The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.

After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative of

with respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.

Several methods have been developed in recent years in the Cavendish Laboratory for the absolute measurement of fast neutron flux. All depend on observing the effects of elastic collisions between neutrons and light nuclei. The methods fall into two categories according as individual recoil nuclei are counted (1, 2), or the total ionization current due to the recoils is measured (3). In order completely to interpret the experimental results, it is necessary to know the cross-section for scattering and the angular distribution of the recoil nuclei. The total number of recoil nuclei and their energy distribution are then determined for a known incident neutron spectrum. For precise neutron flux measurements, recoil protons are invariably studied, as the neutron-proton scattering cross-section has been measured over a wide range of energies (4, 5), and the angular distribution of the recoils is isotropic in the centre of gravity space for neutrons of energy less than about 10 MeV. (6, 7). This makes the reduction of the experimental results particularly simple and certain.

A simplified model of the ice crystal, equivalent to Barnes's, was used. By applying the symmetry operations of this model to the dynamical matrix, it was made to depend on six arbitrary constants. By assuming that the tetrahedra of oxygen atoms which form the lattice are regular, and hence applying a further symmetry transformation to a smaller unit of the crystal, the number of arbitrary constants was reduced to two. The elastic constants were then found in terms of these two atomic constants. Two of the experimental measurements of the elastic constants of polycrystalline (quasiamorphous) ice were used to calculate the atomic constants and hence the elastic constants for single crystals.

The methods of a previous paper (9) are improved and extended. It is assumed that the eigenfunctions and eigenvalues of an eigenvalue problem given by an elliptic differential equation are known subject to given boundary conditions on a finite boundary. It is shown how the corresponding quantities can be obtained for a similar problem in which the original differential equation, boundary and boundary conditions are simultaneously perturbed. The introduction of a surface displacement vector allows of a Taylor expansion of all quantities and a subsequent separation of orders. The problem of finding the perturbed eigenfunctions for each order then reduces to the solution of an inhomogeneous differential equation subject to known boundary conditions. These equations are solved by a variational method. An application of Green's theorem at each stage enables us to find the perturbed eigenvalues. The method is applied to a problem of which an exact solution is known and good agreement is obtained.

The author is greatly indebted to Dr H. Fröhlich for many interesting discussions and some valuable suggestions.

The grand partition function of any statistical assembly may be defined by the equation

where E denotes any value of the energy of the assembly, k is Boltzmann's constant, T the thermodynamic absolute temperature and λi a parameter which is later to be connected with the chemical potential, μi, of the ith species in the assembly by the simple formula

1. The half-life of Th C′ was determined by a new method of coincidence technique.

2. Previous investigators have recognized difficulties in regard to this investigation. These were: (a) random delays in the counters; (b) coincidences due to other disintegrations; (c) possibility of difference in delays in recording α- and β-particles. In this investigation the difficulties arising from these effects have been overcome.

3. From this type of experiment it is possible to determine the order in which the particles are emitted.

4. It would appear that the time of breakdown of a counter depends slightly upon the magnitude of the initialionization.

5. The value for the half-life of Th C′ was found to be 3.0 ± 0.15 × 10−7 sec.

In this paper an approximate formula is developed for the phase shifts required in the calculation of the scattering of particles according to wave mechanics. It is shown to be more accurate than the current approximations and it is not difficult to handle. The same method can also be used to compute any quantity which is sensitive to the form of the wave function near the centre of force (e.g. a matrix element).

It is well known (1) that for an unsaturated hydrocarbon of the type CnHn+2 (where n is the number of carbon atoms in the linear chain), the theory of molecular orbitals applied to calculate the binding energy of the mobile π-electrons leads to the secular determinant, of continuant type,

where β is the resonance integral between adjacent carbon centres and x stands for the quantity generally written E0 − E, E being the energy of a mobile electron, which we seek to calculate. The equation for the allowed energies E is, of course, simply Pn = 0.

A method is described in which the rate at which fast neutrons cross unit area is measured by counting protons projected into a small solid angle in the forwards direction from thin and thick layers of polythene. The protons are detected by triple coincidences between three proportional counters mounted coaxially behind the hydrogenous layers. The method is applicable to neutrons of energies from 1 MeV. upwards, and can be used in the presence of intense γ-rays. The flux of these neutrons is calculated in terms of the rate of detection of the protons, the solid angle for proton collection, the mass per unit area of the polythene layer, and the neutron-proton-scattering cross-section. A study of the behaviour of proportional co-axial counters, used in this manner, has been made. A determination of the angular distribution of fast neutrons produced by a deuteron-deuterium source has been made by our coincidence method, and the results compared with those obtained by the ionization chamber method by Bretscher and French. Absolute values agree to within 10%.

