Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1968"

The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics

Abstract: A new model is proposed to describe the large recoverable extensions which can be observed with high polymers below their glass transition points. The model consists of a Hookean spring in series with an Eyring dashpot and rubber elasticity spring in parallel. Conventional stress-strain curves for cellulose nitrate and cellulose acetate sheet and PVC are compared with those predicted by the model. In this way many of the characteristic features of the stress-strain curves of plastics can be illustrated. Differences between the stress-strain curves of a number of polymers are discussed in terms of the model, and the tendency of a polymer to show necking is related to the ultimate network strain (limiting elongation) under the conditions of the test.

Variational Principles in Continuum Mechanics

Abstract: Variational principles for problems in fluid dynamics, plasma dynamics and elasticity are discussed in the context of the general problem of finding a variational principle for a given system of equations. In continuum mechanics, the difficulties arise when the Eulerian description is used; the extension of Hamilton’s principle is straightforward in the Lagrangian description. It is found that the solution to these difficulties is to represent the Eulerian velocity v by expressions of the type v = ∇ X + λ∇ μ introduced by Clebsch (1859) for the case of isentropic fluid flow. The relation with Hamilton’s principle is elucidated following work by Lin (1963). It is also shown that the potential representation of electromagnetic fields and the variational principle for Maxwell’s equations can be fitted into the same overall scheme. The equations for water waves, waves in rotating and stratified fluids, Rossby waves, and plasma waves are given particular attention since the need for variational formulations of these equations has arisen in recent work on wave propagation (Whitham 1967). The idea of solving some of the equations by ‘potential representations’ (such as the Clebsch representation in continuum mechanics and the scalar and vector potentials in electromagnetism), and then finding a variational principle for the remaining equations, seems to be the crucial one for the general problem. An analogy with Pfaff’s problem in differential forms is given to support this idea.

Time decay for the nonlinear Klein-Gordon equation

Abstract: It is shown that solutions of the nonlinear Klein-Gordon equation u tt - ∆ u + mu + P 9( u ) = 0 decay to zero in the local L 2 mean if the initial energy is bounded provided s P 9) s ) - 2 P ( s ) ≥ a P ( s ) ≥ 0 with a > 0. The local energy also decays. The proof is based on manipulating energy identities and re­quires that u have continuouś first derivatives and piecewise continuous second derivatives. The proof is also applicable to certain systems of equations.

New Conservation Laws for Zero Rest-Mass Fields in Asymptotically Flat Space-Time

Abstract: Some recently discovered exact conservation laws for asymptotically flat gravitational fields are discussed in detail. The analogous conservation laws for zero rest-mass fields of arbitrary spin s(= 0, $\frac{1}{2}$, 1, $\ldots$) in flat or asymptotically flat space-time are also considered and their connexion with a generalization of Kirchoff's integral is pointed out. In flat space-time, an infinite hierarchy of such conservation laws exists for each spin value, but these have a somewhat trivial interpretation, describing the asymptotic incoming field (in fact giving the coefficients of a power series expansion of the incoming field). The Maxwell and linearized Einstein theories are analysed here particularly. In asymptotically flat space-time, only the first set of quantities of the hierarchy remain absolutely conserved. These are 4s + 2 real quantities, for spin s, giving a D(s, 0) representation of the Bondi-Metzner-Sachs group. But even for these quantities the simple interpretation in terms of incoming waves no longer holds good: it emerges from a study of the stationary gravitational fields that a contribution to the quantities involving the gravitational multipole structure of the field must also be present. Only the vacuum Einstein theory is analysed in this connexion here, the corresponding discussions of the Einstein-Maxwell theory (by Exton and the authors) and the Einstein-Maxwell-neutrino theory (by Exton) being given elsewhere. (A discussion of fields of higher spin in curved space-time along these lines would encounter the familiar difficulties first pointed out by Buchdahl.) One consequence of the discussion given here is that a stationary asymptotically flat gravitational field cannot become radiative and then stationary again after a finite time, except possibly if a certain (origin independent) quadratic combination of multipole moments returns to its original value. This indicates the existence of 'tails' to the outgoing waves (or back-scattered field), which destroys the stationary nature of the final field.

