Showing papers in "Physical Review in 1949"

On the Origin of the Cosmic Radiation

Abstract: A theory of the origin of cosmic radiation is proposed according to which cosmic rays are originated and accelerated primarily in the interstellar space of the galaxy by collisions against moving magmetic fields. One of the features of the theory is that it yields naturally an inverse power law for the spectral distribution of the cosmic rays. The chief difficulty is that it fails to explain in a straight-forward way the heavy nuclei observed in the primary radiation.

Space - time approach to quantum electrodynamics

Abstract: In this paper two things are done. (1) It is shown that a considerable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons. The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form. The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used. The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix.

The S Matrix in Quantum Electrodynamics

Abstract: The covariant quantum electrodynamics of Tomonaga, Schwinger, and Feynman is used as the basis for a general treatment of scattering problems involving electrons, positrons, and photons. Scattering processes, including the creation and annihilation of particles, are completely described by the $S$ matrix of Heisenberg. It is shown that the elements of this matrix can be calculated, by a consistent use of perturbation theory, to any desired order in the fine-structure constant. Detailed rules are given for carrying out such calculations, and it is shown that divergences arising from higher order radiative corrections can be removed from the $S$ matrix by a consistent use of the ideas of mass and charge renormalization.Not considered in this paper are the problems of extending the treatment to include bound-state phenomena, and of proving the convergence of the theory as the order of perturbation itself tends to infinity.

The Theory of Positrons

Abstract: The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equation. It is possible to write down a complete solution of the problem in terms of boundary conditions on the wave function, and this solution contains automatically all the possibilities of virtual (and real) pair formation and annihilation together with the ordinary scattering processes, including the correct relative signs of the various terms. In this solution, the "negative energy states" appear in a form which may be pictured (as by Stuckelberg) in space-time as waves traveling away from the external potential backwards in time. Experimentally, such a wave corresponds to a positron approaching the potential and annihilating the electron. A particle moving forward in time (electron) in a potential may be scattered forward in time (ordinary scattering) or backward (pair annihilation). When moving backward (positron) it may be scattered backward in time (positron scattering) or forward (pair production). For such a particle the amplitude for transition from an initial to a final state is analyzed to any order in the potential by considering it to undergo a sequence of such scatterings. The amplitude for a process involving many such particles is the product of the transition amplitudes for each particle. The exclusion principle requires that antisymmetric combinations of amplitudes be chosen for those complete processes which differ only by exchange of particles. It seems that a consistent interpretation is only possible if the exclusion principle is adopted. The exclusion principle need not be taken into account in intermediate states. Vacuum problems do not arise for charges which do not interact with one another, but these are analyzed nevertheless in anticipation of application to quantum electrodynamics. The results are also expressed in momentum-energy variables. Equivalence to the second quantization theory of holes is proved in an appendix.

On the Classical Radiation of Accelerated Electrons

Abstract: This paper is concerned with the properties of the radiation from a high energy accelerated electron, as recently observed in the General Electric synchrotron. An elementary derivation of the total rate of radiation is first presented, based on Larmor's formula for a slowly moving electron, and arguments of relativistic invariance. We then construct an expression for the instantaneous power radiated by an electron moving along an arbitrary, prescribed path. By casting this result into various forms, one obtains the angular distribution, the spectral distribution, or the combined angular and spectral distributions of the radiation. The method is based on an examination of the rate at which the electron irreversibly transfers energy to the electromagnetic field, as determined by half the difference of retarded and advanced electric field intensities. Formulas are obtained for an arbitrary charge-current distribution and then specialized to a point charge. The total radiated power and its angular distribution are obtained for an arbitrary trajectory. It is found that the direction of motion is a strongly preferred direction of emission at high energies. The spectral distribution of the radiation depends upon the detailed motion over a time interval large compared to the period of the radiation. However, the narrow cone of radiation generated by an energetic electron indicates that only a small part of the trajectory is effective in producing radiation observed in a given direction, which also implies that very high frequencies are emitted. Accordingly, we evaluate the spectral and angular distributions of the high frequency radiation by an energetic electron, in their dependence upon the parameters characterizing the instantaneous orbit. The average spectral distribution, as observed in the synchrotron measurements, is obtained by averaging the electron energy over an acceleration cycle. The entire spectrum emitted by an electron moving with constant speed in a circular path is also discussed. Finally, it is observed that quantum effects will modify the classical results here obtained only at extraordinarily large energies.

