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Hughes and Mann(1) have shown that, under certain conditions, the distribution curve of the energies of electrons scattered inelastically by an atom or molecule has the same shape as that of the X-ray Compton line. Hughes and his collaborators (1–5) have applied the method to an examination of a number of atoms and molecules in the gaseous state under which condition it can be assumed that each atom or molecule acts as an independent scattering centre. Coulson and the author (6–11) have applied quantum methods to derive a theoretical shape of the Compton line for certain of the molecules for which wave functions are available and have discussed the discrepancy between the experimental and the theoretical results(10, 11).

Theoretical expressions are obtained for the momentum distribution of the electrons in atoms of different atomic number. This enables density curves to be drawn for the momentum p. The mean momentum, which is also calculated, shows a steady increase with atomic number. Finally, the new calculations of momentum distribution enable earlier work on the shape of the Compton profile to be extended from sodium to potassium, in monatomic form.

Calculations have been made of the momentum distribution and the shape of the Compton profile for gaseous methane, using the Dirac transformation theory and three distinct approximations (molecular orbital, electron-pair and self-consistent-field) to the true molecular wave functions. The bearing of the results upon the validity of the original wave functions is discussed; it is concluded that the exponents in molecular wave functions should be approximately 1·1 times greater than in corresponding atomic wave functions. Comparison is made with experimental electron scattering results.Our thanks are due to Prof. Massey for providing us with the numerical tables of the self-consistent-field for methane in advance of publication, and to Mr E. G: Phillips of the Mathematics Department of Univeristy College of North Wales, for the loan of a calculating machine.

The protons emitted from boron, when it is bombarded by α-particles from radium C′ and polonium, have been examined. All have been ascribed to the disintegration of the B10 isotope following the capture of an α-particle. Four energy changes with values 3·1 m.v., 0·4 m.v., − 0·1 m.v., and − 1·0 m.v. are necessary to explain the range distribution observed. Experiments with α-particles from polonium also show that penetration through the top of the potential barrier stops at an energy of 3·6 m.v. and that below this there is a resonance level at about 2·9 m.v.

Theoretical calculations on the mean radial momentum distribution of the electrons are made for the molecules CH4, C2H6, C2H4 and C2H2; from these distributions the profiles of the Compton line are deduced. It is assumed that each electron acts as a single scattering centre so that the mean radial distribution for the molecule is just the sum of the various “partial distributions” for each electron separately. The electrons are supposed to be paired together in the formation of localized bonds, and the contributions from each type of bond have to be separately determined. When superposed, these give the momentum distribution for the whole molecule. Such distributions are similar in shape, but the peak value of the momentum curves moves to higher values of p as the C—C bond becomes more saturated.A comparison of the various partial distributions for C2H4 shows that the C—H bond is an important factor in the momentum distribution of such molecules. In other hydrocarbon molecules, we may presume that the proportion of C—H bonds will be very significant in determining the breadth of the momentum distribution curve, and hence of the width of the Compton line; in fact the half-width of the Compton line decreases both with an increase in the number of the C—H bonds, and with increased saturation of the C—C bonds.In the only case where experimental results are available, CH4, the discrepancy between theory and experiment amounts to about 30%. Reasons for this discrepancy are discussed; in part the discrepancy may be attributed to the approximations used in the wave functions; but reasons are given for supposing that in part also it arises from an incorrect interpretation of the experimental results.

Recent experiments on α-particles and protons have demanded an accurate knowledge of the range-velocity relation of these particles, more particularly in the case of protons, since in most experimental work their energy can only be determined indirectly by measuring their range and by deducing the velocity from a range-velocity curve.

The momentum distribution for the electron in the hydrogen molecular ion has been calculated for various wave functions, including the one used by James with which he obtained such a good value for the binding energy. The method adopted for this particular wave function is outlined and the results show appreciable change with improvement in the wave function. In conclusion there are discussed the implications of the present calculations on similar work on the H2 molecule.