Sound emission from open turbulent premixed flames

Abstract: A study of the noise produced by turbulent premixed flames stabilized on open burners is described. It is shown that such flames may be represented acoustically as a collection of monopole sound sources in the combustion zone. The radiated sound pressure is dependent on the rate of change of the rate of increase of volume of the gas during combustion, which varies owing to the turbulence in the flow. The rate of volume increase is proportional to the rate of consumption of combustible gas mixture in the flame. To measure this quantity, an optical technique has been developed which relies on observations of changes in the intensity of emission from the free radicals C 2 or CH generated in the reaction zone. Good quantitative agreement is obtained between the radiated sound pressures calculated from these intensity measurements and the values recorded simultaneously with a microphone. A correlation observed between the mean emission intensities of C 2 and CH radicals and the total flow rate of combustible gas mixture both in the laminar and turbulent flames, and in their transition region, supports the wrinkled laminar flame concept of turbulent flame propagation.

Self Similar Solutions to Adhesive Contact Problems with Incremental Loading

Abstract: This paper is concerned with the calculation of contact stress distributions, surface displacements and contact radii when an elastic half space z > 0 is indented by a rigid body, symmetrical about the z axis and exerting a prescribed force P normal to the interface, with friction sufficient to prevent any slip between the surfaces of the body and the half space during the application of the load. Within the framework of small displacement theory, an exact solution is obtained to the problem of static indentation by a body whose shape is given by a general polynomial z(r), where r is the distance from the z axis. The solution is worked out in detail for indentors of three simple shapes, namely a flat-faced cylinder, a sphere and a cone (the first two of which have previously been considered by rather different methods by Mossakovski (I954, I963). A closed solution is also obtained to another problem closely related to that of the spherical stamp: that of calculating the indentation and resulting velocity defect when a rigid sphere rolls on an elastic half space. The treatment in all cases is elastostatic, i.e. it is assumed for the stamp problems that the load is applied slowly enough for static equilibrium to prevail at all stages, and correspondingly in the case of the rolling sphere, that its velocity is low enough for the same to be true. .The results for rigid indentors are readily extended to cover the more general case of adhesive contact between two elastic bodies of different material properties by means of a straightforward linear transformation noted by Mossakovski (I963) and recapitulated in the present notation as appendix A, which reduces the general elastic-elastic case to that of a rigid-elastic indentation with suitably adjusted [ 55 ]

On the Flow Near the Trailing Edge of a Flat Plate

Abstract: It is shown that the boundary layer approximation to the flow of a viscous fluid past a flat plate of length l, generally valid near the plate when the Reynolds number Re is large, fails within a distance O(lRe$^{-\frac{3}{4}}$) of the trailing edge. The appropriate governing equations in this neighbourhood are the full Navier-Stokes equations. On the basis of Imai (1966) these equations are linearized with respect to a uniform shear and are then completely solved by means of a Wiener-Hopf integral equation. The solution so obtained joins smoothly on to that of the boundary layer for a flat plate upstream of the trailing edge and for a wake down-stream of the trailing edge. The contribution to the drag coefficient is found to be O(Re$^{-\frac{3}{4}}$) and the multiplicative constant is explicitly worked out for the linearized equations.

Magnetic Isotherms in the Band Model of Ferromagnetism

Abstract: Calculations are given for the dependence on temperature and magnetic field strength H of the magnetization M of ferromagnetic metals treated on the basis of the itinerant electron model. The spin wave contribution to the low temperature magnetization is considered in the limits of strong and very weak ferromagnetism. For the first limit, problems related to the well-known divergence of the zero field differential susceptibility are briefly discussed. The main part of the paper is concerned with the single particle contributions to the magnetization and associated differential susceptibility for very weak ferromagnetism. A simple equation for the magnetic isotherms is obtained in this limit and shown to be valid over a wide temperature range including 0 °K and the Curie temperature. This equation implies that plots of M 2 against H / M at various temperatures in this range give a series of parallel straight lines. Recently measured isotherms for the material ZrZn 2 are analysed on the basis of the theory, and several characteristic properties of this material are obtained from the analysis.

