Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1957"

The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems

Abstract: It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabu­lated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.

Elastic deformation and the laws of friction

Abstract: This paper examines whether the hypothesis of elastic deformation of surface protuberances is consistent with Amontons’s law, that the friction is proportional to the applied load. For a single elastic contact, the area of contact A is known to be proportional to the ⅔ power of the load W . Since the frictional force is generally assumed to be proportional to A , it has been thought that in elastic deformation Amontons’s law would not be obeyed. However, conforming surfaces usually touch at many points, and it is shown that in these circumstances A and W become nearly proportional. Experiments are described which show that the general law is that the friction is proportional to the true area of contact; whether or not Amontons’s law is obeyed depends upon the surface topography. For highly elastic materials such as Perspex, Amontons’s law is obeyed when contact is made at many points, and other relations between A and W are observed when the contacts are few. Experiments with lubricated brass specimens show that the same conclusions apply to carefully prepared or well run-in metal surfaces running in conditions where the damage is small.

Derivation of the Brueckner Many-Body Theory

Abstract: An exact formal solution is obtained to the problem of a system of fermions in interaction. This solution is expressed in a form which avoids the problem of unlinked clusters in manybody theory. The technique of Feynman graphs is used to derive the series and to define linked terms. The graphs are those appropriate to a system of many fermions and are used to give a new derivation of the Hartree-Fock and Brueckner methods for this problem.

Studies of the Jahn-Teller Effect. I. A Survey of the Static Problem

Abstract: According to the Jahn-Teller theorem, symmetrical molecules with degenerate electronic states are unstable. Such molecules therefore take up a distorted shape. If there is near-degeneracy, the symmetrical shape may also be unstable. We have studied the distortion in some particular cases. The approach is to minimize the total electronic energy with respect to distortions of the nuclear framework, the latter being considered to be static. There are always several equivalent distortions of equal energy, so that a static distortion fails to remove the degeneracy. The discussion of dynamic effects is postponed to a subsequent paper. A linear molecule of formula BAB , for which two electronic states of opposite symmetry are sufficiently nearly degenerate, will be stable in a configuration with unequal A—B separations, and unstable when symmetrical. This example illustrates some of the main physical features of Jahn-Teller distortions in a simple manner. Although we know of no example where the symmetrical structure is actually unstable, there are examples where the tendency towards distortion noticeably reduces the force constant of the asymmetric vibration. Octahedral complexes AB 6 with degenerate electronic states occur in many situations (e. g. paramagnetic crystals, F-centres, luminescent centres and exciton states in cubic crystals). The orbital degeneracy may be threefold ( T 1 and T 2 ) or twofold. In the former case the stable distortion is found to be either of tetragonal symmetry about a [100] direction, or of trigonal symmetry about a [111] direction. The twofold degenerate situation leads to a more complicated situation. If one neglects anharmonic effects there appears to be an infinity of distortions minimizing the energy; a more detailed consideration of the anharmonic terms shows that the stable distortions are of elongated tetragonal character. This result has an important bearing on complexes involving the cupric ion.

The Distribution of Stress and Velocity in Glaciers and Ice-Sheets

Abstract: A block of ice resting upon a rough slope forms a theoretical model of a glacier or an ice-sheet, the sides of the glacier valley being ignored. Previous papers have described two types of steady flow in this model: ( a ) laminar flow, in which the longitudinal velocity gradient r is zero, and ( b ) extending or compressive flow, in which r is non-zero, ( a ) was derived under the assumption of a general flow law for ice, but ( b ) was only derived under the assumption of perfect plasticity. In the present paper a general flow law is used throughout, and the equations for steady flow, with r allowed to be non-zero, are found. The previous results ( a ) and ( b ) appear as special cases. Possible variations of density, temperature or flow law with depth are allowed for. If the density and the flow law are known as functions of depth in any region, and if the surface slope, the surface velocity, and the value of r are known, the equations give the stresses and velocity as functions of depth. The borehole experiment on the Jungfraufirn (1948-50) allows an experimental test. From the observed value of r , and Glen’s laboratory flow law for ice, a theoretical curve for the result of the experiment is calculated which is compared with the experimental curve. At a depth of 50 m the effect of ignoring r , as has been done hitherto, is to underestimate the shear rate by a factor of 50; on the present theory it is overestimated by a factor of 1∙33. The remaining discrepancy is probably mainly due to the effect of the glacier sides.

