Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1953"
TL;DR: In this article, the authors developed a form of theory which appears to be appropriate to measurements of position on a sphere and demonstrated the simultaneous distribution of the amplitude and direction of the vector sum of a number of random unit vectors of given precision.
Abstract: Any topological framework requires the development of a theory of errors of characteristic and appropriate mathematical form. The paper develops a form of theory which appears to be appropriate to measurements of position on a sphere. The primary problems of estimation as applied to the true direction, and the precision of observations, are discussed in the subcases which arise. The simultaneous distribution of the amplitude and direction of the vector sum of a number of random unit vectors of given precision, is demonstrated. From this is derived the test of significance appropriate to a worker whose knowledge of precision lies entirely in the internal evidence of the sample. This is the analogue of ‘Student’s’ test in the Gaussian theory of errors. The general formulae obtained are illustrated using measurements of the direction of remanent magnetization in the directly and inversely magnetized lava flows obtained in Iceland by Mr J. Hospers.
5,482 citations
TL;DR: In this paper, it was shown analytically that the distribution of concentration produced in this way is centred on a point which moves with the mean speed of flow and is symmetrical about it in spite of the asymmetry of the flow.
Abstract: When a soluble substance is introduced into a fluid flowing slowly through a small-bore tube it spreads out under the combined action of molecular diffusion and the variation of velocity over the cross-section. It is shown analytically that the distribution of concentration produced in this way is centred on a point which moves with the mean speed of flow and is symmetrical about it in spite of the asymmetry of the flow. The dispersion along the tube is governed by a virtual coefficient of diffusivity which can be calculated from observed distributions of concentration. Since the analysis relates the longitudinal diffusivity to the coefficient of molecular diffusion, observations of concentration along a tube provide a new method for measuring diffusion coefficients. The coefficient so obtained was found, with potassium permanganate, to agree with that measured in other ways. The results may be useful to physiologists who may wish to know how a soluble salt is dispersed in blood streams.
4,530 citations
TL;DR: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object.
Abstract: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an ‘effective source’, and the complex transmission of the optical system— they are the data initially known and are generally of simple form. A generalized ‘transmission factor’ is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.
550 citations
TL;DR: In this article, the authors tried to relate four recent sets of measurements on the flow laws of ice: (1) Glen's laboratory compression tests, (2) Gerrard, Perutz & Roch's pipe experiment on the Jungfraufirn, (3) measurements of contraction rate in a tunnel in Skauthoe Glacier by McCall, and (4) measurements in tunnels in the Z’Mutt and Arolla Glaciers reported by Haefeli & Kasser.
Abstract: The paper attempts to relate four recent sets of measurements on the flow laws of ice: (1) Glen’s laboratory compression tests, (2) Gerrard, Perutz & Roch’s pipe experiment on the Jungfraufirn, (3) measurements of contraction rate in a tunnel in Skauthoe Glacier by McCall, and (4) measurements of contraction rate in tunnels in the Z’Mutt and Arolla Glaciers reported by Haefeli & Kasser. When they are analyzed within the framework of a general flow theory, satisfactory agreement is found between all these results, with one exception: the reported rate of collapse of the Arolla tunnel is faster than would be given by the present theory. The paper includes the theory of the slow collapse of a cylindrical or a spherical hole in a medium showing non-Newtonian viscosity.
472 citations
TL;DR: In this article, it is shown that the propulsion of a single filament which forms itself into a single wave is very near to that of an infinite filament with the same wave motion, where the velocity field can be described in terms of singularities situated at the centre of the sphere.