Some Thermal Effects in Cholesteric Liquid Crystals

Abstract: On the basis of conservation laws proposed by Ericksen (1961), constitutive equations are formulated for cholesteric liquid crystals. To motivate the choice, an entropy inequality of the type discussed by Muller (1967) is employed. Apart from minor differences, the hydrostatic theory of Ericksen (1962) is obtained, and proposals are made for the non-equilibrium terms. With this theory, the spinning phenomenon noted by Lehmann (1900) is investigated, and it is shown that solutions similar to his observations arise in a natural way.

The propagation of continental shelf waves

Abstract: Taking the case of a continental shelf of exponential slope, and assuming zero horizontal divergence, the authors derive a simple theory of free shelf waves, which, however, is more general than previous theories in that shorter as well as longer waves (in comparison with previous work) are taken into account. The properties of the waves are discussed, and the dispersion curves for each mode are obtained. Although the phase velocities of shelf waves are always in the same sense as those of Kelvin waves, there is a negative group velocity for a range of wavelengths, indicating that energy can propagate in the opposite sense. A similar approach is used to derive a theory for free waves propagating on a shelf between two regions of constant depth. The limiting case of a shelf of zero width is also considered, and is compared with a limiting case of the ‘double-Kelvin’ waves discovered by Longuet-Higgins (1967).

The Coalescence of Closely Spaced Drops when they are at Different Electric Potentials

Abstract: To discuss the coalescence of two neighbouring drops at different electric potentials as a problem of stability it is necessary to assume that they are held on fixed supports, for other­wise there is no equilibrium state to become unstable. In this work these supports are two equal rings. Experiments at first using soap films instead of drops, are in agreement with the analysis when the geometry of the rings conforms to the requirements of the theory. When the distance between the drops is small compared with that between the supporting rings the critical potential difference does not depend on the size or position of the rings but only on the radius of the two drops and the distance between them. Under these circum­stances a simple expression derived from computer solutions of the relevant equation has been found. This limiting formula is in agreement with some unpublished experiments with water drops carried out by J. Latham, and also with experiments in which instability was observed when neighbouring glycerine drops were subjected to potential differences as low as 9·8 V.

Flame structure and flame reaction kinetics II. Transport phenomena in multicomponent systems

Abstract: The transport fluxes in multicomponent flame systems due to diffusion and thermal conduc­tion, and to thermal diffusion and its reciprocal effect, are considered from the standpoint of the extension of the Chapman–Enskog kinetic theory to polyatomic gases by Wang Chang, Uhlenbeck & de Boer (1951, 1964) and the subsequent development by Mason, Monchick and coworkers (1961-66). Equations are given for the various diffusional, thermal diffusional and thermal fluxes which it is necessary to derive in order to obtain reaction rates from experimental temperature and composition profiles in flames; and the organization of com­puter programs for calculation of the multicomponent diffusion and thermal diffusion coefficients and the thermal conductivity is described. The use of matrix partitioning tech­niques in suitable circumstances to reduce the amount of computation is also discussed. The expressions for the transport fluxes are next used to derive equations for the mole fraction and temperature gradients in flowing reaction systems such as flames where trans­port processes and reaction occur side by side. From the mole fraction and temperature at one point in the system it is then possible by a numerical integration method such as the Runge–Kutta procedure to compute the complete composition and temperature profiles. Two methods of obtaining the mole fraction and temperature gradients are described, one of which, the Stefan–Maxwell formulation, leads to the more economical computation. A hydrogen–oxygen–nitrogen–steam mixture was chosen under conditions which simu­lated the pre-reaction region of a hydrogen–oxygen–nitrogen flame that had been studied experimentally, and the detailed composition profiles due to diffusion were computed. The experimental method of measurement involved continuous sampling from the flame and mass spectrometric analysis, a technique which had not previously been checked on a flame system itself. Good agreement between theory and experiment was found when thermal diffusion was considered in the calculation, although the computed hydrogen profile was slightly displaced with respect to the experimental one. This last observation is possibly due to diffusion effects in the pressure gradient at the probe tip. Otherwise the experimental technique seemed to be satisfactory. The computed profiles also showed a number of interesting features such as a maximum in the nitrogen concentration profile caused by thermal diffusion effects.