Pressure Broadening in the Microwave and Infra-Red Regions

Abstract: In this paper a generalized theory of collision broadening is developed, adequate for predicting line breadths in the microwave and infra-red regions. This theory differs from previous ones in taking into account transitions among quantum states caused by collisions, although it is limited to a classical picture of the relative motion of the colliding molecules as a whole. Formulas are derived for computing approximate line broadening collision cross-sections from known intermolecular interactions, and the results of the theory are successfully compared with experiments on self-broadening in the ammonia inversion spectrum and the vibrational band spectra of HCl and HCN. Some cases of foreign gas broadening in the microwave region are examined, but it is concluded that in general the Van der Waals interaction, commonly assumed to be the force important in causing foreign gas broadening, is not adequate to cause the observed broadenings. The more complicated types of forces which become important at short range would have to be taken into account to give good agreement with experiment in these cases.

The Radiation Theories of Tomonaga, Schwinger, and Feynman

Abstract: A unified development of the subject of quantum electrodynamics is outlined, embodying the main features both of the Tomonaga-Schwinger and of the Feynman radiation theory. The theory is carried to a point further than that reached by these authors, in the discussion of higher order radiative reactions and vacuum polarization phenomena. However, the theory of these higher order processes is a program rather than a definitive theory, since no general proof of the convergence of these effects is attempted.The chief results obtained are (a) a demonstration of the equivalence of the Feynman and Schwinger theories, and (b) a considerable simplification of the procedure involved in applying the Schwinger theory to particular problems, the simplification being the greater the more complicated the problem.

Electrical Properties of Pure Silicon and Silicon Alloys Containing Boron and Phosphorus

Abstract: Electrical resistivity and Hall measurements have been made over the temperature range from 87\ifmmode^\circ\else\textdegree\fi{} to 900\ifmmode^\circ\else\textdegree\fi{}K on pure silicon and on silicon alloys containing from 0.0005 to 1.0 percent boron ($p$-type impurity) or phosphorus ($n$-type impurity). X-ray measurements indicate that both elements replace silicon in the lattice. It is shown that each added boron atom contributes one acceptor level, and it is likely that each added phosphorous contributes a donor level.The temperature variation of the concentrations of carriers, electrons and holes, and of their mobilities, are determined from the resistivity and Hall data for the different samples. In the intrinsic range, at high temperatures, conductivity results from electrons thermally excited from the filled band to the conduction band. The energy gap is about 1.12 ev. The product of electron and hole concentration at any temperature is ${n}_{e}{n}_{h}=7.8\ifmmode\times\else\texttimes\fi{}{10}^{32}{T}^{3}\mathrm{exp}(\frac{\ensuremath{-}12,900}{T})$In the saturation range, which occurs just below the intrinsic range, the concentrations are independent of temperature. All donors (or acceptors) are ionized and the concentration of carriers is equal to the net concentration of significant impurities ($P$ or $B$).The energy, ${E}_{A}$, required to ionize an acceptor by exciting an electron from the filled band, as determined from the temperature variation of concentration at lower temperatures, decreases with increasing impurity concentration and vanishes for concentrations above 5\ifmmode\times\else\texttimes\fi{}${10}^{18}$/${\mathrm{cm}}^{3}$. The value of ${E}_{A}$ at high dilution, 0.08 ev, is about what is expected for a hole moving in a hydrogen-like orbit about a substitutional ${B}^{\ensuremath{-}}$ ion. The decrease in ${E}_{A}$ with increase in concentration is attributed to a residual potential energy of attraction between the holes and impurity ions. The ionization energy of donors is less than that of acceptors, probably because conduction electrons have a smaller effective mass than holes. In samples with large impurity concentrations the carriers form a degenerate gas at low temperatures, and the resistivity and Hall coefficient become independent of temperature.At high temperatures the mobilities of electrons and holes approach the values ${\ensuremath{\mu}}_{e}=3.0{\ensuremath{\mu}}_{h}=15\ifmmode\times\else\texttimes\fi{}{10}^{5}{T}^{\ensuremath{-}\frac{3}{2}}(\frac{{\mathrm{cm}}^{2}}{\mathrm{volt}sec.}).$These values are determined by lattice scattering and are independent of impurity concentration. At lower temperatures scattering by both ionized and neutral impurity centers contribute, and the mobility is largest for the more pure samples. Impurity scattering increases rapidly with decrease in temperature and the mobility passes through a maximum which depends on impurity concentration. Theories of impurity scattering of Conwell and Weisskopf, of Johnson and Lark-Horovitz, and of Mott give mobilities which agree as to order of magnitude with the observed.