Abstract: Since the Reynolds number of motion of microscopic organisms through liquids, defined as L ρ V /μ, where L is the length of the organism, V the velocity with which it moves, ρ the density of the liquid and μ the viscosity, is small, propulsion is due predominantly to the viscous forces, the effect of the inertial forces being negligible. The best-known problem that neglects all inertial forces is Stokes’s solution for the slow steady fluid motion past a sphere, in which the velocity field can be described in terms of singularities situated at the centre of the sphere. The movement of microscopic organisms is determined by placing distributions of these singularities inside the surface of the organism and satisfying all boundary conditions. The motions that are considered are restricted to organisms which propagate some kind of disturbance along filaments of circular cross-section with small radius. The first problem to be considered is that of an infinite thin filament along which are propagated plane waves of lateral displacement. Formulae for the velocity of propulsion are obtained for (i) the limiting case of zero radius and (ii) the case when the amplitude of the displacement is small compared to the wave-length. Computations have been carried out to estimate the propulsion in the case of small non-zero filament radius when the amplitude is larger than that allowed for in case (ii) above. It is also shown that the propulsion of a finite filament which forms itself into a single wave is very near to that of an infinite filament with the same wave motion. The second problem is that of an infinite filament along which any general three-dimensional disturbance is propagated. The movement is then deduced for the propagation of a spiral wave along an infinite filament, and also for the propagation of longitudinal waves along a finite filament.
470 citations
TL;DR: A chemical actinometer has been devised whose high sensitivity and precision depends upon the spectrophotometric determination of the photolysis products of potassium ferrioxalate with 1:10 phenanthroline as discussed by the authors.
Abstract: A chemical actinometer has been devised whose high sensitivity and precision depends upon the spectrophotometric determination of the photolysis products of potassium ferrioxalate with 1:10 phenanthroline. The sensitivity is some hundreds of times greater than that of the uranyl oxalate actinometer. The behaviour of the actinometer has been investigated in detail at 3650 to 3663 A, and tests also show that it is suitable for use over a wide range of wavelengths. The approximate quantum efficiency has been determined at twelve wave-lengths between 4900 and 2537 A. The advantages and possible applications of the actinometer in the measurement of small doses of radiation are discussed.
385 citations
TL;DR: In this paper, a further step is made in the development of the orbital theory of molecular structure, the object being to allow for the tendency of two electrons in the same space orbital to keep apart because of the repulsive field between them.
Abstract: A further step is made in the development of the orbital theory of molecular structure, the object being to allow for the tendency of two electrons in the same space orbital to keep apart because of the repulsive field between them. The description of a molecule is generalized in such a way as to bracket together electrons in pairs, the simple theory of molecular orbitals being used as a starting point for a wider treatment. If two orbitals are used for each electron pair (paired electron orbitals), it is shown that the theory contains as special cases both previous results of the present series of papers and also the valence-bond wave function which is shown to be an alternative particular approximation to the general function.
382 citations
TL;DR: In this article, a new theory for the supercurrent was proposed, in which the current is related to an average of the vector potential over a region around the point considered, and the size of the region is governed by a parameter ξ, which is dependent, in a similar way to the mean free path in a normal metal, on the degree of purity.
Abstract: The penetration depth, λ, in superconducting tin at 0° K has been found to be capable of variation by the addition of impurity, 3% of indium causing it to be nearly doubled, although the thermodynamical properties such as T c are hardly affected. It is suggested that this result throws doubt on the phenomenological theory of F. and H. London, and a new equation for the supercurrent is proposed, in which the current is related to an average of the vector potential over a region around the point considered. The size of the region is governed by a parameter ξ, which is dependent, in a similar way to the mean free path in a normal metal, on the degree of purity . The new theory agrees in a satisfactory manner with the experimental results, and also provides an explanation of the magnitude of λ in a pure metal. It has been found that the unusual anisotropy of λ in pure tin is absent in tin + 3 % indium , and this too is in agreement with the prediction of the new theory. The fact that the temperature variation of λ appears to have the same form independent of the degree of purity implies that ξ depends on temperature in the same way as λ; this result agrees with the observed behaviour of the interphase surface energy, α ns , if ξ is regarded as determining the width of the interface between superconducting and normal regions. The paper ends with a discussion of the relation of the new theory to microscopic theories of superconductivity.
312 citations
TL;DR: In this article, the Boltzmann equation for conduction electrons in a crystal was solved by means of a variational method, and exact expressions for the electrical conductivity and the thermo-electric power were obtained in the form of ratios of infinite determinants.