A $^{13}$ and $^{31}$P Magnetic Double Resonance Study of Organo-Phosphorus Compounds

Abstract: Twenty-two organo-phosphorus compounds of a variety of structural types have been examined by $^1$H-{$^{13}$C} and $^1$H-{$^{31}$P} magnetic double resonance spectroscopy. The signs and magnitudes of the $^{31}$P-H and $^{31}$P-$^{13}$C spin-spin coupling constants are sensitive to the valency of the phosphorus atom, and the nature of the groups attached to it. Parallel behaviour is noted between two types of coupling constant. The $^{31}$P chemical shifts agree with results obtained by conventional $^{31}$P single resonance spectroscopy, and the $^{13}$C chemical shifts depend on the polarizability of the phosphorus atom and its associated groups.

Motion of a Fluid in a Curved Tube

Abstract: Dean's work on the steady motion of an incompressible fluid through a curved tube of circular cross-section is extended. A method using a Fourier-series development with respect to the polar angle in the plane of cross-section is formulated and the resulting coupled non-linear equations solved numerically. The results are presented in terms of a single variable D = 4R$\surd$(2a/L), where R is the Reynolds number, a the radius of cross-section of the tube, and L the radius of the curve. The results cover the range of D from 96 (the upper limit of Dean's work) to over 600. From these it is found that the secondary flow becomes very appreciable for D = 600, moving the position of maximum axial velocity to a distance less than 0.38 a from the outer boundary, and decreasing the flux by 28% of its value for the straight tube. These calculations fill a large part of the gap in existing knowledge of secondary flow patterns, which lies in the upper range of Reynolds number for which flow is laminar. This range is of particular interest in the investigation of the cardiovascular system.

High resolution molecular photoelectron spectroscopy II. Water and deuterium oxide

Abstract: Helium 584 $\overset{\circ}{\mathrm A}$ radiation produces photoelectron spectra from water and deuterium oxide, which show extensive vibrational fine structure. The results are discussed in terms of the electronic structure of the molecule, and the bonding characteristics of the various molecular orbitals. Water is inferred to have only three configurationally distinct ionization potentials smaller than 21.22 eV.

The Microstructure and Crystallography of Aluminium-Silicon Eutectic Alloys

Abstract: Aluminium-silicon alloys in the composition range 12 to 16 wt.% silicon have been frozen unidirectionally over a range of rates from ca . 0.3 to 30 μ m/s and with imposed temperature gradients ranging from 0.35 to ca . 40°C/mm. The detailed morphology and crystallography of silicon in these alloys are described and the various structures are rationalized in terms of the variables, composition, freezing rate and temperature gradient. Three growth processes are distinguished in which (A) massive silicon crystals grow at a planar aluminium front by a relatively long range diffusion process; (B) short range dif­fusion occurs at the growth front between silicon crystals which develop a preferred fibre texture; and (C) silicon crystals are more or less heavily twinned and grow by steady or fluctuating short-range diffusion processes. There do not appear to be any preferred epitaxial orientations between the phases. Similarities are noted between the microstructures of eutectic alloys in the systems Al-Si and Ag-Si, and it is shown that in chilled or ‘modified’ alloys the silicon occurs in an irregular fibrous form rather than as isolated globules. The mechanisms are discussed for (I) the growth kinetics of silicon, (II) the transition from a long-range to a short-range diffusion process, (III) steady-state, and (IV) non-steady-state growth processes, and (V) repeated nucleation of silicon crystals from the liquid.