Mutual friction in a heat current in liquid helium II III. Theory of the mutual friction

Abstract: As was shown in part II, the Gorter-Mellink mutual friction force in a heat current is probably associated with turbulence in the superfluid. Following Feynman, it is suggested that this turbulence takes the form of a tangled mass of quantized vortex lines, so that the mutual friction probably arises from collisions between thermal excitations and these vortex lines. From the observed properties of the mutual friction it is deduced that the walls of the channel carrying the heat current play no essential role in the generation, maintenance or decay of the turbulence, but merely introduce a number of incidental complications; the present paper ignores these complications and deals therefore with the idealized case of a homogeneous heat current in an unbounded volume of helium. The turbulence in this idealized case must be homogeneous, and it is shown from experimental evidence that it is probably also isotropic. Values of the force exerted on unit length of a vortex line, which have been derived from the study of the attenuation of second sound in uniformly rotating helium, are used to calculate the Gorter-Mellink force per unit volume in terms of the length of line per unit volume; then by a simple dimensional argument it is shown that the force must depend on (v s — v n ) in a manner agreeing with experiment. An attempt is made to produce a detailed theory of the generation and decay of superfluid turbulence: it is shown first that owing to the Magnus effect the turbulence can probably be built up by the action of the mutual friction force exerted on the individual lines, although the way in which turbulence can be initiated in undisturbed helium is not known, and secondly that the turbulence can probably decay in a manner closely analogous to the decay of homogeneous turbulence in an ordinary fluid. Equations for the rate of generation and decay of turbulence are obtained by dimensional arguments, and by analogy with formulae known to apply to turbulence in an ordinary fluid. Comparison of the equations with the experimental results described in parts I and II reveals good agreement, and makes it possible to deduce the form and magnitude of a term describing the effect of the unknown initiation process.

Interferometry of the intensity fluctuations in light - I. Basic theory: the correlation between photons in coherent beams of radiation

Abstract: It is shown by a quantum-mechanical treatment that the emission times of photoelectrons at different points illuminated by a plane wave of light are partially correlated, and identical results are obtained by a classical theory in which the photocathode is regarded as a squarelaw detector of suitable conversion efficiency. It is argued that the phenomenon exemplifies the wave rather than the particle aspect of light and that it may most easily be interpreted as a correlation between the intensity fluctuations at different points on a wavefront which arise because of interference between different frequency components of the light. From the point of view of the corpuscular picture the interpretation is much less straight-forward but it is shown that the correlation is directly related to the so-called bunching of photons which arises because light quanta are mutually indistinguishable and obey Bose-Einstein statistics. However, it is stressed that the use of the photon concept before the light energy is actually detected is highly misleading since, in an interference experiment, the electromagnetic field behaves in a manner which cannot be explained in terms of classical particles. The quantitative predictions of the theory have been confirmed by laboratory experiments and the phenomenon has been used, in an interferometer, to measure the apparent angular diameter of Sirius: these results, together with further applications to astronomy, will be discussed in detail in later papers. It is shown that the classical and quantum treatments give identical results when applied to find the fluctuations in the photoemission current produced by a single light beam, and the connexion between these fluctuations and the correlation between photons in coherent beams is pointed out. The results given here are in full agreement with those obtained by Kahn from an analysis based on quantum statistics: however, they differ from those derived on thermodynamical grounds by Fellgett and by Clark Jones and the reasons for this discrepancy are discussed.