Abstract: The Boltzmann equation is set up for the conduction electrons in a crystal in which the scattering is due to the polarization waves of the lattice, and it is pointed out that at low temperatures it is impossible to define a unique time of relaxation for the scattering process. The Boltzmann equation is solved by means of a variational method, and exact expressions for the electrical conductivity and the thermo-electric power are obtained in the form of ratios of infinite determinants. By approximating to the exact solutions, relatively simple expressions are derived which are used to discuss the dependence of the conduction phenomena upon the temperature and upon the degree of degeneracy of the electron gas.
280 citations
TL;DR: In this paper, the elastic properties of a dilute emulsion of one incompressible viscous liquid in another, arising from the interfacial tension between the two phases, are investigated.
Abstract: A calculation is made of the elastic properties of a dilute emulsion of one incompressible viscous liquid in another, arising from the interfacial tension between the two phases. A linear relation between the stress tensor, the rate-of-strain tensor and their first time derivatives defines the behaviour at small rates of strain; the three constants involved are expressed as functions of the viscosities of the two components, the drop size (assumed small and uniform), the interfacial tension and the concentration. The relaxation time and retardation time for the system vary directly as the drop diameter and inversely as the interfacial tension. The effect of slip at the interfaces, which might be associated with the presence of a film of a third component introduced as a stabilizer, is also calculated. The values of the rheological constants are appreciably altered if the frictional coefficient specifying the degree of slipping is sufficiently small, but the type of elastic behaviour is unchanged. In the case of a suspension of elastic solid particles in a viscous liquid, slip at the solid-liquid interfaces can cause a change in the type of elastic behaviour.
272 citations
TL;DR: In this article, it is shown that the effect of the inner viscous sublayer on the behaviour of each harmonic component outside it is exactly as if there were a solid wall at a certain position in the stream, with no flow across it and inviscid flow outside it.
Abstract: When any boundary layer on a straight wall is subjected to an expansive steady disturbance (either an incident wave, or departure of the wall shape from straight), or when a turbulent layer is subjected to a fairly weak compressive one, an interaction between the main stream (if supersonic) and the boundary layer is set up but the layer does not separate. Such an interaction is here treated mathematically by an extension of the author’s method (Lighthill 1950) of perturbing a parallel flow and neglecting the disturbances of the viscous forces. It is shown that these are non-negligible only in an ‘inner viscous sublayer’, in which the Mach number remains small, and which for the turbulent layer is inside the so-called laminar sublayer. Further, if the disturbances are Fourier-analyzed longitudinally, then the effect of the inner viscous sublayer on the behaviour of each harmonic component outside it is exactly as if there were a solid wall at a certain position in the stream, with no flow across it and inviscid flow outside it. This position depends on the wave number k and on the skin friction in the undisturbed layer. This new boundary condition is now inserted into the old theory. Solutions are obtained for large and small k and used to deduce, respectively, (i) the principal local features of both the wall-pressure distribution and the flow outside the boundary layer, and (ii) the extent of upstream influence. An interpolation between them is used in an attempt to predict the complete form of the reflected wave when a shock is incident upon a turbulent layer. Reasonable agreement with experiment is obtained. The paper ends with a discussion of the skin friction distribution and how it influences the onset of non-linearity.
TL;DR: In this paper, an account of present ideas about the electron orbits in octahedral complexes containing 4 d and 5 d transition elements, with particular reference to paramagnetic resonance, is given.
Abstract: An account is given of present ideas about the electron orbits in octahedral complexes containing 4 d and 5 d transition elements, with particular reference to paramagnetic resonance. The complex is treated by the method of molecular orbitals with the assumption that the magnetic carriers are partly in central d -orbits and partly in p π -orbits round the outer nuclei. This is shown to lead to a reduction in the orbital g -factor and to the possibility of observing the hyperfine structure from the outer nuclei, thus testing the initial assumptions.
TL;DR: In this paper, a general account of paramagnetic resonance in rare-earth crystals is given, assuming that the crystalline electric field splitting is less than the spin-orbit splitting.
Abstract: A general account is given of the theory of paramagnetic resonance in rare-earth crystals, assuming that the crystalline electric field splitting is less than the spin-orbit splitting. The treatment is related, as far as possible, to numerical parameters, which are sufficiently comprehensive for it to be a straightforward process to pass from a given crystalline field to the spin-Hamiltonian.