Improved calculations of leading-edge separation from slender, thin, delta wings

Abstract: In previous calculations (Mangler & Smith 1959) of the vortex-sheet model of leading-edge separation, only qualitative agreement was found with experimental observations. Because the numerical treatment of the model was then necessarily incomplete, it was uncertain how far the lack of quantitative agreement was to be attributed to the limitations of the model. The use of an automatic digital computer has now made it possible to reduce the uncertainties in the calculation to a negligible level. The features of interest in the real flow are more accurately predicted and the remaining discrepancies can be ascribed to the deficiencies in the model. The paper describes the method used to locate the vortex sheet and determine its strength in terms of the two boundary conditions on it; assesses the credibility of the results; and relates them to the observations. It is concluded that the model successfully predicts the observed height of the vortex above the wing, though the predicted lateral position is in error by up to 6% of the semi-span of the wing. This error falls as the incidence increases and is less when transition occurs in the boundary-layer upstream of secondary separation. Normal force is predicted accurately as is the distribution of pressure on the lower surface and the inboard part of the upper surface. The observed suction peak below the vortex changes its character when transition occurs in the boundary-layer upstream of secondary separation. The model predicts the suction peak in the turbulent case fairly well, but it is clear that detailed prediction of the suction peak is not possible by a model which is wholly inviscid.

Cylindrically symmetric charged dust distributions in rigid rotation in general relativity

Abstract: The paper presents a family of stationary cylindrically symmetric solutions of the Einstein-Maxwell equations corresponding to a charged dust distribution in rigid rotation. The interesting feature of the solution is that the Lorentz force vanishes everywhere and the ratio of the charge density and mass density may assume arbitrary value. The solutions do not seem to have any classical analogue.

Single Crystal Laser Raman Spectroscopy

Abstract: The chemical implications of single crystal Raman spectroscopy by means of laser excitation are briefly outlined. Unambiguous assignments are presented for PdCl 2- 4 , PtCl 2- 4 , PtCl 2- 6 , PdCl 2- 6 , GeCl 2- 6 and TeCl 2- 6 species. Detailed scattering equations are applied to mercurous chloride. The vibrational spectra of cassiterite, anataseand rutile are discussed in terms of the factor group. Gallium trichloride and aluminium tribromide spectra are discussed from the viewpoint of a perturbed D 2h Raman tensor. The complex spectrum of molybdenum trioxide is interpreted in terms of a site-factor group analysis. Problems of birefringence, crystal imperfection, dichroism, the resonance Raman effect, twinning, enantiomorphism, internal reflexion and dependence of relative intensity on physical state are noted. The methods discussed are applied to the first-order spectra of centrosymmetric crystals and rely on the validity of the Raman tensor predictions resulting from factor group analysis.

Resonant Oscillations of Water Waves. I. Theory

Abstract: A theory is presented to describe the oscillations of a liquid in a tank near a resonant frequency, where linearized theory is invalid. It is shown that although the oscillations are described adequately by the classical wave equation, the boundary conditions cannot be properly satisfied unless the non-linear terms are included. The effects of dissipation and dispersion are also significant in the determination of the oscillations, even though the terms to which they give rise in the equations are multiplied by small parameters under normal laboratory conditions. When the former is dominant a weak bore is formed which travels to and fro in the tank and is continually reflected at either end. When dispersion is significant the surface profile can be likened to a series of cnoidal waves which also travel along the tank and suffer reflexion. Several novel features appear. The amplitude does not increase monotonically as the nominal resonant frequency is approached. There are several distinct frequencies at which there is a sharp change in amplitude and in the form of the profile. More than one stable oscillation is possible at some frequencies. Near a resonant frequency higher than the fundamental, subharmonic oscillations are possible over part of the range.