The Description of Collective Motions in Terms of Many-Body Perturbation Theory

Abstract: In this and a succeeding paper it is shown how a theory equivalent to the Bohm & Pines collective motion theory of the electron plasma can be derived directly from a perturbation series which gives in principle an exact solution of the many-body problem. This result is attained by making use of a diagrammatic method of analysis of the perturbation series. By a process analogous to the elimination of photon self-energy parts from the electrodynamic S matrix it is found possible to simplify the perturbation series, introducing a modified interaction between the particles. A useful integral equation for this modified interaction can be set up, and it is shown how the energy of the system can be expressed in terms of the modified interaction. The close connexion between this approach and the dielectric theory of plasma oscillations is indicated.

A theory of inflammability limits and flame-quenching

Abstract: The energy and diffusion equations are solved for a one-dimensional, laminar, steady combustion wave in a pre-mixed gas which is in heat-transfer communication with solid surroundings. It is shown that in general a given combustible mixture has two possible flame speeds, but the lower one represents a normally unstable condition. If the heat-loss parameter increases, the two flame speeds first become coincident and then imaginary. The condition of coincidence is identified with the inflammability limit. Particular examples are evaluated in which heat loss is in turn by radiation and by conduction to the walls of a tube. Agreement with experiment is good for quenching in tubes. Results for the flame speed at the inflammability limit are of the correct order of magnitude, but precise experimental data are lacking. Experimental evidence for the existence of two flame speeds is adduced. 1. THE PROBLEM

The Effect of a Tangential Force on the Contact of Metallic Bodies

Abstract: When two solid bodies are brought together, they are usually very heavily deformed in their region of contact Bowden & Tabor (1954) have discussed very fully how a consideration of this deformation may be used to explain friction They have shown that the interaction between the two bodies in the area of contact, where they come very close together on an atomic scale, is so strong that the application of a tangential force tending to slide two normally loaded bodies over each other nearly always produces deformation and rupture in the bodies themselves, rather than slip in the original interface We have studied experimentally the tangential movement of two bodies relative to each other and the size of the area of contact between them, when they are first loaded normally and then subjected to tangential forces too small to cause sliding We have examined the contact of like metallic specimens, using gold, platinum, tin, indium and mild steel The experiments showed that both the relative displacement of the two bodies and the area of contact between them are smooth, increasing functions of the tangential force, as long as this is increased monotonically from zero Any value of the tangential force, less than the force of static friction, gives rise to a certain equilibrium displacement Such displacements cannot therefore be called ‘sliding’ in the ordinary sense As the displacement increases the tangential force increases more and more slowly, tending asymptotically to the force of static friction We found that the tangential force usually closely approaches the force of static friction while the displacement is still small compared with the diameter of the area of contact The area of contact was found by measuring the electrical contact resistance Only if there are no insulating layers between the bodies does this resistance give a direct measure of the contact area The increase in contact area could therefore be clearly observed only with the noble metals With these the behaviour can be described in terms of the simple analysis given by McFarlane & Tabor (1950) of the yielding in the area of contact under combined normal and tangential loading With metals that bear an insulating oxide layer, the contact resistance gives information on the disruption of the layer We found that purely normal loading causes very little mechanical breakdown; marked breakdown occurs only when the tangential force is increased to a large fraction of that needed to cause sliding It was found that the changes both in displacement and area of contact produced by tangential loading are essentially irreversible A reversible (elastic) component of the displacements exists, but it is a small part of the total except when the tangential force is small Release and even reversal of the tangential force produces no further irreversible changes until the force is increased again, in either direction, to a numerical value as high as the highest it has previously reached It was further found that a lubricant does not essentially affect the deformation process for tangential forces less than those required to cause slip The lubricant acts by weakening the surface-interaction, so that slip occurs for a smaller value of tangential force Displacements corresponding to forces less than those required to cause sliding are approximately equal for lubricated and unlubricated specimens The experiments provide quantitative data on the micro-displacements before sliding The results can be interpreted in terms of the adhesion theory of friction They confirm the analysis of combined stresses given by McFarlane & Tabor and extend its applicability

Mutual friction in a heat current in liquid helium ii I. Experiments on steady heat currents