TL;DR: In this article, the infra-red and Raman selection rules for an infinite long-chain molecule having helical symmetry are discussed, and the normal modes of vibration are classified by the use of group theory.
Abstract: The infra-red and Raman selection rules for an infinite long-chain molecule having helical symmetry are discussed, and the normal modes of vibration are classified by the use of group theory. A weak-coupling approximation is used to derive relations between the intensities of the variously polarized spectral components arising from a single vibration frequency of the unit out of which the helix is built. A form of matrix perturbation theory is used to obtain some approximate relations between the frequencies of these components; these relations involve the angle ψ relative to the screw axis which separates successive units of the helix and provide, in principle, a method by which this angle may be measured. It is suggested that the methods discussed here may serve as a basis for the interpretation of the spectra of fibrous proteins.
TL;DR: In this paper, the second moments of the Raman spectrum of polycrystalline benzene molecules have been measured and the spin-lattice relaxation time has been analyzed. But the authors did not consider the intermolecular contribution to the second moment.
Abstract: The nuclear magnetic resonance absorption spectrum and the spin-lattice relaxation time have been measured for the protons in three isotopic species of benzene in polycrystalline form between 75 and 278 degrees K. The three species were C$\_{6}$H$\_{6}$, C$\_{6}$H$\_{5}$D and 1.3.5-C$\_{6}$H$\_{3}$D$\_{3}$. For all three species the measured spectrum has its full rigid lattice width below 90 degrees K. A method of analysis is developed which makes it possible to derive separately the intramolecular and the intermolecular contributions to the second moment (mean square width) of the spectrum from the measured second moments, without the necessity of knowing the crystal structure. From the intramolecular contribution it is found that the separation of neighbouring protons in the C$\_{6}$H$_{6}$ molecule is 2$\cdot $495 $\pm $ 0$\cdot $018 angstrom. The intermolecular contribution is in agreement with a value calculated from a knowledge of the crystal structure. On warming from 90 to 120 degrees K the spectrum for all three species narrows considerably. From 120 degrees K to the melting-point (278$\cdot $7 degrees K) the second moments remain almost constant. The second moment separation procedure is also applied in this range and leads to the conclusion that the narrowing is caused by reorientation of the molecules about their hexad axes in the crystal lattice. Analysis of the measurements of the spin-lattice relaxation time shows that for all three species the reorientation process is governed by an activation energy of 3$\cdot $7 $\pm $ 0$\cdot $2 kcal/mole. The reorientation frequency is of the order of 10$^{4}$ c/s at 85 degrees K and rises to a value of the order of 10$^{11}$ c/s just below the melting-point. The relationship between the present experimental results and recent measurements of the Raman spectrum of solid benzene is discussed. Finally, consideration is given to the application to other materials of methods of separating the intra- and intermolecular contributions to the second moment.
TL;DR: In this article, the amplitude-frequency curve of a wave with a small amplitude and at frequencies where great amplification occurred, owing to resonance, is shown to have two non-intersecting branches, a result which can be explained theoretically.
Abstract: The experiments here described were designed to test experimentally some conclusions about free standing waves recently reached analytically by Penney & Price. A close approximation to free oscillations was produced in a tank by wave makers operating with small amplitude and at frequencies where great amplification occurred, owing to resonance. The amplitude-frequency curve proved to consist of two non-intersecting branches, a result which can be explained theoretically. A striking prediction made by Penney & Price was that when the height of the crests of standing waves reaches about 0·15 wave-length they will become pointed, in the form of a 90° ridge. Higher waves were expected to be unstable because the downward acceleration of the free surface near the crest would exceed that of gravity. The experimental conditions necessary for producing a crest in the form of an angled ridge were found and the wave photographed in this condition. Good agreement was found with the calculated form of the profile of the highest wave, which had an angle very near to 90°. The predicted instability for two-dimensional waves was found to begin at the moment the crest became a sharp ridge. It rapidly assumed a three-dimensional character which was revealed by two photographic techniques. Even when the amplitude of oscillation of the wave makers was only 0·85°, violent types of instability developed which produced effects that are here recorded.
