In Ewald's theory of crystal optics, based on classical principles, an optical wave through a crystal lattice of polarizable atoms (idealized as isotropic oscillators) is a self-sustaining system of vibrations so constituted that, on the one hand, the electric moment of each atom is caused to oscillate by the electromagnetic field and, on the other hand, the electromagnetic field is itself the resultant field due to the superposition of the dipole waves produced by the lattice atoms. Born extended the theory to the case of movable lattice ions and showed that in an optical wave the electromagnetic field is so coupled to the lattice vibrations that each lattice ion vibrates in phase with the local field which, as in Ewald's case, is itself produced by the vibrating ions. In Born's original theory, the motion of the lattice particles had to be treated by classical mechanics. It is shown that the results of the quantum-mechanical treatment of the lattice motion agrees exactly with the classical theory. Not only is the induced current at each lattice point in phase with the local field, but the magnitude of the current is also identical with the classical value, completely independent of whichever vibrational state the crystal might be in.

The use of artificially pulsed sources of particles in experiments in nuclear physics has already given rise to powerful methods of investigating short-period phenomena and of studying the effects of ‘slow’ neutrons of different energies. On the other hand, the use of such sources introduces complications in counting experiments which are not present when steady sources are employed and it is possible to assume that the time distribution of ionizing events in any counter is a strictly random one. Again, sources which are effectively steady in respect of one counting arrangement are not necessarily so in respect of all. Ordinary cyclotron sources (i.e. cyclotron sources which are not ‘artificially’ pulsed) belong to this class; they have justifiably been considered as steady in the majority of experiments hitherto performed, but the flux of primary particles reaching the target is definitely periodic in time (with the period of the R.F. Voltage applied to the dees), thus a cyclotron will appear as ‘naturally’ pulsed whenever it is used with a counting system for which the effective resolving time is sufficiently short (say < 10−7 sec.).

The search for a theory of the elementary particles which is founded on the well-established principles of quantum mechanics and conforms at the same time with the requirements of the principle of relativity has, in recent years, taken several divergent directions. On the one hand, the second quantization of wave fields derived from a Lagrangian by a variational procedure(1) has succeeded in accounting for the existence and most of the properties of the electron, the photon, and the meson. On the other hand, many generalizations of the Dirac wave equation of the electron(2) have been attempted, with applications to the meson(3) and the proton(4). Heisenberg(5) has considered the much more difficult problem of the interaction between different particles, and has found that the key to the situation is the so-called ‘scattering matrix’, which is nothing other than a limiting form of the relativistic density matrix, as defined in § 2 of this paper. It seems probable that the relativistic density matrix ρ; or statistical operator, as it may be called without reference to representation, will play an important part in relativistic quantum mechanics in the future. It satisfies the same equation as the wave function, but differs from it in being a real linear operator, or a dynamical variable, in the terminology of Dirac.

This paper deals with some details of the application of the Lorentz-Dirac equations of motion of an electron to two simple cases, (a) with no incident field, and (b) with an incident pulse of radiation. In case (a), the field-energy distribution in the self-accelerating motion of the electron when the electron has built up a velocity close to the velocity of light is considered. Numerical and graphical methods are used to form a picture showing how the field energy tends to be concentrated in a disk for this state of motion. A discontinuity of the advanced field variables of the self-accelerating electron is also examined. In case (b), the general solution of the equations of motion for the velocity of the electron is worked out accurately. In the physical solution, this brings out the effects of radiation damping when the pulse has great intensity. Some aspects of the non-physical solutions are pointed out.

The object of the paper was to find the forces acting in bubble rafts and to use them for a description of the mechanical properties of these rafts. The first section is based entirely on Nicolson's work and gives expressions for the cohesive forces in the lattice. The second and third sections deal with the repulsive forces arising when a bubble is compressed uniaxially, and are really the basis of the paper. The fourth section introduces the idea of ‘interaction’ between compressions in the three close-packed directions—a conception which explains the breakdown of Cauchy's relations. The conditions of equilibrium in the lattice are investigated in § 5, and in § 6 all these forces are combined to give a complete account of the elastic properties of the rafts.

The remaining section gives a brief account of the experimental checks of the theory; it appears to be a reasonable approximation for small bubbles, and, indeed, we could certainly not expect it to apply to large ones, since for them not only are all the calculations of the attractive potentials invalid because of the large local slopes of the surface, but also the ‘flats’ and ‘caps’ become so large as to interfere with one another.

The forces are considered well enough supported to justify their further use in an investigation of the plastic behaviour of the bubble rafts—an investigation which has explained several interesting experimental facts but brought us little nearer an understanding of the low yield strains of metallic single crystals (see (2)).