Abstract: Experiments have been carried out on the conduction of heat through helium II in channels of large rectangular cross-section ($\sim $ 2 $\times $ 6 mm) for small heat current densities. The observed relationship between temperature gradient and heat-current density can be interpreted phenomenologically in terms of the Gorter-Mellink (1949) mutual friction force, F$\_{sn}\approx $ A$\rho \_{s}\rho \_{n}$(v$\_{s}$-v$\_{n}$)$^{3}$ per unit volume, in the two-fluid model, and observed values of A have been found to agree fairly well with those deduced from earlier measurements. Evidence is presented to show that the magnitude of the mutual friction is determined entirely by the value of (v$\_{s}$ - v$\_{n}$), independently of the boundary conditions imposed on the flow. A study of the propagation of second sound across the heat currents has shown that, while the presence of the heat current leads to no observable change in the velocity of the second sound, it does lead to an attenuation; the attenuation is linear and approximately proportional to the square of the heat-current density. This behaviour can be described phenomenologically in terms of the two-fluid model, if it is assumed that, in the presence of both a steady heat current and a second sound wave, the Gorter-Mellink mutual friction must be generalized to the form F$\_{sn}$ = A$\rho \_{s}\rho \_{n}$ U$^{2}$u, where u is the instantaneous relative velocity between the two fluids and U is the time-average of this relative velocity. This result shows that in the presence of a steady heat current one or both of the fluids must become modified in some way, and that an essentially linear mutual friction is associated with this modification. Observation of changes in the attenuation of second sound provides a more sensitive method of measuring mutual friction than does the observation of temperature gradients, and it has been shown by the former technique that in the channels used in the present work there is a critical heat current below which the mutual friction is either absent or very small.

On the Susceptibility of a Ferromagnetic above the Curie Point

Abstract: Series expansions for the susceptibility of a ferromagnetic at high temperatures are examined in detail to estimate the curvature of the reciprocal of the susceptibility immediately above the Curie point. For an Ising model with spin ${\textstyle\frac{1}{2}}$ a substantial number of terms of the series are available, and the series is well behaved (i.e. the coefficients vary smoothly). The asymptotic behaviour of the coefficients can be conjectured with fair confidence, and a closed formula can be derived based on this conjecture. For an Ising model with spin greater than ${\textstyle\frac{1}{2}}$ fewer terms of the series are available, but the series is still well behaved, and an extrapolation can again be undertaken fairly confidently. For the Heisenberg model with spin ${\textstyle\frac{1}{2}}$ the behaviour of the coefficients is considerably more erratic and the predictions are more tentative. It is concluded generally that the curvature of the reciprocal susceptibility depends primarily on the lattice structure and little on the type of interaction; that there is little variation between the different types of two-dimensional, and different types of three-dimensional lattice, although there is a marked difference between two- and three-dimensional lattices. The experimental curve of Weiss & Forrer for nickel is examined, and it is found that the data can be fitted quite well by the extrapolation formula for a three-dimensional lattice. It is thus possible to account for the experimental results assuming only short-range interatomic forces.

Mutual Friction in a Heat Current in Liquid Helium II. II. Experiments on Transient Effects

Abstract: It was shown in part I that, when helium II is carrying a steady heat current on which is superimposed a second-sound wave, the mutual friction acting between the two fluids (the Gorter-Mellink force) is of the form G (v s - v n ), where (v s - v n ) is the instantaneous relative velocity between the fluids, and the factor G is proportional to the square of the time average of this relative velocity. The present paper describes some experimental studies that have been made of the manner in which G changes when the heat current in a wide (~ 2 mm) channel is suddenly changed from one steady value to another; the changes in G have been observed as changes in the attenuation of second sound, and, where possible, as changes in the temperature gradient in the helium. It has been found, for example, that, when a steady supercritical heat current is suddenly switched on in initially undisturbed helium, G rises to its equilibrium value only after a delay time which is of the order of 1s, and that, when the heat current is removed, a non-zero value of G persists for at least 30s. The results indicate that the Gorter-Mellink force is probably associated with turbulence in the superfluid. It is suggested that the force may therefore be due fundamentally to the presence in the superfluid of motions for which curl v s ≠ 0, and it is recalled that experimental evidence in favour of this view has been provided by the recent discovery (Hall & Vinen 1956 a ) that a mutual friction acts in helium that is simply in a state of uniform rotation.

