Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1951"

Analysis of the Swimming of Microscopic Organisms

Abstract: Large objects which propel themselves in air or water make use of inertia in the surrounding fluid. The propulsive organ pushes the fluid backwards, while the resistance of the body gives the fluid a forward momentum. The forward and backward momenta exactly balance, but the propulsive organ and the resistance can be thought about as acting separately. This conception cannot be transferred to problems of propulsion in microscopic bodies for which the stresses due to viscosity may be many thousands of times as great as those due to inertia. No case of self-propulsion in a viscous fluid due to purely viscous forces seems to have been discussed. The motion of a fluid near a sheet down which waves of lateral displacement are propagated is described. It is found that the sheet moves forwards at a rate 2π 2 b 2 /λ 2 times the velocity of propagation of the waves. Here b is the amplitude and λ the wave-length. This analysis seems to explain how a propulsive tail can move a body through a viscous fluid without relying on reaction due to inertia. The energy dissipation and stress in the tail are also calculated. The work is extended to explore the reaction between the tails of two neighbouring small organisms with propulsive tails. It is found that if the waves down neighbouring tails are in phase very much less energy is dissipated in the fluid between them than when the waves are in opposite phase. It is also found that when the phase of the wave in one tail lags behind that in the other there is a strong reaction, due to the viscous stress in the fluid between them, which tends to force the two wave trains into phase. It is in fact observed that the tails of spermatozoa wave in unison when they are close to one another and pointing the same way.

Spinning Test-Particles in General Relativity. II

Abstract: A method for the derivation of the equations of motion of test particles in a given gravitational field is developed. The equations of motion of spinning test particles are derived. The transformation properties are discussed and the equations of motion are written in a covariant form.

Crystallite Growth in Graphitizing and Non-Graphitizing Carbons

Abstract: An investigation of the structure of carbons of different origin treated at temperatures between 1000 and 3000°C has shown that the graphitizing and non-graphitizing carbons form two distinct and well-defined classes. The differences in structure are apparent from the earliest stages of carbonization, and may be attributed mainly to the formation at low temperatures, in the non-graphitizing carbons, of a strong system of cross-linking uniting the crystallites. This leads to a random orientation of the crystallites in a rigid, finely porous mass. In the graphitizing carbons the cross-linking is much weaker, the structure is more compact, and neighbouring crystallites have a strong tendency to lie in nearly parallel orientation. It is shown that crystallite growth occurs by the gradual displacement of whole layer-planes or even of groups of layer-planes. The pre-orientation existing in the graphitizing carbons facilitates this process, enabling the rearrangement of the layer-planes to take place by small stages. It is the principal factor favouring crystallite growth in the graphitizing carbons. In the non-graphitizing carbons crystallite growth is impeded both by the strong cross-linking between neighbouring crystallites and by their random orientation.

Theory of the nuclear hyperfine structure of paramagnetic resonance spectra in crystals