The space wave functions for the π-electrons in a graphite layer, as found in Part I, are transformed into momentum coordinates, and the radial distribution function, including σ-electrons, and π-electrons, is calculated. Using the theory of Jauncey and DuMond the shape of the Compton profile in x-ray scattering is predicted. Both this, and the momentum distribution, are in excellent agreement with experimentally determined values. The introduction of a plausible scale factor in the atomic wave functions would make the agreement almost perfect

As a basis for the discussion of the dimensions of physical quantities, the three usual quantities (length, time and mass) are taken as indefinables and the mechanical quantities are defined in terms of these. The dimensions follow directly from the definition of a quantity and indicate the type of measurements that have to be made to determine the magnitude of those quantities. It is shown that a further indefinable is necessary for defining electrical and magnetic quantities, and reasons are given for choosing electric charge for this purpose. It is pointed out that when the dielectric constant and the permeability are equated to unity for a vacuum it amounts to using these two quantities as indefinables, which, although in some cases convenient from practical considerations, is not necessary. For defining thermodynamical quantities, only the three indefinables, mass, length and time, are used. The introduction of Boltzmann's constant, as a numerical scale factor, is not necessary, but gives a more convenient unit of temperature. The significance of the introduction of such scale factors is also discussed from a more general point of view. Finally, it is shown that three indefinables are not necessary for measuring the magnitude of mechanical, and hence thermodynamical quantities, and that these can be quite adequately defined in terms of length and time only, while electrical and magnetic quantities are measured in terms of length, time and charge.

Closed analytical expressions are obtained for the shapes of the modified Compton lines scattered from the elements H to Ne, in atomic form. The widths of the Compton lines are calculated for these atoms, and conditions of resolution from the unmodified lines are obtained. Certain errors in other work on this subject are noted and corrected. The width of the Compton line, as determined experimentally, for diatomic molecules is about 15 to 25% greater than for isolated monatomic atoms.

Examination of recent papers on dimensions and physical quantities suggests a need for more careful definition of these quantities. This matter was dealt with in an earlier paper, but is now expanded and treated in more detail. It is maintained that if definitions were more precise there would not be the same confusion about the dimensions of certain quantities. There may still be differences of opinion on the most appropriate definition of a particular quantity, but if the definition is adequate then there is no doubt about the dimensions of the quantity as deduced from the definition. In this connection there is discussed the use of quantities defined as ratios, such as specific gravity (or relative density), and their more fundamental equivalents, such as density. Further, a logical set of definitions leads to the conclusion that the introduction of certain universal constants is unnecessary - for example, the Newtonian constant N in the equation for force, F=Nmf. Finally, there is discussed briefly the definition of the intensive quantity, temperature, in terms of the extensive quantity, energy.

The momentum distribution has been calculated for an electron in one of the hybridized orbitals of a carbon atom, and is shown as a function of the degree of hybridization. The mean momentum shows a steady increase from pure s to pure p binding. Similar calculations have been made for the momentum distribution in the C—H bond, and it is shown that this also depends upon the degree of s-p mixing. The mean momentum for the bond is less than for an isolated hybridized carbon orbital, but behaves similarly to such an orbital when the degree of hybridization is varied. The differences are about within the limit of experimental determination.

The method of molecular orbitals has been applied to a discussion of the ground state of Li 2 . Both the simple product-type functions and the determinantal functions have been used. The final energy value has an error of about 1·3 %. The changes in screening constants in passing from atomic to molecular states is discussed, and empirical rules deduced from this and other work. Provided that all the molecular orbitals used are mutually orthogonal, there is not as much electron exchange between inner and valence electrons as has often been supposed. The importance of this orthogonality in discussion of exchange is emphasized.

A study of the protons emitted from certain elements when bombarded by α-particles has yielded valuable information on the structure of the atomic nuclei of light elements. The development of electrical counters for α-particles and protons has provided a method by which the emitted protons can be analysed in a more detailed manner than was possible by the scintillation method. These electrical methods have been applied to the examination of the protons emitted from many elements when bombarded by α-particles. For several elements the absorption curve of these protons reveals the presence of a number of discrete groups of protons each ending at a definite range. It is a found that an α-particle of given energy may give rise to one or more groups of protons, each group corresponding to a transmutation process in which a definite amount of nuclear energy is either absorbed or released. It also appears that α-particles of all energies are not equally effective in producing protons, since the energies of the α-particles which produce protons fall into definite discrete groups. The experimental evidence so far obtained has been co-ordinated by adopting the picture of the nucleus shown in fig. 1.