A further study has been made of the disintegration of boron by slow neutrons, B10 (n, α) Li, in the cloud chamber. The total ranges of the α-particles and the lithium particles were measured and two groups found. 91·4 % of the disintegrations form the main group, which was previously measured, and 8·6 % a long-range group corresponding to the lithium being left in the ground state. The results of the previous measurements are corrected for the variation of stopping power with velocity and the ranges in standard air of the particles of the two groups are now given as 7·7 mm. α-particle and 4·8 mm. lithium particle, total 12·5 mm. for the main group, and 9·3 mm. α-particle and 5·6 mm. lithium particle, total 14·9 mm. for the long-range group.

These results, taken together with those of other workers, make it certain that in about 92 % of disintegrations the lithium is left in an excited state of about 0·47 MeV. energy and in the remaining 8 % in the ground state. The energies calculated from this scheme by using the known masses do not agree with those deduced from the measured ranges and the range-energy relation for a-particles. It is suggested that the rangeenergy relation is at fault and that these measurements provide two experimental points to which a range-energy relation should be fitted.

1. Shastri (1) has shown that if and then

But (Goldstein (2))

hence substituting from (1·2) in (1·1) and changing the order of integration, which we suppose to be permissible, we find that if

then

assuming of course that the integrals converge.

During the past few years, several methods for the measurement of fast neutron flux have been developed in the Cavendish Laboratory. A critical inter-comparison is given in a forthcoming paper by Allen, Livesey and Wilkinson. Some of the methods depend on the counting of protons which are projected by the neutrons from a film of hydrogen-rich material. Polythene is usually employed because its hydrogen content is high and accurately known. If the thickness of the film is comparable with the range of the fastest proton produced by the incident neutrons, it becomes necessary to know the range-energy relation for protons in polythene. This is particularly so in the ‘thick film chamber’ method to be described by Allen and Wilkinson. Another method, the ‘homogeneous ionization chamber’ method to be described by Bretscher and French makes use of the well-known principle that the average density of ionization obtaining in a chamber whose walls have an atomic composition identical with that of the gas which fills the chamber is the same as that in an infinite extent of that gas. This principle itself depends on the identity of mass stopping power of solid and gas of the same atomic composition. The chamber usually used has polythene walls and is filled with ethylene.

A method is described for measuring thermal conductivities of metals at liquid helium temperatures. It is shown that the method of determining the temperature gradient by directly measuring the temperature at two points along a rod is probably free from systematic error, but that the end of a heated rod in direct contact with a helium bath is not at the temperature of the surface of the bath, even in the He ii region. A description is also given of a technique for using mercury as a low-temperature solder, tight to He ii. The results for copper are in reasonable agreement with Makinson's theory of metallic conduction. Tentative extrapolation to very low temperatures in the presence of fairly large magnetic fields shows that pure copper is a reasonably good material for thermal transfer in demagnetization experiments. The results for german silver seem to indicate that only the lattice conductivity is important since it follows a T3 law, and that scattering takes place with a mean free path of the order of 10−3 cm.

It is shown how the configurational partition functions of certain assemblies can be rigorously constructed. From them it is proved that the terms in the expression for the free energy of the assembly depending on the surface enclosing the assembly are of a smaller, order than N, N being the total number of atoms. This justifies the usual ignoration of surface effects in studying the bulk properties of a large assembly of atoms.

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that

(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we write

where λ > 1, then the equation

defines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such that

The problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

The yields of 139Ba per fission in natural uranium by unslowed neutrons from a mixed radium-beryllium source and by C-group neutrons were compared. In each case the fission rates measured with an ionization chamber were correlated with the β-ray activity of 139Ba, chemically extracted from a mass of uranium oxide exposed to an identical neutron spectrum. The ratio of the two fission yields is found to be 0·84 ± 0·03. The possibility of using 139Ba as a local or integrating indicator of fission in extended uraniferous media, and of distinguishing between the contributions by fast neutrons and by slow neutrons is discussed.