Folding Instability of a Layered Viscoelastic Medium under Compression

Abstract: A preliminary theory is established for the stability of a viscoelastic layer sandwiched in an infinitely extended medium of another viscoelastic material when a compressive force is acting in a direction parallel with the layer. The instability is manifested by a folding of the layer. It is shown that in general there exists a lower and a higher critical load between which folding occurs at a finite rate with a dominant wavelength. This is the wavelength whose amplitude increases at the fastest rate. Special cases are also discussed in more detail such as that of a purely viscous layer in a viscous fluid, an elastic layer in a viscous fluid, a viscous layer in an elastic medium, and of two Maxwell materials. Results indicate that the ratio of the relaxation times of the two materials is an important parameter.

The instability of plastic flow of metals at very low temperatures

Abstract: The low-temperature unstable plastic deformation of aluminium alloys is described. It is shown that discontinuities in the stress-strain curve are caused by a localized temperature rise produced during the deformation. The calculated magnitudes of the drops in load and the transition temperature between smooth and discontinuous flow agree reasonably well with the experimental observations. It is believed that all metals should exhibit unstable deformation at sufficiently low temperatures.

Effect of a repulsive core in the theory of complex nuclei

Abstract: Using Brueckner’s method for the treatment of complex nuclei, the effect of an infinite repulsive core in the interaction between nucleons is studied. The Pauli principle is taken into account from the beginning. A spatial wave function for two nucleons is defined, and an integro-differential equation for this function is derived. Owing to the Pauli principle, the wave function contains no outgoing spherical waves. A solution is given for the case when only a repulsive core potential acts. The effective-mass approximation is investigated for virtual states of very large momentum.

The Dynamics of Twinning and the Interrelation of Slip and Twinning in Zinc Crystals

Abstract: Zinc crystal wires in which the basal plane was nearly parallel to the wire axis were found to twin at abnormally high stresses. No single critical resolved shear stress exists; resolved shear stresses near 3$\cdot $7 Kg/mm$^{2}$ were common for short crystals, while longer crystals (which often fractured at the instant of twinning) had an average critical stress of 2$\cdot $9 Kg/mm$^{2}$. All crystals experienced detectable plastic deformation before twinning, by slip on the plane (11$\overline{2}$2) in the direction [$\overline{11}$23], for which the critical resolved shear stress in tension was 1$\cdot $0 to 1$\cdot $5 Kg/mm$^{2}$. The existence of this slip system was placed beyond doubt by observations on slip lines and lattice rotations. The twinning stress is determined by the work-hardening accompanying the slip on (11$\overline{2}$2), which continues until the lattice has rotated sufficiently for slip to begin on the (0001) planes, originally impeded by the specimen grips. The observations suggest that, quite generally, slip is the necessary prelude to the nucleation of a twin. A model for twin nucleation is developed in detail; it postulates the creation, by homogeneous lattice shear, of a twin nucleus about 250 angstrom in diameter, at a site where the applied stress is magnified by an array of dislocations. Twins are nucleated at a stress which depends on the slip characteristics of the particular specimen; hence the great spread in published twinning stresses. Crystals which were indented under stress twinned immediately if the resolved stress exceeded about 2 Kg/mm$^{2}$, but indentations made before the test were ineffective, probably because of a strain-ageing effect. Ageing also led to a slow reduction of the high stresses needed to produce basal slip in the twinned crystals. Immediately after the twin had been created, a shear stress, resolved in the basal plane, of 550 g/mm$^{2}$ was required; this fell to 200 g/mm$^{2}$ two years later, which is still six times the normal basal slip stress. Basal slip in twinned crystals was accompanied by considerable work-softening. Stresses needed to thicken existing twins were very variable but much lower than the stresses for twin creation; thickening of twins always took place smoothly and was never accompanied by stress relaxation.