The Structure of Vitamin B$_{12}$. VII. The X-ray Analysis of the Vitamin B$_{12}$ Coenzyme

Abstract: The three-dimensional molecular structure of coenzyme B 12 (59-deoxyadenosylcobalamin) has been determined by X-ray diffraction. The crystals, as grown from an acetone-water solution and photographed wet, are orthorhombic (space group P 2 1 2 1 2 1 ) with a = 27·93, b = 21·73 and c = 15·34 A. Four coenzyme molecules (C 72 H 100 O 17 N 18 PCo) and about 68 water molecules make up the unit cell. 3068 Bragg reflexions, extending to a spacing of 0·9 A, were measured with the crystals in contact with their mother liquor. The intensities were estimated visually from Weissenberg films taken with Cu Kα radiation. The cobalt atoms were easily located from the Patterson synthesis. The structure was solved in three steps, using first cobalt alone, then cobalt and 53 light atoms, and in the third approximation, 106 atoms, which included nearly the full asymmetric unit, except for water and hydrogen. Refinement of the atomic coordinates was accomplished initially by calculation of difference syntheses and finally by differential synthesis. The atomic positions have standard deviations of about 0·04 A. The conformation of the molecule is very similar to cyanocobalamin. The principal differences are in the orientation of the acetamide and propionamide side chains. Factors which influence the conformation of the corrin nucleus are analysed by comparing several corrinoids of known structure. Features of the molecule which have been examined in detail include the pucker in the pyrroline rings, the bend in the corrin macrocycle, the conformation of the nucleotide and nucleoside moieties and the orientation of the deoxyadenosine moiety with respect to the corrin nucleus. The packing of the molecules and the hydrogen bonding is discussed and compared with that found in the wet and dry vitamin B 12 crystals. Each coenzyme molecule participates in 18 direct intermolecular hydrogen bonds.

Physical properties of a transition metal oxide: optical and photoelectric properties of single crystal and thin film molybdenum trioxide

Abstract: The optical absorption spectra of single crystals and thin films of MoO 3 have been measured in the temperature range 150 to 340 °K with polarized light. At room temperature, the absorption spectrum of a single crystal consists of two peaks at λ 4130 and λ 3930 A with E || C and only one absorption peak at 4250 A with E ⊥ C followed by a rapid rise in absorption. The temperature and frequency dependence of the absorption coefficient in the edge over a range of absorption magnitudes 10 2 -10 5 cm -1 are described by expressions of the form K(v, T) = K 0 exp[— (β/kT) (E 0 — hv )]. The temperature dependence of the absorption edge was found to be linear from 340 to 150 °K with a temperature coefficient of — 6.2 x 10 -4 and — 9.3 x 10 -4 eV/°K for E || C and E ⊥ C , respectively. The corresponding temperature co­efficients in thin films are — 2.7 x 10 -4 and —4.0 x 10 -4 eV/°K for E|| and ⊥ to the film surface, respectively. Measurements have been made of the refractive indices of a single crystal and thin films. Ultraviolet irradiation of a thin film of MoO 3 produces a broad colour centre band having a maximum at 8700 A. Several absorption peaks are resolvable in polarized light. On cooling down to 150 °K, the peak position shifts to shorter wavelength by 400 A (4.67 x 10 -4 eV/°K). An e. s. r. signal with g = 2.001 ± 0.005 has been observed in the colour film. The electrical conductivity has been measured on single crystals and polycrystalline samples in the temperature range 25 to 500 °C, and the activation energies are found to be 1.83 ± 0.01 eV (intrinsic) and 0.29 to 0.70 eV (extrinsic). Photoconductivity has been measured in single crystals and thin films as a function of photon energy, temperature, and irradiation intensity. Trapping plays a significant role in the conduction phenomena. The thermal activation energies associated with different trapping levels were determined from the photoconductive decay curves and the electrical glow peaks measurements and were found to be in the range 0.16 to 0.64 eV.