Abstract: A theory is given of the nuclear hyperfine structure of the paramagnetic resonance in hydrated crystals. Starting from the states of a free ion, neglecting spin-dependent terms in the energy, successive steps of perturbation calculation introduce the effect of the crystalline electric field, the spin-orbit and spin-spin interactions, the coupling with an external magnetic field, and the coupling between the nucleus of the ion and the electrons. The self-consistent field approximation, in which the state of the free ion is described by a single electronic configuration, is found to be inadequate, and it is found necessary to consider admixtures, through configurational coupling, of configurations in which there are unpaired 8-electrons, as these contribute significantly to the hyperfine splitting. A review of the application of the theory to the ions of the iron transition group is given. The phenomenon of paramagnetic resonance, discovered by Zavoisky (1945, 1946), is by now well known. In crystals it normally takes the form of a number of lines, whose breadth is partly determined by the random distribution of the magnetic fields exerted on one ion by its neighbours, and partly by the rapid interchange of magnetic energy and the thermal energy of the lattice, and there may be considerable overlapping of the lines. In particular, the nuclear hyperfine structure, i.e. the splitting of the lines by the action of the nucleus of the ion, though present when the spin of the nucleus differs from zero, is not normally resolved. The line breadth can be substantially reduced by working at low temperatures using magnetically dilute crystals in which most of the magnetic ions are replaced by chemically similar non-magnetic ions forming an isomorphous crystal; and under these conditions the hyperfine structure can be resolved (Penrose I949). The purpose of this paper is to discuss the theory of the nuclear hyperfine structure of the paramagnetic resonance in salts of the iron transition group. The main reasons for confining the discussion to this group are that most of the experimental work has been done on it; and that although the general theory is common to all paramagnetic substances, the application to particular cases involves approximations which are different for the different groups of paramagnetics. In particular, for the iron group the dominating effect on the energy levels of an ion is the crystalline electric field; next in order of importance comes the spin-orbit coupling; then, under usual experimental conditions, the coupling of the ion with the external magnetic field; and last, the interaction with the nucleus. For the rare earths, on the other hand, the spin-orbit coupling may be the dominant effect, and the nuclear interaction may come second. Since up to the present little experimental work has been done on the hyperfine struc

The molecular orbital theory of chemical valency VIII. A method of calculating ionization potentials

Abstract: An analysis of the ‘linear combination of atomic orbitals’ approximation using the accurate molecular orbital equations shows that it does not lead to equations of the form usually assumed in the semi-empirical molecular orbital method. A new semi-empirical method is proposed, therefore, in terms of equivalent orbitals. The equations obtained, which do have the usual form, are applicable to a large class of molecules and do not involve the approximations that were thought necessary. In this method the ionization potentials are calculated by treating certain integrals as semi-empirical parameters. The value of these parameters is discussed in terms of the localization of equivalent orbitals and some approximate rules are suggested. As an illustration the ionization potentials of the paraffin series are considered and good agreement between the observed and calculated values is found.

On the Interaction of Two Electrons

Abstract: A relativistic wave equation for helium-like systems which gives energy levels correct to within α 2 Ry is derived from quantum electrodynamics, care being taken in the handling of pair-production processes. Calculations made with it agree to this accuracy with Breit’s calculations.

The thermal conductivity of dielectric solids at low temperatures (Theoretical)

Abstract: The statistical equilibrium of phonons in the presence of a temperature gradient is investigated, and a general formula for the thermal conductivity of dielectric solids is obtained. This is applied to quartz glass, giving good agreement with values observed by Berman (1951), and to crystals. Comparison between theory and experiment are made for alkali-halides and quartz, and conclusions are drawn about the defects in neutron-irradiated quartz.

The Theory of Paramagnetic Resonance in Hydrated Cobalt Salts

Abstract: The theory of a previous paper is applied to the paramagnetic resonance of certain cobalt salts. The splitting of the orbital levels by the crystalline field is first investigated, and then the further splitting by the spin-orbit interaction. The g -values so derived are in good agreement with experiment. The magnitude of the hyperfine splitting is also in good agreement.

Molecular Association in Liquids. II. A Theory of the Structure of Water

Abstract: Previous theories of the structure of liquid water have either placed undue emphasis on crystalline structures or have postulated that too many hydrogen bonds connecting neigh­bouring molecules are broken. In this paper a theory is developed in which the majority of hydrogen bonds are regarded as distorted rather than as broken. The structural changes are examined quantitatively with the aid of a simple expression for the energy of a bent hydrogen bond. It is shown that the use of a bending force constant consistent with electrostatic calculations on simple models of water molecules leads to a satisfactory quantitative explana­tion of the observed radial distribution function, the discontinuity in density associated with the melting of ice, and the observed values of the dielectric constant of water.