TL;DR: In this article, the second moment (mean square width) of the measured spectrum for temperatures at which the lattice is effectively rigid, namely, below 150° K, is consistent with a molecular structure having D 3d symmetry, tetrahedral bond angles, C-C bond lengths of 1.54 A and C-H bond lengths with 1.10 A.
Abstract: The nuclear magnetic resonance absorption spectrum and the spin-lattice relaxation time have been measured for the protons in polycrystalline cyclo hexane between 100° K and its freezing-point (279·6° K). It has been found that the second moment (mean square width) of the measured spectrum for temperatures at which the lattice is effectively rigid, namely, below 150° K, is consistent with a molecular structure having D 3d symmetry, tetrahedral bond angles, C—C bond lengths of 1.54 A and C—H bond lengths of 1.10 A. If the HCH angle is treated as a parameter to be determined, it is found to be 1071/2 ± 3°. On warming from 155 to 180° K the second moment decreases to a value which indicates the reorientation of the molecules about their triad axes. Analysis of the spin-lattice relaxation time, which falls rapidly in this temperature range, shows that the height of the barrier hindering this reorientation is 11 ± 1 kcal/mole. Just below 186° K, the temperature at which there is a polymorphic change, the reorientation frequency is of the order 10 6 c/s. The polymorphic transformation is accompanied by discontinuous changes in the second moment and the relaxation time. It is concluded that in the higher temperature modification the molecules have a considerable freedom of reorientation, such that the intramolecular contribution to the second moment becomes negligibly small. Just above 186° K the mean reorientation frequency exceeds 3 x 10 7 c/s. A final narrowing of the line between 220 and 240° K is thought to be due to vacancy diffusion of the molecules within the lattice, causing the intermolecular contribution to the second moment to vanish also. Details are given of the gas-flow cryostat used in this work. The theoretical formulation of the second moment has been extended to include the modification of the intermolecular contribution during reorientation.
TL;DR: In this paper, an approximate thermodynamic description of the behavior can be given by introducing an ordering parameter, z, to represent the configuration, and the value of z remains fixed for rapid changes in temperature and pressure.
Abstract: In liquids the molecular configuration is a function of temperature. If in a given substance super-cooling of the liquid is possible to such a low temperature—the ‘transformation temperature’ ( T g )—that the viscosity reaches about 10 13 poises, the required changes in configuration become impossible and at all lower temperatures the structure remains frozen. This causes apparent discontinuities in derivative properties (such as the specific heat) and departures from the Nernst heat theorem. Below T g the substance is said to be in the glassy state. In the neighbourhood of T g relaxation towards the appropriate equilibrium structure can be easily observed. An approximate thermodynamic description of the behaviour can be given by introducing an ordering parameter, z , to represent the configuration. The value of z remains fixed for rapid changes in temperature and pressure. This leads to a number of relations between certain properties of the equilibrium liquid and of the glass. One of these (given by Prigogine & Defay) is Δ K Δ C p = TV (Δα) 2 , where k , C p and α are the compressibility, heat capacity and expansivity, and Δα= α liquid —α glass , etc. New results obtained with glucose and analysis of existing data suggest that the relations are obeyed only very approximately and that the one-parameter model is therefore inadequate. Near equilibrium it is convenient to use—instead of z — a 9fictive temperature’, T ¯, defined so that with fixed p and z the glass would be in equilibrium at T = T ¯. A fictive pressure, p ¯ can similarly be defined. To a first approximation, the approach to equilibrium is described by means of a volume viscosity, ἠ v , such that (at constant p and T )1/V d V /d t =- p - p ¯/ἠ v . The relaxation time for observations of the approach to equilibrium is then given by r = const, ἠ v ∆k, where the constant depends on the experimental arrangement. r has been measured for glycerol and glucose near their transformation temperatures and the results used to find ἠ v and its dependence on temperature. It was found that: (i) The activation energies of volume viscosity for glycerol (25 kcal/mole) and glucose (130 kcal/mole) are equal to those of the shear viscosity (ἠ). Thus ἠ v /ἠ is independent of temperature. (ii) ἠ v /ἠ is is of the order 10 for glycerol and is about 200 for glucose. (iii) d V /d t is proportional to ( p - p )¯ for only very small displacements from equilibrium. It follows that the concept of volume viscosity must be used with caution.