Propagation in slowly varying waveguides

Abstract: The W. K. B. approximation is applied to a general system of linear partial differential equations which may be derived from a variational principle of a certain type. The theory describes slowly varying wavetrains, with the oscillation locally in one of the normal modes of a waveguide of quite general structure. The governing equations need not be hyperbolic; the wavelike character of the solution may be imparted by the lateral boundary conditions in the waveguide (e. g. surface waves on water). Variations in amplitude of the waves along rays are governed by conservation of an adiabatic invariant, as suggested by Whitham’s averaged variational principle. Higher order approximations may be constructed and the equations integrated b y quadrature. The averaged variational principle is also derived directly, in a manner applicable also to general nonlinear systems. It is shown to be a necessary condition governing the lowest order approximation for an asymptotic expansion of the same type as that for linear systems, provided such an expansion exists. However, it is not clear from this second approach how to construct higher order approximations.

High Resolution Molecular Photoelectron Spectroscopy. I. Fine Structure in the Spectra of Hydrogen and Oxygen

Abstract: The processes which set fundamental limits to the minimum line widths observable in photo-electron spectroscopy are outlined. An over-all limit of about 0.002 eV is deduced as the smallest attainable line width in ordinary circumstances. A photoelectron spectrometer is described which uses focusing deflexion in a radial electrostatic field for energy analysis. This instrument has energy resolution of 0.015 eV, smaller than the normal width of rotational envelopes in spectra of polyatomic molecules. The applications to the study of molecular hydrogen and oxygen is described.

The photoelectron spectra and ionized states of the halogen acids

Abstract: Photoelectron spectra have been obtained for HF, HCI, HBr and HI. These yield the following values (eV) for ionization to the $^2$II and $^2\Sigma^+$ states; HF, 16.05, 18.6, (20.0); HCl, 12.74, 12.82, 16.23 (16.6); HBr; 11.67, 12.00, 15.29 (15.6); HI 10.38, 11.05, 13.85 (14.25), the experimental error being $\pm$ 0.01 eV in most cases. The values are adiabatic except those in brackets which represent the vertical ionization associated with the values immediately preceding them which are minimum ionization energies to the $^2\Sigma^+$ states. The ionized states of HF are appreciably affected by interaction between states of the same species in a manner directly related to the mechanism of covalent-ionic resonance in the neutral molecule. Vibrational structure is found in the $^2\Sigma^+$ states of HCI and HBr but not in those of HF and HI. Potential energy diagrams are derived to explain this. Diagrams are also given correlating the geometrical and energetic parameters of this set of related molecules.

Extinction coefficients of triplet--triplet transitions.

Abstract: A new method for determining these coefficients has been developed. The method employs pulse radiolysis for triplet excitation and, by means of triplet-triplet (T-T) energy transfer, leads to a comparison between the extinction coefficients of the benzophenone ketyl radical ($\phi\_2$COH) and the triplet states of several aromatic compounds whose triplet levels are lower than that of benzophenone. Since the ketyl extinction coefficient can be determined independently, the actual T-T extinction coefficients can be obtained. Values of the extinction coefficient at the absorption maximum were obtained for anthracene (57 200 M$^{-1}$ cm$^{-1}$), naphthalene (22 600 M$^{-1}$ cm$^{-1}$), phenanthrene (21 000 M$^{-1}$ cm$^{-1}$) and 1,2-benzanthracene (25 100 M$^{-1}$ cm$^{-1}$), all relative to an extinction coefficient for the benzophenone ketyl maximum around 540 nm determined to be 3220 M$^{-1}$ cm$^{-1}$. For several other compounds the above method for various reasons could not be applied. In such cases estimates were obtained by energy transfer experiments, leading to comparisons between their triplet absorptions and those of the compounds already determined. Estimates were thus obtained for the maximum T-T extinction coefficients of benzophenone (10 300 M$^{-1}$ cm$^{-1}$), biacetyl (6400 M$^{-1}$ cm$^{-1}$), diphenyl (35 400 M$^{-1}$ cm$^{-1}$), and acridine (28 800 M$^{-1}$ cm$^{-1}$). During the course of these experiments estimates were also obtained for the rates of the following reactions: e$^-\_{aq.}$ + benzophenone (k$\_{II}$ = 3.0 x 10$^{10}$ M$^{-1}$ $s^{-1}$), OH$^.$ + benzophenone (k$\_{II}$ = 8.7 x 10$^9$ M$^{-1}$ s$^{-1}$) and hydrogen abstraction by triplet benzophenone from cyclohexane, to give $\phi\_2$COH (k$\_{II}\sim$ 8 x 10$^5$ M$^{-1}$ s$^{-1}$).