The occurrence and the crystal structure of α"-iron nitride; a new type of interstitial alloy formed during the tempering of nitrogen-martensite

Abstract: A new iron-nitrogen phase, α99 , occurs as an intermediate in the decomposition nitrogen-martensite ( α9 ) → Fe 4 N( γ9 ). Although it is a transition phase, α99 is relatively stable and has a structural unit which contains eight (2 x 2 x 2) distorted and expanded body-centred tetragonal units of the original martensite structure. The contents of the unit cell, which has dimensions a ═ 5⋅72, c ═ 6⋅29 A, c/a ═ 1⋅10, are ideally Fe 16 N 2 with the atoms occupying special positions of space group I 4/ mmm . The iron atoms are displaced from sites occupied in the ferrite lattice by the presence of nitrogen atoms which occupy one twenty-fourth of the number of octahedral interstices in a completely ordered manner. Up to one-half of the number of nitrogen-atom sites may be vacant. The structure is also simply related to that of Fe 4 N, and the mechanisms of its formation from nitrogen-martensite and of its subsequent transformation to Fe 4 N are discussed. For a case where solute atoms are separated by such relatively large distances, the ordering observed in α99 -iron nitride appears to be unique. The unknown nitride reported by previous workers to be precipitated from nitrogen-ferrite is identified as the α99 -phase. Its relationship with the transition phase of tempered carbon-martensite and the technical significance of its existence are mentioned.

The Iron-Nitrogen System: The Preparation and the Crystal Structures of Nitrogen-Austenite (γ ) and Nitrogen-Martensite (α ')

Abstract: The nitrogen atoms of nitrogen-austenite occupy up to a maximum of about one in ten of the octahedral interstices of the face-centred cubic arrangement of iron atoms in a completely random manner. In nitrogen-martensite the same proportion of the octahedral interstices at ½, ½, 0 and 0, 0, ½ of the body-centred tetragonal iron-atom lattice is randomly occupied. Nitrogen-austenite and nitrogen-martensite are therefore isostructural with the corresponding iron-carbon alloys. The observed unit-cell dimensions vary with interstitial-atom concentration in the same way as for carbon-austenite and carbon-martensite, and for a given concentration are no smaller than in the latter alloys in spite of the smaller size usually assumed for the nitrogen atom. The displacements of iron atoms from the lattice points of martensite, due to the presence of nitrogen atoms are considerably smaller than those for carbon-martensite calculated from X-ray data given by previous workers. The values now found for nitrogen-martensite are consistent with the observed cell dimensions and with a nitrogen-atom radius of about 0⋅7 A. A modified interpretation of the martensite structure accounts for all the X-ray diffraction data.

The Flow of Glaciers and Ice-Sheets as a Problem in Plasticity

Abstract: A calculation is made of the distribution of stress and velocity in an ideal glacier and in an ideal ice-sheet. The ice is assumed to have a constant yield stress and to obey, like other polycrystalline plastic aggregates, the Levy-Mises equations of flow and either the Mises or the Tresca criterion of yielding. The solution obtained for an ideal glacier represents the two-dimensional flow of a long slab of ice down a gently undulating rough slope. The addition of ice to the upper surface by snowfall and the removal of ice by ablation are allowed for, but the frictional resistance of the sides of the glacier valley is neglected. Two states of flow are possible, 'active' and 'passive', corresponding to the active and passive Rankine states in soil mechanics. Which of these states occurs at a given place depends upon the relative magnitudes of the curvature of the bed and the rate of snowfall or ablation; a simple algebraic expression of this dependence is obtained. In both states of flow the velocity is greatest at the surface and decreases with depth according to an elliptical law. It is shown that, in accordance with observation, crevasses of limited depth can open in active flow but not in passive flow. The slip-line field for the problem has a close connexion with the directions and positions of shear faults (although the laminated structure of a glacier is doubtless also an important factor here). In passive flow the faults to be expected are similar to the 'thrust planes' often seen on glaciers. The theory suggests that in active flow a complementary sort of shear fault with the opposite direction of movement may occur-and there is some observational evidence for this. The tendency of glaciers to accentuate hollows in their beds is connected with the suggestion that erosion should proceed faster under passive flow than under active flow. The second solution obtained is formally similar but represents the two-dimensional flow of a large ice-sheet, such as the Greenland ice-cap. If a horizontal bed is assumed the profile is calculated to be formed from parts of two parabolas, the maximum height being given by the yield stress of ice. In the accumulation area flow is active. The maximum velocity is everywhere at the surface while the maximum shear rate is on the bed. The solution thus gives no support to the belief that the weight of ice above squeezes out the underlying ice at a faster rate.