The Thermal Properties of Alkali Halide Crystals. II. Analysis of Experimental Results

Abstract: The results reported in part I, together with similar results for sodium chloride, have been analyzed in terms of the spectrum of harmonic vibrations of the crystalline lattice; since the zero-point energy proves to be small, the analysis should conform fairly closely to that of a static lattice. ʘ 0 and the coefficients of the terms in v 2 , v 4 and v 6 in the low-frequency expansion of the spectrum have been derived from the data for the region T D /20. The values of ʘ 0 agree well with ʘ (elastic) calculated from the elastic properties of the crystals. After correction for thermal expansion, the results in the temperature range immediately above ʘ D /6 yield ʘ ∞ and the first three even moments of the spectrum ( μ 2 , μ 4 , and μ 6 ) when fitted to the Thirring expansion for ʘ D . For the three potassium salts, and again for the two sodium salts, the ratio ʘ 0 /ʘ ∞ appears to depend almost entirely upon the mass ratio of the ions. Values of this ratio suggest that the type of interatomic force is determined primarily by the alkali ion. Negative moments of the spectrum, together with μ 1 and the geometric mean frequency v g , have been derived from integrals of the form ʃ F 0 ( C v / T s )d T , with an accuracy comparable to that of the primary experimental heat capacities. Explicit spectra have not been com­puted, but instead v g , Θ 0 and the μ n have all been correlated in a graph of the function v D (n) = {1/3( n + 3) μ n } 1/ n . Potassium bromide is used as an illustrative example. The sharp curvature of the function v D (n) for negative values of n indicates that moments for n 1 and compared with values derived by the approximate method of Domb & Salter (1952). The estimated increase in the volume of the crystal caused by zero-point energy ranges from 0.23% for potassium iodide to 0.37% for sodium chloride. By subtracting the heat capacity given by the Thirring expansion we may estimate the effect of anharmonicity of the vibrations. This seems to be roughly determined by the ratio of the amplitude of atomic vibrations to the interatomic distance.

Colour centres in irradiated diamonds. I

Abstract: A theoretical calculation of the energy levels, and hence absorption spectrum, of an isolated vacancy in an otherwise perfect diamond lattice has been made. The concept of a defect molecule is introduced. This enables familiar molecular orbital theory to be applied in calculating the electronic structure of the defect. The quantitative results suggest that the observed band at 2·0 eV causing irradiated diamonds to appear blue, is due to spin and orbitally allowed electronic transitions of symmetry 1 E → 1 T 2 in the neighbourhood of isolated neutral vacancies.

Buoyant vortex rings

Abstract: It is pointed out that there is a fundamental difference in the behaviour of vortex rings projected upwards, according as they do or do not contain fluid which is lighter than the surroundings. A theory based on the perfect fluid approximation is developed to describe the motion of buoyant rings in a uniform fluid. The essential assumption is that the circulation remains constant with time while the buoyancy force acts to increase the impulse of a ring. This leads to the prediction that increasing the buoyancy will give a greater rate of expansion and a lower velocity of rise. The theory is extended to the case of a ring rising through a stably stratified fluid having a constant density gradient; in this case increasing the buoyancy should lead to a lower final height. The predictions of the theory have been verified by carrying out experiments in the laboratory with small vortex rings formed in water, using methylated spirits and salt to produce the density differences. The observations suggest that substantially the same analysis may be applicable to phenomena on a larger scale in the atmosphere; the velocity and final height of an explosion cloud should be determined by the buoyancy and circulation generated near the ground and the stability conditions of the atmosphere.

Dislocations and stacking faults in stainless steel

Abstract: Further experiments by transmission electron microscopy on thin sections of stainless steel deformed by small amounts have enabled extended dislocations to be observed directly. The arrangement and motion of whole and partial dislocations have been followed in detail. Many of the dislocations are found to have piled up against grain boundaries. Other observations include the formation of wide stacking faults, the interaction of dislocations with twin boundaries, and the formation of dislocations at thin edges of the foils. An estimate is made of the stacking-fault energy from a consideration of the stresses present, and the properties of the dislocations are found to be in agreement with those expected from a metal of low stacking-fault energy.