In this paper, it has been shown that the usual demonstrations of the phenomenon of Bose-Einstein condensation are invalid. Exact formal solutions have been derived for the partition functions and the mean occupation numbers in classical and Bose statistics, and the theory of fluctuations in the case of Bose statistics has been given.

In a second paper, convenient approximations will be derived for the mean occupation numbers in Bose statistics, and it will be shown that for temperatures well above the critical temperature of the supposed Bose-Einstein condensation the well-known result for the mean occupation numbers is valid, whilst for temperatures rather lower than this critical value it will be shown that particles will condense into the lowest available state. The critical temperature T0 is given by where the summation is over all energy levels except the lowest.

A quantitative study of the law of force between bubbles floating on a fluid has been made and the results exhibited in the form of a potential energy-distance curve. The main approximations made are that the inclination of the surface of the fluid is small (so that the equations for the surface become linear), and that the submerged portion of the bubble remains spherical. The latter assumption restricts the validity of the results to small bubbles of about 3·0 mm. diameter and less.

One object of the work was to determine the size of bubble which simulated most nearly a typical atomic interaction curve. This application arises in connexion with Prof. Bragg's ‘bubble’ model of crystal lattices. Exact simulation cannot be expected, for the bubble model is two-dimensional, and also does not exert forces analogous either to Coulomb repulsion or to the metallic bond. It can be shown, however, that the law of force between bubbles of a particular size is a remarkably close fit to the law of force between inert gas atoms, that is, to an interaction consisting of a van der Waals attraction and a short-range or ‘overlap’ repulsion. Fig. 4 illustrates this point. The two full curves are the energy-distance curves for krypton and argon atoms (plotted from Fowler, Statistical Mechanics, 2nd ed., Chapter x, Table 29). The circles mark the corresponding curve for bubbles with α = 1·0 (i.e. of diameter 3 mm. in soap solution). The units of energy and distance are chosen so as to bring the minimum of each curve to the point (1, − 50). It will be seen that the curves are remarkably similar over the range shown.

A paper under the above title (1) by G. H. Livens was published recently in this journal. The main object of this paper appears to be to show that the treatment of the same subject (2) by myself is unsound. I shall not attempt to refute Professor Livens's opinions. His views concerning magnetic energy have already been criticized by others (3) more expert than myself. I merely wish to draw attention to two completely untrue statements of fact in Professor Livens's paper, namely the following:

(a) On p. 538 “Guggenheim writes therefore

It has recently been shown (4, 5) that in order to give a true account of the magnetization processes in a ferromagnetic it is necessary to consider the influence of the demagnetizing field of a finite specimen not merely in a quantitative way, affecting the magnitude of the field acting on the specimen, but also in a qualitative way, affecting the actual arrangement of the domains of the ferromagnetic. It is the purpose of this paper to give an account of the process of magnetization in a specimen which has a tension applied in the same direction as the magnetic field. It will be shown that by considering fully the effects of demagnetizing fields results are obtained which agree well with experiment and remove some discrepancies of the older domain theory.

In applying the general theory of intermolecular forces to the complicated problem of ionic solvation, the minimum requirements are an exact knowledge of (1) the force laws governing the interaction of two isolated solvent molecules as a function of their separation and the various angles of mutual inclination, (2) the force laws governing the interaction of an ion with one solvent molecule, also as a function of their distance apart and of their mutual inclinations, (3) the average disposition of solvent molecules in the pure liquid and (4) the disposition of the solvent molecules in the force field of the ion, particularly in its immediate vicinity. None of these theoretical requirements is available for systems of practical interest. The established laws of intermolecular force are those relating to spherically symmetrical fields, applied to gases (J. E. Lennard-Jones; see (1)), in which the molecular arrangements are random, and to crystals (M. Born (2)), in which the arrangements are orderly. In seeking to extend these laws to liquids and solutions, we are hampered at the start by our ignorance of directional forces in general and of the degree of disorder prevailing in condensed fluid systems. What is attempted here is a simplification of the problem of applying intermolecular force theory to ionic hydration by the introduction of assumptions which can claim some experimental warrant. How far the introduction of these assumptions may swamp other factors must remain an open question. At any rate, the attempt removes many anomalies, and allows of the absolute calculation of ionic dimensions in aqueous solution from intermolecular force constants determined without reference to the condensed states of matter.

Relations are derived between the dose in r./min. delivered by the γ-ray beam from a betasynchrotron and the mean electron current which is incident on the internal target. Comparison with experiment shows that multiple passage of electrons through the target probably plays an important part in production of the γ-ray beam. Some possible effects connected with the multiple passage of electrons are discussed.