The Concept of Partial Coherence in Optics

Abstract: It is shown that a 'phase-coherence factor' may be defined in a manner which leads, without recourse to explicit statistical analysis, to the theorems established by van Cittert (1934) and Zernike (1938) for analogous factors. An invalid approximation made in their calculations of the phase-coherence factor for a plane illuminated directly by a source is corrected. The new treatment is applied to the theory of Young's experiment, the stellar interferometer, and illumination in the microscope. The phase-coherence factor defined here enables a general theory of the formation of optical images to be formulated. Further, it is shown that the diameter of the area of coherence on a plane illuminated by a source of angular radius $\alpha $ is given by d = $\frac{0\cdot 16\lambda}{N\,\sin \,\alpha}$, where N is the refractive index of the intervening medium.

The Statistical Thermodynamics of Multicomponent Systems

Abstract: This paper describes a new statistical approach to the theory of multicomponent systems. A ‘conformal solution’ is defined as one satisfying the following conditions: (i) The mutual potential energy of a molecule of species L r and one of species L s at a distance ρ is given by the expression u rs (ρ) = f rs u 00 ( g rs ρ ), where u 00 is the mutual potential energy of two molecules of some reference species L 0 at a distance ρ , and f rs and g rs are constants depending only on the chemical nature of L r and L s . (ii) If L 0 is taken to be one of the components of the solution, then f rs and g rs are close to unity for every pair of components. (iii) The constant g rs equals ½( g rr + g ss ). From these assumptions it is possible to calculate rigorously the thermodynamic properties of a conformal solution in terms of those of the components and their interaction constants. The non-ideal free energy of mixing is given by the equation ∆* G = E 0 ƩƩ rs x r x s d rs , where E 0 equals RT minus the latent heat of vaporization of L 0 , x r is the mole fraction of L r and d rs denotes 2 f rs — f rr — f ss . This equation resembles that defining a regular solution, with the important difference that E 0 is a measurable function of T and p , which makes it possible to relate the free energy, entropy, heat and volume of mixing to the thermodynamic properties of the reference species; and the predicted relationships between these quantities agree well with available data on non-polar solutions. The theory makes no appeal to a lattice model or any other model of the liquid state, and can therefore be applied both to liquids and to imperfect gases, and to two-phase two-component systems near the critical point.

On the Fine-Scale Structure of Turbulence

Abstract: Recent experimental measurements of the energy spectrum of turbulence at very large wave-numbers have shown that the motion represented by these wave-number components is determined by the kinematic viscosity and the total turbulent energy dissipation, although the range of wave-numbers responsible for the bulk of the viscous dissipation is far from the condition of absolute equilibrium postulated in the theory of local similarity. To account for this anomaly and for the observed spatial inhomogeneity of the motion, it is suggested that the motion is essentially a random distribution of vortex sheets and lines, in which the vorticity distribution is effectively stationary in time, due to balance between the opposing effects of vorticity diffusion by molecular viscosity, and vorticity production and convection by the turbulent shear. The spectrum functions due to stationary vorticity distributions in two types of shear field, viz. plane shear and axial extension, have been computed and compared with experiment. A better fit is obtained for plane shear, which produces a vortex sheet, than for axial extension, which leads to a vortex line, and there is some experimental evidence that conditions suitable for the production of vortex sheets are more common than for production of vortex lines. Finally, the implications of the validity of this hypothesis are considered in relation to the applicability of the Heisenberg form for the turbulent energy transfer.