The antiferromagnetism of manganese copper alloys

Abstract: Manganese-rich $\gamma $ phase alloys of manganese + copper have been investigated by neutron and X-ray diffraction techniques, and the temperature variations of magnetic susceptibility and electrical resistivity have been measured. The disappearance of long-range antiferromagnetic ordering has been correlated with the face-centred-tetragonal $\rightarrow $ face-centred-cubic martensitic transformation of these alloys. The Neel temperatures and the magnetic moments of the manganese ions, for alloys of various compositions, have been determined. The Neel temperature increases approximately linearly with increasing manganese content and extrapolates to 660 degrees K at 100% Mn; the value of 2$\cdot $4 $\pm $ 0$\cdot $1 $\mu \_{\text{B}}$ has been obtained for the ionic moment of the manganese ions in pure $\gamma $-Mn. Experiments on plastically strained specimens show that compressive stress results in alinement of the c axes of the crystallites along the [100]-type direction most nearly parallel to the stress axis. The ionic moments are alined along the c axis and the susceptibility of plastically strained specimens is anisotropic. An analysis is given of the results of measurements of the anisotropy of susceptibility from which may be calculated $\chi \_{\|}$ and $\chi _{\perp}$, the susceptibilities of a single antiferromagnetic domain measured respectively parallel and perpendicular to the direction of antiferromagnetism. The results of the investigations are discussed in terms of an electron band model of an antiferromagnetic metal.

On steady-state bubbles generated by Taylor instability

Abstract: This paper is concerned with the flow of an incompressible heavy liquid past a gas bubble in an infinitely long vertical tube. Attention is confined to plane flow, and it is assumed that the bubble extends downwards without limit, so that the motion is steady. A co-ordinate system attached to the bubble is chosen, with the liquid falling around the bubble, instead of the bubble rising in the liquid. It is supposed that the flow is irrotational, whence it can be described in terms of a complex potential C = 0 + i3f which is an analytic function of the complex variable z = x + iy in the physical plane. The gas in the bubble is considered to be at rest, in a state of constant pressure. The vertical tube is represented in the z plane as an infinite strip - -h < x < Ih of width h. The complex potential = (z) is normalized so that it transforms the region of flowing liquid in this tube conformally on to the infinite strip - 1 < /i < 1, with the free boundary of the bubble mapping on to a slit along the positive 0 axis. If the origin is placed in the physical plane at the vertex of the bubble, Bernoulli's equation shows that the requirement of constant pressure along the free boundary of the bubble takes the form

The Theory of Chemical Shifts in Nuclear Magnetic Resonance. I. Induced Current Densities

Abstract: The chemical shifts of nuclear magnetic resonance frequencies are determined by the secondary magnetic field due to the electronic current induced by the applied field. This paper is concerned with the general problem of finding the current density vector field. An approximate orbital theory is developed which enables this to be divided up, under certain conditions, into local diamagnetic and paramagnetic circulations about individual atoms. The method is applied to the acetylene molecule as an example.

Energy Levels in Rare-Earth Ions

Abstract: The tensor-operator and group-theoretical methods of Racah are used to examine the following contributions to the Hamiltonian of a rare-earth ion: ( a ) the Coulomb interaction between the electrons, ( b ) the spin-orbit coupling of the electrons, and ( c ) the term arising from the influence of an external electrostatic field such as occurs when the rare-earth ion is situated in a crystal lattice. With regard to ( a ), the positions of all the terms of the configurations f 2 , f 3 and f 4 , as well as all the terms of other configurations f n whose multiplicities are the highest or the highest but one, are tabulated on the basis of a hydrogenic 4 f wave function. The theory of ( b ) is developed, and the parameter λ defining the spin-orbit splitting of a term is given for all terms whose energies have been found, ( c ) is treated by similar techniques and the theory is related to the one at present in use. All the general methods are illustrated by examples.