TL;DR: In this paper, the renewal theorems for non-discrete non-negative random variables with mean k > o were studied. But the main emphasis was on processes which are not discrete.
Abstract: A sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete processwhere Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under whichThese new conditions are much weaker than those of previous theorems by Feller, Tacklind, and Cox and Smith.
TL;DR: In this paper, the authors compared the results of Gronwall et al. with those obtained from the new theory of conductivity, and showed that the treatment gave no improvement on the Falkenhagen results.
Abstract: In the theories of the conductivity and viscosity of electrolyte solutions originally developed by Debye & Hϋckel, and Falkenhagen and his co-workers, the ions were regarded as dimensionless point charges. In the theory of activity and osmotic coefficients developed by Gronwall, La Mer & Sandved the ions were assumed to be of finite size, having a distance of closest approach, a . Making use of the Gronwall et al . results, the concept of closest approach has been introduced into the theories of conductivity and viscosity. These quantities are obtained in the form of infinite series, of which only the first few terms have been calculated, owing to the great complexity of the work. In the case of conductivity, for 1-1 salts in water the approximate range of validity of the results is for concentrations less than N/10. Complete quantitative agreement with experiment is found. Under other experimental conditions the calculated conductivity shows the correct qualitative behaviour. The new theory of viscosity should possess a similar range of validity, but comparison with experiment shows that the treatment gives no improvement on the Falkenhagen results. In their paper, Gronwall et al . gave only a very limited comparison between their theory and experiment. It has been necessary to extend this comparison, and values of a found thereby are compared in table 2 with those obtained from the new theory of conductivity. The significance of these values is discussed in the last section of the paper, together with reasons for the success of the conductivity theory and the absence of any improvement in the viscosity theory.
TL;DR: In this article, the infra-red spectra of carboxylic acids and their COOD derivatives have been investigated between 1500 and 500 cm -1, as dimeric units in the liquid or crystalline solid states.
Abstract: The infra-red spectra of a considerable number of carboxylic acids and their COOD derivatives have been investigated between 1500 and 500 cm -1, as dimeric units in the liquid or crystalline solid states. Under these conditions the COOH group is shown usually to give rise to strong absorption bands in the regions 1420 ±20, 1300 ± 15 and 935 ± 15 cm -1 . The first two of these are found to correspond to closely coupled OH deformation and C—O stretching vibrations occurring in the plane of the (COOH) 2 dimeric ring; the latter is caused by the out-of-plane OH deformation vibration. COOD groups have absorption bands in the ranges 1350 ±50, 1050 ± 10 and 675 ±25 cm -1 , which can be assigned respectively to the C—O stretching mode and the in-plane and out-of-plane OD deformation vibrations. Less constant absorption bands of the COOH group between 700 and 575 cm -1 are attributed to O—C = O skeletal deformation vibrations, and the corresponding bands are found at slightly lower frequencies in the spectra of the COOD derivatives. The results of the infra-red investigation are compared with the Raman spectra of such acids and with related infra-red and Raman frequencies of other molecules. Infra-red spectra of some equimolecuiar mixtures of acids with water have also been studied.
TL;DR: The theory of the magnetic properties of rare-earth ions in crystals, which has been developed in preceding papers, is here applied to the ethyl sulphates, and the magnetic resonance and susceptibility results are discussed at length, and it is shown that the observations can be explained by using fields of C 3۸ symmetry which vary in a systematic way as one rare earth ion is replaced by another as mentioned in this paper.
Abstract: The theory of the magnetic properties of rare-earth ions in crystals, which has been developed in preceding papers, is here applied to the ethyl sulphates. The magnetic resonance and susceptibility results are discussed at length, and it is shown that the observations can be explained by using fields of C 3۸ symmetry which vary in a systematic way as one rare-earth ion is replaced by another.
TL;DR: In this article, the distribution of atomic nuclei in potassium dihydrogen phosphate at room temperature has been obtained by Fourier synthesis of single-crystal neutron diffraction data, and series termination effects have been examined and corrected for, either by specific assessment of the diffraction rings or by difference syntheses.