On the interaction between the radiation field and ionic crystals

Abstract: A symmetric treatment of the interaction between the radiation field and a diatomic ionic crystal with optical isotropy is presented; the explicit solutions of the problem can be readily obtained, which describe all the possible modes of vibration of the combined system of the radiation field and the ionic lattice. An unexpected conclusion is that the optical waves immediately below the infra-red dispersion frequency are in fact none other than what are normally considered as the transverse lattice vibrations of long wave-length in the theory of ionic crystals. It is shown that such waves consist mainly of mechanical vibrational energy rather than radiative energy.

Secondary Circulation in Fluid Flow

Abstract: Secondary circulation appears after fluid with a non-uniform velocity distribution passes round a bend. It alters the character of the flow and is a source of loss. A general expression is developed for its change along a streamline in a perfect, incompressible fluid. The flow in bent circular pipes is analyzed and the theory is compared with experiments on bent pipes and rectangular ducts. In bends the secondary flow is not spiral but oscillatory, the direction of the circulation changing periodically. The theory shows that secondary circulation remains unchanged if streamlines are geodesics on surfaces of constant total pressure.

Atoms in Molecules and Crystals

Abstract: A new approach to the problems of the electronic structure of molecules is proposed. The basis for this treatment is the observation that the energies of atomization of molecules are small relative to their total energies. Accordingly, a perturbation theory is developed in which the unperturbed states consist of isolated atoms and ions, and the mutual approach of these atomic species is treated as the result of a perturbation. In common with other theories, the total molecular eigenfunction is expressed linearly in terms of more simple functions. It is argued that when these more simple functions are of the orbital type, this expansion converges only slowly. On the other hand, if the orbital approach is abandoned, an expansion in terms of so-called composite functions will converge much more rapidly. This representation has the advantage that, at infinite internuclear separations, the composite functions are eigenfunctions. A new use is found for atomic orbitals. They are used in order to construct approximate atomic functions, for which purpose they were originally intended. These approximate atomic functions are then employed to estimate only the perturbation or interaction terms. The energies of the unperturbed states (isolated atoms and ions) must either be determined by the use of more complex atomic functions, or by an appeal to experimentally determined atomic term values.

On higher Born approximations in potential scattering

Abstract: Many discussions of higher Born approximations have followed the work of Distel (1932) and Sauter (1933b), which contains an error in its development, causing many conclusions presented in the literature to be incorrect. In this paper, the second Born approximation for scattering of a Dirac electron by a Yukawa potential is calculated by a correct method. For the limiting case of the Coulomb potential, the cross-section thus obtained is just the expression obtained by McKinley & Feshbach (1948) by expansion of Mott's complete solution. A corresponding calculation is sketched for the meson, using $\beta $-matrix formalism, and its physical interpretation considered. For the non-relativistic case, the behaviour of the Born series (up to third order) is discussed for the transition from Yukawa to Coulomb potential, disproving the conclusions of Distel (1932), Moller (1930) and Urban (1943) that contributions to the cross-section from higher Born approximations become infinite in this limit.

A new classical theory of electrons

Abstract: In the theory of the electromagnetic field without charges, the potentials are not fixed by the field, but are subject to gauge transformations. The theory thus involves more dynamical variables than are physically needed. It is possible by destroying the gauge transformations to make the superfluous variables acquire a physical significance and describe electric charges. One gets in this way a simplified classical theory of electrons, which appears to be more suitable than the usual one as a basis for a passage to the quantum theory.

Friction of diamond, graphite, and carbon and the influence of surface films

Abstract: This paper describes an experimental study of the frictional behaviour of diamond, graphite and of carbon which have been outgassed in vacuo . The removal of surface films which are normally present causes a large increase in the friction. The admission of a small amount of oxygen, water vapour or other contaminant will reduce the friction. Both physical adsorption and chemical adsorption are important. There is evidence that with clean graphite surfaces there is strong adhesion at the interface, so that when sliding takes place slip and shearing occurs beneath the surface. Carbon and graphite have a negative tem perature coefficient of friction. The low friction normally observed with diamond is due to the presence of adsorbed oxygen and other gases. The friction of clean diamond on diamond is high, and the shear strength at the interface is comparable with the shear strength of diamond. Large-scale seizure does not occur because the deformation of the diamond in the region of contact is elastic and the real area of contact necessarily remains small.