Extrusion and intrusion by cyclic slip in copper

Abstract: Extrusions and intrusions have been observed on slip bands in copper fatigued at 300, 90 and 20 degrees K. A model based on a sequence of slip movements during a fatigue cycle is proposed to explain their formation.

Cobalt Nuclear Resonance Spectra

Abstract: The frequencies of the cobalt nuclear resonances of solutions of fourteen cobalt (III) complexes have been measured in a magnetic field of 4370.9G. The temperature coefficient of the nuclear resonance frequency of two of the complexes has been measured, and the effect of variation of solvent studied. The electronic absorption spectra of the complexes have also been measured, and the absorption band shifts which occur when the temperature is changed have been studied for two of the complexes. The nuclear resonance frequencies show a close correlation with the spectrochemical series of the ligands. Following a suggestion by Orgel, a simple interpretation of the chemical shifts is developed in terms of crystal field theory. The theory predicts a linear relation between the cobalt nuclear resonance frequency and the wavelength of the lowest frequency optical absorption maximum of the octahedral complexes. The compounds studied support this prediction. Using measured values of the temperature coefficient of the nuclear resonance frequency and the optical absorption wavelength, the theory permits the temperature dependence of the optical spectrum to be calculated. In the two cases studied the calculations are in satisfactory agreement with experiment. The results provide information about the ‘electronic shielding’ of the cobalt nucleus which leads to an improved value of the cobalt nuclear moment.

The Odd-Parity States of $^{16}$O and $^{16}$N

Abstract: The properties of the odd-parity states of O16 and N16 have been calculated insofar as they may be considered to arise from the configurations p-1d and p-12s. The calculations were performed in intermediate coupling, including the spin-orbit splittings of the p and d shells, and full account was taken of the mixing of the two configurations. A satisfactory picture can be given of the known odd-parity levels of both nuclei, although some anomalous features, apparently associated with the detailed nature of nuclear forces, remain without full explanation. Some remarks are made regarding the photodisintegration of oxygen and it is found that the shell model can account satisfactorily for the observed photon widths which constitute the giant resonance.

The infra-red spectrum and molecular configuration of sodium deoxyribonucleate

Abstract: The infra-red spectra of oriented films of sodium deoxyribonucleate have been investigated between 700 and 4000 cm$^{-1}$ using polarized radiation and under varying degrees of relative humidity. Similar spectra have been obtained when the films have been deuterated by vapour-phase exchange with heavy water. It is found that the infra-red dichroism of nearly every band increases with the relative humidity. Many of the principal absorption bands can be assigned to separable modes of vibration in the bases, the phosphate groups or the absorbed water. Measurement of the dichroic ratios of certain of these bands indicates that at high relative himidity the bases must be nearly perpendicular to the orientation direction. The configuration deduced for the phosphate groups is unlike that proposed in the Crick-Watson model, but is in essential agreement with that recently proposed by Wilkins and his co-workers. Some observations are also reported on the corresponding spectra of sodium ribonucleate. Since no dichroism was observed, no conclusions can be drawn regarding the molecular configuration of this polymer.

The Influence of Interelectronic Collisions on Conduction and Breakdown in Covalent Semi-Conductors

Abstract: The energy and momentum distributions of electrons (or holes) in high electric fields are largely determined by interelectronic collisions if the electron densities are sufficiently high; but at lower densities only the energy distribution is affected. These distributions are used to evaluate the field for which a steady state becomes impossible (breakdown), and to calculate the variation of the mobility ($\mu $) with the applied field (F). Approximate results, obtained by neglecting interelectronic collisions entirely are also presented. However, it is shown that the latter calculations are really invalid since electrons which, by a fluctuation, achieve energies exceeding a certain critical energy, of order 1 eV, will increase their energy indefinitely under the action of the field. Both acoustic and optical lattice mode scattering are considered for various temperature ranges. Beside the well-established proportionality of $\mu $ to F$^{-\frac{1}{2}}$ and to F$^{-1}$ at high temperatures there are regions of field strengths at low temperatures where $\mu $ varies as F$^{-\frac{4}{5}}$ and F$^{-\frac{4}{3}}$, respectively.