Abstract: Projections of the distribution of atomic nuclei in potassium dihydrogen phosphate at room temperature have been obtained by Fourier synthesis of single-crystal neutron diffraction data. Series termination effects have been examined and corrected for, either by specific assessment of the diffraction rings or by difference syntheses. The parameters so obtained are confirmed by least-squares analyses and give a reliability index of 7%. Hydrogen peaks occur midway between two oxygen atoms and appear circular when viewed along the O$\chembond{1,0} $H$\chembond{1,0} $O bond, but elliptical when viewed across it. The PO$_{4}$ tetrahedra are much more nearly regular than previously supposed, and the O$\chembond{1,0} $H$\chembond{1,0} $O bonds are inclined at not more than $\frac{1}{2}$ degrees to the xy-plane.
TL;DR: In this paper, an experimental study has been made of the friction of real ski and of smaller models sliding on snow and ice at various temperatures, and experiments support the view, put forward earlier, that this low friction is due to localized surface melting produced by frictional heating.
Abstract: An experimental study has been made of the friction of real ski and of smaller models sliding on snow and ice at various temperatures. On cold snow the static friction is high. When the sliding speed is appreciable the friction falls to a low value, and experiments support the view, put forward earlier, that this low friction is due to a localized surface melting produced by frictional heating. Measurements are made on a variety of surfaces including metals, synthetic polymers and waxes. The contact angle which water makes with the surface is important, and there is evidence that this can decrease during sliding. In general, the solids with a high contact angle give a lower friction. The behaviour is also influenced by the relative hardness of ice and of the ski surface at the temperature of sliding. Polytetrafluoroethylene gives a very low friction on snow and ice under all conditions.
TL;DR: In this paper, the interaction of twenty metals with clean evaporated films of some twenty metals has been investigated between 0 and - 183 degrees C, and the results are expressed as gas chemisorbed or gas not chemiscorbed.
Abstract: The interaction of N$\_{2}$, H$\_{2}$, CO, C$\_{2}$H$\_{4}$, C$\_{2}$H$\_{2}$ and O$\_{2}$ with clean evaporated films of some twenty metals has been investigated between 0 and - 183 degrees C. The results are expressed as gas chemisorbed or gas not chemisorbed. O$\_{2}$ chemisorption is universal among all the metals studied, with the single exception of Au. Chemisorption of N$\_{2}$ and H$\_{2}$ is limited to transition metals and alkaline earth metals, and chemisorption of CO, C$\_{2}$H$\_{4}$ and C$\_{2}$H$\_{2}$ to these and to Al, Cu and Au. O$\_{2}$ chemisorption is believed to take place with formation of O$^{2^{-}}$ ions at the surface, with electron donation from the s and p bands of the metals. The inactivity of Au towards O$\_{2}$ is, however, only partly understandable in terms of this mechanism. The remaining five gases are believed to form covalent bonds with the metal d bands in chemisorption; with Cu and Au it is suggested that d-s promotion takes place in the act of chemisorption to allow such covalence. Al is exceptional, and with the light metals, s and p band electrons may be active in chemisorption: a further exception is the interaction of K with C$\_{2}$H$\_{2}$. Some applications of these results to catalytic systems are discussed.
TL;DR: The thermal conductivity of a type I diamond has been measured between 2 degrees K and room temperature, and the experimental results show that at 3 degrees K the proportion of specular reflexions is as great as 40% as discussed by the authors.
Abstract: The thermal conductivity of a type I diamond has been measured between 2 degrees K and room temperature. Above 100 degrees K the temperature variation of conductivity accords with Peierls's theory of Umklapp processes. To account for the results at the lowest temperatures it is necessary to assume that some specular reflexion of phonons occurs at the crystal boundaries. Casimir's theoretical treatment of boundary scattering has been extended to take into account the finite length of specimens and the smoothness of the crystal walls, and the experimental results show that at 3 degrees K the proportion of specular reflexions is as great as 40%. Between 50 and 100 degrees K the conductivity is considerably less than that calculated for an ideal diamond, and some conclusions are drawn as to the imperfections which cause this discrepancy.