Resonant Oscillations of Water Waves. II. Experiment

Abstract: An experimental investigation of the oscillations of water in a tank forced to oscillate at near resonant frequencies is described. The results are given in a form which enables a direct comparison to be made with the predictions of the theory of part I. A description of the apparatus and methods of measurement is presented in §§ 2 and 3. In § 4 the results of both the theory of part I and the experiments are reported, compared and discussed. It is shown that there is reasonable agreement between theory and experiment over the range of the parameters investigated.

The Hyperfine Structure Stark Effect. I. Theory

Abstract: A theory of the quadratic Stark effect is presented. It is aimed at a description of the hyperfine structure of a free atom in a uniform electric field. A perturbation theory approach is adopted and extensive use is made of effective operators. In spherical tensor form these can be written as the sum of a scalar and a tensor of rank two. Associated scalar and tensor polarizabilities are defined and their properties are discussed. A variety of applications of the theory are given.

The photoionization absorption continua for alkali metal vapours

Abstract: Absorption measurements are reported for the metal vapours potassium, rubidium and caesium and existing data for lithium and sodium are reviewed. The separation of atomic and molecular absorption is discussed and the data presented as absolute cross-section curves as a function of wavelength. The shapes of the atomic photoionization continua and their absolute values are discussed in relation to existing theoretical cross-sections data. Best experimental values of the cross-sections at the series limits are found to be : Li = 1.9 ± 0.2 Mb, Na = 0.125 ± 0.01 Mb, K = 0.007 ± 0.002 Mb, Rb = 0.10 ± 0.015 Mb and Cs = 0.20 ± 0.03 Mb. The curves for potassium, rubidium and caesium have non-zero minima and that for sodium has a minimum which is near zero. Experimental minimum values of 4 ± 2 x 10 -3 Mb, 8 ± 3 x 10 -3 Mb and 6 ± 1 x 10 -2 Mb for potassium, rubidium and caesium respectively are obtained.

Magnetic and Electrical Properties of Ferromagnetic Alloys Near the Critical Concentration

Abstract: A simple molecular field theory is developed to explain at least qualitatively the behaviour of weakly ferromagnetic and strongly paramagnetic alloys near their critical concentration c F . The free energy of a weakly ferromagnetic alloy can be expanded in powers of magnetization ζ and a modified Landau theory is used to calculate the concentration dependence of the magnetization, specific heat and the susceptibility near c F . It is believed that such a theory can work better in this case than does the usual Landau theory as fluctuations in ζ are relatively small for low temperatures. A change in low temperature resistivity around c F is also expected as a result of ciritcal scattering of conduction electrons. This change was first calculated by Rice (1967) for strongly paramagnetic alloys giving a large T 2 term. However, this calculation is not applicable for c very close to c F . It is shown that for c → c F the paramagnetic resistivity remains finite and has a maximum for c ═ c F , ρ Para c ═ c F ∝ T 5/3 . It is also found that the ferromagnetic resistivity ρ ferro has a similar dependence on concentration and temperature as ρ Para and ρ ferro c ═ c F ═ ρ Para c ═ c F , i.e. the resisitivity is symmetrical around c F .