Unified Interactions of Leptons and Hadrons

Neutrino astronomy and lepton charge

It is shown that for aluminum absorbers, a single range-energy equation $R=412$ ${{E}_{0}}^{1.265\ensuremath{-}0.0954\mathrm{ln}{E}_{0}}$ (mg/${\mathrm{cm}}^{2}$) will fit the most reliable published values of practical ranges of monoenergetic electrons and the maximum ranges of nuclear beta-rays in the energy region $0.01l~{E}_{0}l~2.5$ Mev. The average deviation of 59 monoenergetic measurements from this equation is +0.08 percent in energy, and -0.05 percent in energy for 35 beta-ray measurements. The mean deviation is 4.1 percent in each case. There are few ranges for energies above 2.5 Mev. All the higher energy values found in the literature and four new measurements on monoenergetic electrons are presented and are shown to be consistent with the range-energy equation $R=530$ ${E}_{0}\ensuremath{-}106$ (mg/${\mathrm{cm}}^{2}$) for ${E}_{0}g~2.5$ Mev. It is shown that the curve ($\frac{d{E}_{0}}{\mathrm{dR}}$) is nearly parallel to the theoretical curve for the rate of energy loss by ionization in the region between 0.01 and 20 Mev and is about 25 percent larger. The reason for this discrepancy is not known. All the methods commonly used to determine the ranges of beta-rays from absorption curves are discussed and a new method developed by the authors is presented.

Range-Energy Relations for Electrons and the Determination of Beta-Ray End-Point Energies by Absorption

Log-f tables for beta decay

Beta-decay half-lives calculated on the gross theory

On the basis of the hypothesis that the same form of interaction acts among any spin-½ particles, it is interesting to apply the interaction law found for β decay to the muon processes. The application is beset by two types of ambiguity. The first is due to the uncertainty in measured values of coupling constants, and particularly their signs. The second arises from the various ways in which the correspondence between the particles of μ and β decay may be taken. Arguments are presented that the unique correspondence established if two like neutrinos are ejected in μ decay is the correct one.
It is argued that previous interpretations of the Universal Fermi Interaction have been unjustifiably broad. Only processes in which two normal particles (vs antiparticles) are annihilated, and two created, should be expected. The positive muon must be treated as the normal-particle (if the neutron, proton and negatron are) in order to avoid the expectation that muon capture by a proton may yield electrons, contrary to experimental facts. The conclusion that two like neutrinos are ejected in μ decay follows immediately.

The Universal Fermi Interaction

Fermi's theory of the energy distribution of $\ensuremath{\beta}$-particles is extended to first and second forbidden transitions for arbitrarily charged nuclei. The calculations are done not only for Fermi's original "polar vector" form of the theory but also for the scalar, tensor, axial vector and pseudoscalar forms. Selection rules appropriate for these are given in Table I. The final results are given in the form of a "correction factor" $C$, by which the allowed distribution must be multiplied to give a forbidden spectrum. They are listed in \textsection{}4. The energy dependence of the correction factors was found to be completely independent of knowledge concerning the details of the nuclear states only for the scalar and pseudo-scalar interactions (which give identical results) and for certain special selection rules in the other interactions. Comparison with experimental data on ${\mathrm{Na}}^{24}$, ${\mathrm{P}}^{32}$ and RaE seems to eliminate the scalar, pseudo-scalar and axial vector possibilities, all of which yield results independent of detailed knowledge concerning the nuclear states involved in these cases. The polar vector and tensor results depend on the unknown ratio of the magnitudes of certain nuclear matrix elements. Arbitrary adjustment of the unknown ratios allows fairly good fitting of the data. Especially striking is the reproduction of a "K-U type" shape for the RaE spectrum. Although the tensor and polar vector theories are equally favored by the evidence of the energy distributions, the fact that the tensor theory leads to Gamow-Teller selection rules perhaps make it preferable.

On the Fermi Theory of β -Radioactivity. II. The "Forbidden" Spectra

The variations with energy of the newly extended observations on the D+D reaction products are analyzed It is found adequate to assume that all these variations, including details of the angular distributions, are due to differences in centrifugal barriers met by different components of the incident deuteron waves This is found possible only if considerable spin-orbit interaction is allowed for The possibility is investigated that numerical measures of the interactions can be obtained from the experimental data Such conclusions as are permitted by the nature of the data are presented

Theory of the D + D Reactions Part I. Analysis of the Energy Dependence

Some possibilities are explored for the theoretical explanation of the angular distribution of $D+D$ reaction products. The variation with energy is ascribed entirely to differences in the centrifugal barriers encountered by the bombarding particles responsible for the isotropic and non-isotropic components. The detailed data on the proton distributions at low energies appear to be explained in terms of an asymmetric part produced by $P$-waves superposed on isotropic emissions caused by $S$-waves. However, ordinary extrapolation of this to higher energies gives much less symmetric distributions than experimentally found at such energies for the emission of neutrons. If the experiments are correct as interpreted, then they appear to show that spin-orbit coupling plays a large part in the reactions. In that case, an isotropic component produced by incoming $P$-waves grows in importance with energy and accounts for the increasingly parallel growth of the isotropic and asymmetric emissions.

Theoretical Considerations Concerning the D + D Reactions

A series of empirical arguments has developed which leave two alternatives for the law of $\ensuremath{\beta}$-interaction: an $\mathrm{STP}$ (scalar+tensor pseudoscalar) or a $\mathrm{VTP}$ ($V\ensuremath{\equiv}\mathrm{vector}$) combination. We offer a new argument, based on an interpretation of the reported "allowed" shapes of once-forbidden spectra, which favors the $\mathrm{STP}$ over the $\mathrm{VTP}$ form. This new development may be important because it is a possible last step in arriving at a unique law of $\ensuremath{\beta}$-decay (i.e., all three components of the $\mathrm{STP}$ form would become both admissible and necessary).A critical stage in the new argument is the explanation of the "allowed" shapes. The theory may yield such shapes if the Coulomb potential energy at the nucleus is sufficiently larger than the kinetic energy with which electron and neutrino are emitted. That condition happens to be best fulfilled just for the cases in which the "allowed" shapes are the most accurately observed. However, the dominance of the Coulomb effect cannot be a sufficient condition except with the $\mathrm{STP}$ or $\mathrm{VA}$ forms of $\ensuremath{\beta}$-interaction. Other forms make deviations from the allowed shape possible even when the Coulomb energy is supposed indefinitely large.The last type of deviation has the same origin as the well-known Fierz interference in allowed spectra. The absence of the latter effect has been widely quoted but apparently never heretofore examined in detail. It is important in one of the series of arguments mentioned above. Hence, we investigate the empirical limits on Fierz-type deviations not only in once-forbidden but also in allowed spectra.

E. J. Konopinski

Papers

The Universal Fermi Interaction

On the Fermi Theory of β -Radioactivity. II. The "Forbidden" Spectra

Theory of the D + D Reactions Part I. Analysis of the Energy Dependence

Theoretical Considerations Concerning the D + D Reactions

The Evidence of the Once-Forbidden Spectra for the Law of β -Decay