The Paramagnetic Resonance Spectra of Two Salts of Manganese

Abstract: The magnetic resonance spectra of manganese fluosilicate and manganese ammonium sulphate have been analyzed using centimetre wave-lengths. It is shown that the splitting of the $^{6}$S state is due mainly to the non-cubic symmetry of the crystalline field; the effect of the cubic field, hitherto assumed to be dominant, appears only as a small correction. The spectra show that the splitting is consistent with a mechanism suggested by Abragam & Pryce (1951), where it is attributed to magnetic interaction between the five electrons constituting the $^{6}$S state. An extensive hyperfine structure is resolved, and the nuclear spin ($\frac{5}{2}$) of $^{55}$Mn is confirmed. The spectroscopic splitting factor, g, is 2$\cdot $000 $\pm $ 0$\cdot $001, very close to the free-spin value. The energy levels in zero magnetic field are evaluated for manganese ammonium sulphate.

The theory of the nuclear hyperfine structure of paramagnetic resonance spectra in the copper Tutton salts

Abstract: The general theory of a previous paper is applied to the copper Tutton salts. It is found that unless account is taken of the effect of admixed configurations containing unpaired s-electrons, agreement with experiment is impossible. Allowance for unpaired s-electron configurations gives good agreement, both for anisotropy and magnitude of the effect. The effect of the nuclear quadrupole moment, though small, is important, and the paramagnetic resonance method is in principle perhaps capable of giving a more accurate value of this nuclear constant than optical methods.

The Volta effect as a cause of static electrification

Abstract: An experimental investigation of the static electrification of metal/metal surfaces shows that the erratic results normally obtained are owing to there being more than one cause, and that the complete elimination of rubbing leaves an effect that is amenable to quantitative measurement. This ‘separation charging’ is shown to be related to the contact potential, and to have the characteristics to be expected if the Volta-Helmholz hypothesis of its origin is correct. The importance of this explanation of tribo-electrification has long been debated. The supposition that the hypothesis predicts negligible charging of conductor/conductor surfaces is shown to be based on an inadequate understanding of the significance of surface topography at the points of contact. By paying proper attention to this, and revising the hypothesis to allow for the transfer of electrons by tunnel effect, a precise theory of separation charging is derived. When applied to the experiments, agreement within the limits of experimental error is obtained, without the introduction of disposable constants.

The diamagnetism of free electrons

Abstract: A rigorous formulation is given of the quantum theory of the diamagnetism of free electrons. It is shown that Z , the partition function in classical statistics, may be calculated for arbitrary magnetic fields and temperatures without explicit knowledge of the energy levels, and complicated arguments involving boundary electrons are therefore unnecessary. It is further shown that the phenomena which arise in Fermi-Dirac statistics are determined by the singularities of Z regarded as a function of a complex variable, and, in particular, that the poles of Z give rise to the de Haas-van Alphen effect (the periodic field dependence of the susceptibility at low temperatures). The theory confirms the results obtained in earlier treatments.

Relativity theory and the creation of matter

Abstract: It is shown that the continuous creation of matter can be treated by the standard methods of general relativity provided that the possibility of a ‘zero-point’ stress is allowed. The matter ‘created’ is the mass-equivalent of the work done by this stress in the expansion of the universe. According to relativity theory this stress also makes a contribution to the density of gravitational mass. When these two effects are taken into account it is shown that the Newtonian analogues of relativistic models of the universe discovered by Milne and the writer can be extended to cover all relativistic models. This result makes it possible to obtain a unified physical interpretation of Hoyle’s treatment of the cosmological problem. It is suggested that the interpretation in terms of a zero-point stress may lead to a connexion with the quantum theory of fields.