The integral absorption curve of cosmic rays has been examined at sea level up to a thickness of 365 cm of lead. In the first place, the absorption curve was examined in sections using a control telescope method to reduce the effects of atmospheric variation. It was found, to an accuracy of within about 1%, that there was no deviation from a monotonically decreasing curve. By a second method two regions of the curve, namely 0-30 cm and 71-193 cm, were examined in greater detail. Five Geiger-Muller counter telescopes were employed so that five consecutive points on the absorption curve were determined simultaneously. No irregularities in the absorption curves for these two regions were found, and it was also shown for the 0-30 cm region that this was true whether the absorber was wholly above or wholly within the telescope. The fact that no irregularities were found in the 71-193 cm region (corresponding to a momentum range of about 1-3 kMeV/C for μ-mesons) indicates that the irregularities in the meson momentum spectrum suggested by Blackett, and by Glaser, Hamermesh and Safonov are probably due to poor statistics.

Before trustworthy predictions can be made concerning the reliability of various approximate wave-functions in the case of complicated molecules, it is necessary to study the simplest molecules in as great detail as possible. Such a study enables one to assess the merits and inaccuracies of the different approximations in a way that is impossible with the more complex systems. The simplest of all molecular problems is H + 2 , and this ion has been studied thoroughly by several writers (e.g. Dickinson 1933; Sandemann I935; Steensholt 1936 a, b ). The simplest two-electron problem is that of H 2 , and a very complete knowledge of the wave-functions for this molecule has been obtained (e.g. Weinbaum 1933; Coolidge and James 1933; Coulson 1937 a ). In order of increasing complexity the next molecule is the two-electron ion H 3 + , which has been discussed by Coulson (1935), by Eyring, Rosen and Hirschfelder (1936) and Hirschfelder, Diamond and Eyring (1937). All these molecules are homonuclear, however, so that the binding is predominantly covalent; but the majority of molecules experimentally observed are heteropolar, and then the binding is largely ionic. The present paper, therefore, extends the calculations already made for H + 2 , H 2 and H + 3 , and discusses in detail the two simplest heteronuclear molecules, viz. the ground states of the single-electron ion HeH ++ and of the double-electron ion HeH + . The object of the paper is primarily to compare the different types of wave-functions, and for this purpose as many diverse methods as possible have been employed; it is not important, from this point of view, that the HeH ++ ion is unstable and that the HeH + ion is stable. This work may be regarded as the analogue, in the molecular sphere, of a recent paper by Baber and Hassé (1937) on He, in the atomic sphere. Some of the methods used for the single-electron bond have no immediate counterpart in the discussion of the two-electron bond, and accordingly the two problems are treated independently.

SummaryWave functions have been obtained for the ground states of the atoms Li to Ne. These are of the simple analytical type proposed by Morse, Young, and Haurwitz. Some errors in the latter work are corrected.

THE absorption curve for cosmic radiation in lead for small thicknesses of absorber (0–30 cm.) has been measured by many observers. The general shape of the curve is not in doubt, but several observers have obtained differences in detail. Clay1 and co-workers, using ionization chambers, have obtained three maxima at 1.5, 17 and 25 cm. These maxima disappeared when a Geiger–Muller telescope was employed and the absorber was placed within the telescope. With the absorber above the telescope, two rather extended maxima appeared at about 15 and 30 cm. Clay associates these with maxima observed in the absorption curve for showers. George and Appapillai2 have used a counter telescope with the absorber placed within it, and find a plateau region at about 10–17 cm., but no maxima. More recently, Fenyves and Haiman3 have obtained pronounced maxima in the absorption curve at 17 and 25 cm. They used counter telescopes and determined two consecutive points simultaneously. (The position of the absorber is not mentioned.) George, Janossy and McCaig4 report that the second maximum in the shower curve is instrumental and not real, unless it be very small.

IN an earlier paper1, we published the results of some calculations of analytical atomic wave functions for the ground-states of atoms lithium to neon. Since that tune, we have had several requests for the values of the associated total electronic energies. These energies are therefore presented in the accompanying table, where they may be compared with those deduced from experiment. For the lighter atoms these experimental values have been obtained by summing the first, second, . . . ionization potentials of the atoms. For nitrogen, oxygen and fluorine, however, it has been necessary to estimate the last ionization potential by means of a small extrapolation. But the ‘experimental’ values quoted should all be accurate to about 1 part in 10,000. In no case do the theoretical values differ from the experimental ones by more than 1 per cent. This would still be true if an allowance were made for the motion of the nucleus.