF SURFACE TENSION: HISTORICAL BACKGROUND The classical theory of capillarity is concerned with general consequences of attractive and repulsive forces acting between particles very small in comparison with the range of the attractive forces. It is a mathematical theory that does not contain unknown physical factors; it could have been completely developed 150 years ago. In spite of the simplicity of its basis, however, it has been notorious for obscurities and misunderstandings which have persisted ever since the disputes between the schools of Young (I805) and Laplace (i8o6). They are illustrated by Maxwell's famous article on 'Capillary action' in the 9th edition of the Encyclopaedia Britannica (I878). He wrote that Thomas Young's 'methods of demonstration, though always correct, and often extremely elegant, are sometimes rendered obscure by his scrupulous avoidance of mathematical symbols'; as regards Laplace, 'his results are in many respects identical with those of Young, but his methods of arriving at them are very different, being conducted entirely by mathematical calculations. The form into which he has thrown his investigation seems to have deterred many able physicists from inquiry into the ulterior cause of capillary phenomena, and induced them to rest content with deriving them from the fact of surface-tension'. A few pages later Maxwell gave Young's derivation of the equation for the angle of contact between a liquid and a solid from the equilibrium between three surface tensions, in the original form which is still customary in textbooks today, although it is manifestly unsound already because the surface layer of a solid does not slide freely on its interior, and so any condition of equilibrium between the surface tensions would have to contain the shear stresses between the skin and the interior of the solid.t That Young's equation is nevertheless correct shows that it is a relation between surface energies, not surface tensions: these are numeric

Inversion of the geomagnetic induction problem

Abstract: (Communicated by Sir Edward Bullard, F.R.S.-Received 26 February 1969) An algorithm has been found for inverting the problem of geomagnetic induction in a concentrically stratified Earth. It determines the (radial) conductivity distribution from the frequency spectrum of the ratio of internal to external magnetic potentials of any surface harmonic mode. The derivation combines the magnetic induction equation with the principle of causality in the form of an integral constraint on the frequency spectrum. This algorithm generates a single solution for the conductivity. This solution is here proved unique if the conductivity is a bounded, real analytic function with no zeros. Suggestions are made regarding the numerical application of the algorithm to real data. A number of investigators (Chapman i919; Lahiri & Price I939; Rikitake I950; E:ckhardt, Larner & Madden I963) have made estimates of the electrical conductivity of the Earth as a function of radius by studying natural geomagnetic variations. Varying magnetic fields of external origin (i.e. those generated by electric currents in the ionosphere) induce eddy currents in the conducting Earth, which in turn produce magnetic fields of internal origin. The magnetic fields observed at the Earth's surface can be mathematically separated (Chapman & Bartels I940) into those of external origin and those of internal origin. More precisely, one can calculate from the available experimental observations the ratio of internal and external magnetic potentials over a range of frequencies for a number of different spatial distributions of source field. This observed induction response ratio has been used in an indirect way in the work mentioned above; the calculated behaviour of different conductivity models has been compared with the behaviour of the real Earth, and the best fitting model selected as the solution. This method provides geophysically useful information, but it lacks the elegance of a direct method. More important, one does not know if an appreciably different but untried conductivity model also fits the experimental data well.

The Direct Measurement of the Strength of Metals on a Sub-Micrometre Scale

Abstract: Four types of experiment have been carried out to investigate the strength properties of annealed metals when the stressed volume is small enough to lie between the existing dis­locations in a crystal. These are (i) indentation experiments of a soft metal surface with a hard stylus, (ii) blunting of a soft metal tip against a hard surface, (iii) compression of individual metal crystals, and (iv) bending of thin filaments. The experiments were performed in either a scanning electron microscope or a transmission electron microscope with the use of micro­-loading devices capable of applying loads down to 0.1 mgf (10 -6 N). In the blunting experi­ments carried out in the transmission electron microscope it was possible to observe disloca­tions directly in the tip during loading. The majority of the experiments were carried out on gold. The results showed that strengths similar to the theoretical value can be achieved but in the experiments in which the stress was applied at an external surface (experiments (i), (ii) and (iii) above) the strength was strongly dependent on the condition of the interface. The strength that could be sustained by a region of perfect crystal in contact with a hard metallic surface was about five times less than the theoretical strength. These relatively low strengths are probably due to interfacial tractions producing very high local stresses. The introduction of some polymeric or amorphous layer at the interface raised the strength to the theoretical level. It is suggested that this is due to the fact that the polymeric layer removes most of the stress concentration. Transmission electron microscopy through aluminium tips during blunting showed that plastic deformation could take place at quite low stresses in a dislocation-free crystal with no dislocations remaining in the crystal during the early stages of blunting. Dislocation build up only occurred in the later stages of deformation. The yield stress was found to decrease with plastic strain in all the experiments, and could fall to values which were not substantially greater than those observed in large specimens. Comparison of the compressive strength of two spherical gold crystals, 0.5 μ m and 2 mm in diameter respectively, showed that the small crystal was only twice as strong as the large crystal after they had both undergone equal amounts of compression. The maximum strengths observed for gold ( μ /20) are greater than those calculated by Kelly (1966) ( ca . μ /50) and are nearer the value deduced by Brown & Woolhouse (1970) for the generation of dislocations around precipitate particles in alloys. The low strengths observed on deformed crystals are considered in the context of dislocation generation in small volumes and it is concluded that although a source mechanism of the Frank–Read type may be able to operate on this scale, some other source mechanism may also exist.

The deformation of single crystals of copper and copper-zinc alloys containing alumina particles - II. Microstructure and dislocation-particle interactions

Abstract: Electron microscope studies of deformed single crystals of Cu and Cu + Zn alloys of various compositions containing small alumina particles show that the dislocation structures generated and associated with the particles consist generally of prismatic loops, Orowan loops as well as prismatic loops in some brass alloys, and groups of dipoles. The formation of predominantly interstitial prismatic loops is explained in terms of a cross-slip mechanism controlled by the interstitial misfit strains around the particles. The transition from prismatic loop to prismatic plus Orowan loop structures in the brass alloys for Zn contents greater than ca . 20 wt % is considered to be due to the dependence of the activation energy for cross-slip on the stacking fault energy and to the solution hardening friction stress. In the Cu + 20% Zn alloys, measurements of the height of the loops and their numbers show that some dislocations by-pass the particles without forming loops. The mechanism of cross-slip leading to the formation of prismatic loops or ‘by-passing’ is discussed in some detail. The number of prismatic loops generated decreases with temperature. This fact is explained in terms of pipe or interface diffusion controlled climb of dislocations at the particles.

Dielectric Constants of Ionic Crystals and their Variations with Temperature and Pressure

Abstract: Measurements of the audio- and radio-frequency dielectric constants of the alkali and thallium halides have been made over the range of temperature 1.5 to 350 K and at pressure up to 4 kbar (400 MN m$^{-2}$), on crystals of high purity. The results are used to separate, for each compound, that part of the variation in dielectric constant which is explicitly dependent on temperature from that part which is explicitly dependent on volume. The observed variations at constant temperature are used to provide measures of the Gruneisen parameters for the long-wave transverse-optic phonons, and the variations at constant volume are discussed in terms of the anharmonic self-energies of these phonons. The materials studied here are all simple highly ionic compounds but, at room temperature, the self-energies of some prove to be positive while for others they are negative. It is shown how this can be interpreted in terms of competing three-phonon and four-phonon decay and scattering processes.