The Interaction of Screw Dislocations and Sound Waves

Abstract: In Part I, Peierls’s model of a dislocation is adapted to dynamical problems by taking account of the inertia of the matter which in a continuum would lie between the two sheets of atoms defining the glide plane. The scattering of long shear waves by a screw dislocation is examined. The scattering cross-section is nearly proportional to the wave-length. The force acting on the dislocation in the direction of propagation of the wave depends on a specific interaction between a moving dislocation and a varying displacement field. This interaction is derived formally from an electromagnetic analogy, and its meaning is discussed. By its use, the momentum transfer cross-section is calculated, and shown to equal the scattering cross-section. In Part II, the results of Part I are applied to analyze Leibfried’s estimate of the resistance to the motion of a dislocation caused by its interaction with lattice vibrations. The estimate depends on a confusion of two effects. It is shown for each effect that the contribution of long waves to the resistance is smaller than that calculated by Leibfried, but no quantitative estimate is given for the contribution of waves near the top of the Debye spectrum. It is concluded that Leibfried’s result is correct in form and order of magnitude, but numerically too high.

The Gravitational Instability of an Infinite Homogeneous Turbulent Medium

Abstract: Jeans's analysis of the problem of the gravitational stability of an infinite homogeneous medium is examined and the need for including turbulence in the discussion is pointed out. The subject is then reconsidered from the point of view of the modern theories of turbulence, and it is shown how eddies in the density fluctuations which have wave numbers k < $\surd $\[4$\pi $G$\overline{\rho}$/(c$^{2}$+$\frac{1}{3}\overline{u^{2}}$)\] (where c denotes the velocity of sound and $\overline{u^{2}}$ the mean square velocity of turbulence) are unstable in the sense that these fluctuations will grow with time.

The Invariant Theory of Isotropic Turbulence in Magneto-Hydrodynamics

Abstract: In this paper the invariant theory of isotropic turbulence in magneto-hydrodynamics is developed on the basis of the equations of motion recently derived by Batchelor to describe the hydrodynamics of an incompressible fluid which is also a good electrical conductor. The theory allows for the interaction between the electromagnetic field and the turbulent motion when there is no externally imposed electric or magnetic field. Various double and triple correlation tensors involving the components of the velocity and the magnetic field intensity are defined, and three equations governing the scalars defining these tensors are derived; these latter equations admit integrals of Loitsiansky's type. The equations governing the dissipation of energy by viscosity and conductivity are also derived; they exhibit the manner in which energy is exchanged between the velocity and the magnetic fields. Finally, the equations appropriate for the case, when an external agency supplies kinetic energy to the system at a constant rate and a stationary condition prevails, are obtained; they suggest that the energy in the magnetic field is contained, principally, in the eddies with large wave numbers.

The Structure of Synthetic Polypeptides. II. Investigation with Polarized Infra-Red Spectroscopy

Abstract: Methods of producing oriented films of synthetic polypeptides when prepared from various solvents are described. The polarized infra-red spectra of a number of these films have been measured. It is shown, in agreement with earlier work, that the polypeptides can exist in a folded ($\alpha $) and an extended ($\beta $) configuration. In the $\alpha $ form the N$\chembond{1,0} $H and C\chembond{2,0} 0 stretching modes of the polypeptide chain show parallel dichroism while the N$\chembond{1,0} $H deformation mode shows perpendicular dichroism. In the $\beta $ form these conditions are reversed. The $\alpha $ form has characteristic frequencies of about 1660 and 1560 cm.$^{-1}$, while the $\beta $ form has characteristic frequencies of about 1640 and 1530 cm.$^{-1}$. When highly dispersed in an inert polymer or liquid solvent it is shown that all the hydrogen bonds of the polypeptide chain can be internally satisfied. It is concluded that in the folded state the chain is maintained in the form of seven-membered rings by intra-molecular hydrogen bonds.