TL;DR: In this article, a mathematical discussion of the exchange processes, of heat or of matter (as in ion exchange or adsorption), that arise when a fluid flows through the pores or voids along a column containing matter in the solid state is presented.
Abstract: The paper is intended as a contribution to the mathematical discussion of the exchange processes, of heat or of matter (as in ion exchange or adsorption), that arise when a fluid flows through the pores or voids along a column containing matter in the solid state. The exchange is taken to be governed by a linear or bilinear or Langmuir-type kinetic exchange equation. The matters dealt with include the solution of the exchange and conservation equations for any initial conditions, mathematical properties of and asymptotic expansions for the functions arising, and the appearance of the functions in the solutions of other physical problems. The discussion of the mathematical properties of the functions and their asymptotic expansions is needed to establish the relationship between solutions on the equilibrium theory and those on the kinetic theory, and to find closer approximations to the kinetic-theory solutions.
TL;DR: In this article, the perturbed stationary state method (p.s. method) was extended to include resonance degeneracy and when the transition concerned involves a change of the axial component of angular momentum.
Abstract: The probability of slow inelastic collisions between two atomic systems is investigated theoretically using the perturbed stationary state method (p.s.s. method) in which the kinetic energy of the relative motion is treated as the perturbation responsible for the transitions. The p.s.s. method has been extended to include cases in which resonance degeneracy occurs and when the transition concerned involves a change of the axial component of angular momentum. The scope and limitations of the method are examined in detail and compared with those of Born’s approximation. It is shown that two modifications of the latter approximation are of comparable importance in most cases, one involving a change in the effective transition potential, the other the distortion of the plane waves representing the relative motion of the colliding systems. The first of these is the only one included in previous applications. It is shown that, under semi-classical conditions, the formulae of the p.s.s. method reduce to Mott’s impact parameter formulae provided the important distortion effect is neglected. New semi-classical formulae, including this effect as well as the effect of resonance degeneracy, are derived in a form suitable for numerical application.
TL;DR: In this article, a matrix method for calculating the derived invariants of the molecular polarizability in terms of polarisation functions characteristic of bonds and general coefficients defining the normal mode of vibration is presented.
Abstract: Theories of Raman intensities are reviewed, and it is shown that no satisfactory theory exists which enables relative intensities to be predicted. A matrix method for calculating the derived invariants of the molecular polarizability in terms of polarizability functions characteristic of bonds and general coefficients defining the normal mode of vibration is presented. The method is an extension of the well-known matrix techniques for the vibrational problem. The general theory is simplified by making approximations first suggested by Wolkenstein (1941). In this approximation each distinct bond is characterized by three polarizability functions; these may be deduced from measured intensities in much the same way as force constants may be deduced from frequencies. General formulae for setting up the matrices are given, and completely general formulae for intensities in terms of bond polarizabilities, to the Wolkenstein approximation, are quoted in full for molecules of the types MX 2 (C 2 v ), MX 4 ( T d ), MX 3 Y ( C 3 v ) and MX 2 Y 2 ( C 2 v ).
TL;DR: In this article, a study was made of the yielding behavior of single crystals of brass grown in either argon or nitrogen, and the yield strength showed a maximum at 200 degrees C. Conditions for producing yield points in face-centred cubic metals are briefly discussed.
Abstract: A study has been made of the yielding behaviour of single crystals of brass grown in either argon or nitrogen. Crystals of $\beta $-brass showed sharp yield points when grown in nitrogen, but not otherwise. In all $\beta $-brass crystals, plastic flow occurred jerkily, each jerk being accompanied by the formation of a group of slip bands, and the yield strength showed a maximum at 200 degrees C. Crystals of $\alpha $-brass showed yield points at room temperature when they contained 1% zinc or more, and repeated yielding at 200 degrees C when they contained 10% zinc or more, irrespective of whether they were grown in argon or nitrogen; these yield points are considered to be caused by the segregation of zinc atoms to dislocations and a theoretical estimate is made of the critical amount of zinc needed for the yield phenomenon. Conditions for producing yield points in face-centred cubic metals are briefly discussed.