On Closed Shells in Nuclei. II

Abstract: It has been suggested in the past that special numbers of neutrons or protons in the nucleus form a particularly stable configuration.{sup1} The complete evidence for this has never been summarized, nor is it generally recognized how convincing this evidence is. That 20 neutrons or protons (Ca{sup40}) form a closed shell is predicted by the Hartree model. A number of calculations support this fact.{sup2} These considerations will not be repeated here. In this paper, the experimental facts indicating a particular stability of shells of 50 and 82 protons and of 50, 82, and 126 neutrons will be listed.

Equations of State of Elements Based on the Generalized Fermi-Thomas Theory

Abstract: The Fermi-Thomas model has been used to derive the equation of state of matter at high pressures and at various temperatures. Calculations have been carried out both without and with the exchange terms. Discussion of similarity transformations lead to the virial theorem and to correlation of solutions for different Z values.

Interstitial Atomic Diffusion Coefficients

Abstract: An attempt is made to interpret the temperature independent factor ${D}_{0}$ of the previously determined diffusion coefficients of interstitial solute atoms in metals. The primary uncertainty in the value of ${D}_{0}$ given by the standard reaction rate theory resides in an entropy factor $\mathrm{exp}(\frac{\ensuremath{\Delta}S}{R})$. When cognizance is taken of an additional strain in the lattice surrounding a solute atom as it passes over a potential energy divide, and of the increase in entropy associated with an increase in lattice strain energy, one can estimate a "theoretical" range within which these entropy factors should lie. All past observations except for C and N in $\ensuremath{\alpha}\ensuremath{-}\mathrm{Fe}$ are consistent with this theoretical range. The ${D}_{0}'\mathrm{s}$ for these two systems were, therefore, redetermined by more precise measurements, and are found to be an order of magnitude higher than the original values. The associated entropy factors are consistent with the theoretical range.

Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis

Abstract: The partition function for a two-dimensional binary lattice is evaluated in terms of the eigenvalues of the ${2}^{n}$-dimensional matrix V characteristic for the lattice. Use is made of the properties of the ${2}^{n}$-dimensional "spin"-representation of the group of rotations in $2n$-dimensions. In consequence of these properties, it is shown that the eigenvalues of V are known as soon as one knows the angles of the $2n$-dimensional rotation represented by V.Together with the eigenvalues of V, the matrix $\ensuremath{\Psi}$ which diagonalizes V is obtained as a spin-representation of a known rotation. The determination of $\ensuremath{\Psi}$ is needed for the calculation of the degree of order.The approximation, in which all the eigenvalues of V but the largest are neglected, is discussed, and it is shown that the exact partition function does not differ much from the approximate result.

Transient Nutations in Nuclear Magnetic Resonance

Abstract: Transient nutations of the resultant nuclear magnetic moment vector are set up by applying radiofrequency power in the form of pulses in the neighborhood of resonance ($\ensuremath{\omega}=\ensuremath{\gamma}{H}_{0}$). The nutations have an initial amplitude depending on the state of magnetization at the start of a pulse and on the proximity to resonance, and are damped by spin-spin and spin-lattice interaction. The thermal relaxation time can be directly found by observing the dependence of initial amplitude on the time between pulses. The spin-spin time constant ${T}_{2}$ can be found from the rate of decay even in the presence of normally disturbing inhomogeneity in magnetic field. Sensitivity is in many cases comparable to that obtained in the modulation method with narrow band amplifiers. The fast response due to the relatively wide band widths used can be applied to a rapid search for unknown resonances. The effects observed are in qualitative accord with predictions based on the Bloch theory.

Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior

Abstract: A theory of electron oscillations of an unbounded plasma of uniform ion density is developed, taking into account the effects of random thermal motions, but neglecting collisions.The first problem considered is that of finding the frequencies at which a plasma can undergo organized steady-state oscillations of small enough amplitude so that a linear approximation applies. It is found that long wave-length oscillations of plasmas with a Maxwell distribution of electron velocities are characterized by the steady-state dispersion relation ${\ensuremath{\omega}}^{2}=\ensuremath{\omega}_{P}^{}{}_{}{}^{2}+(\frac{3\ensuremath{\kappa}T}{m}){(\frac{2\ensuremath{\pi}}{\ensuremath{\lambda}})}^{2}$. Here ${\ensuremath{\omega}}_{P}$ is the plasma frequency, $T$ the absolute temperature of the electron gas, $\ensuremath{\lambda}$ the wave-length, and $\ensuremath{\omega}$ the angular frequency of oscillation. It is also shown that organized oscillations of wave-lengths smaller than the Debye length for the electron gas are not possible.The theory is then extended to describe the processes by which oscillations are set up. It is found that, for a given wave-length, a plasma can oscillate with arbitrary frequency, but that those frequencies not given by the steady-state dispersion relation describe motions in which, after some time, there is no contribution to macroscopic averages. These additional frequencies lead asymptotically only to microscopic fluctuations of the charge density about the organized oscillation of the plasma. In this way, one can describe the manner in which the system develops organized behavior.The treatment is then applied to large steady-state oscillations for which the equations are non-linear. One obtains solutions in which particles close to the wave velocity are trapped in the trough of the potential, oscillating back and forth about a mean velocity equal to that of the wave. One can also obtain non-linear traveling pulse solutions in which a group of particles, moving as a pulse, creates a reaction on the surrounding charge, which traps the particles and holds them together.

The Electric and Optical Behavior of BaTi O 3 Single-Domain Crystals

Abstract: The dielectric constants, the spontaneous polarization, and the optical properties of BaTi${\mathrm{O}}_{3}$ single-domain crystals have been measured as a function of temperature. It is shown that the deformation of the crystal in the tetragonal state is proportional to the square of the spontaneous polarization, in analogy to the behavior of the other ferroelectrics, and that the temperature dependence of the birefringence is proportional to the square of the spontaneous polarization. The change of the polarization at the two lower transition points is consistent with the assumption that the Ti ions in the oxygen octahedra are displaced from the [001] to the [011] and later to the [111] direction.

Theory of the Effective Range in Nuclear Scattering

Abstract: The scattering of neutrons up to about 10 or 20 Mev by protons can be described by two parameters, the scattering length at zero energy, $a$, and the effective range, ${r}_{0}$. A formula (16), expressing the phase shift in terms of $a$ and ${r}_{0}$ is derived; it is identical with one previously derived by Schwinger but the derivation is very much simpler. Reasons are given why the deviations from the simple formula are very small, as shown by the explicit calculations by Blatt and Jackson.The theory is then applied to proton-proton scattering, with a similarly simple result. Moreover, a method is developed to compare proton-proton and proton-neutron scattering without explicit calculation of a nuclear potential.The most recent experimental results are evaluated on the basis of the theory, and accurate values for the effective ranges are obtained for the triplet scattering of neutrons, and for proton-proton scattering. The nuclear force between two protons is found to differ by a slight amount, but beyond doubt, from that between neutron and proton in the singlet state. All actual results agree with those obtained by Breit and collaborators, and by Blatt and Jackson.

Crystal Statistics. III. Short-Range Order in a Binary Ising Lattice

Abstract: The degree of order in a binary lattice is described in terms of a family of "correlation" functions. The correlation function for two given lattice sites states what is the probability that the spins of the two sites are the same; this probability is, of course, a function of temperature, as well as of the distance and orientation of the atoms in the pair. It is shown that each correlation function is given by the trace of a corresponding ${2}^{n}$-dimensional matrix. To evaluate this trace, we make use of the apparatus of spinor analysis, which was employed in a previous paper to evaluate the partition function for the lattice. The trace is found in terms of certain functions of temperature, ${\ensuremath{\Sigma}}_{a}$, and these are then calculated with the aid of an elliptic substitution.Correlations for the five shortest distances (without restriction as to the orientation of the pair within the plane) are plotted as functions of temperature. In addition, the correlation for sites lying within the same row is given to any distance. For the critical temperature this correlation is plotted as a function of distance. It is shown that the correlation tends to zero as the distance increases, that is to say: there is no long-range order at the critical temperature.

Interpretation of the Thermal Conductivity of Glasses

Abstract: The thermal conductivity of glasses decreases with decreasing temperature, while the conductivity of crystalline substances increases with decreasing temperature. The behavior of glasses is interpreted in terms of an approximately constant free path for the lattice phonons, so that the conductivity decreases roughly with the specific heat. The value of the phonon mean free path at room temperature is of the order of magnitude of the scale of the disorder in the structure of glasses as determined from x-ray evidence---that is, of the order of 7A. This is about the size of the unit cell of the crystalline forms of silica.

Motion in a Constant Magnetic Field

Abstract: The motion of a charged particle in a constant magnetic field is treated in both relativistic and non-relativistic quantum theory. Operators representing the center of the orbit, which obey the commutation law for conjugate variables, are introduced and their connections with energy, angular momentum, and magnetic moment studied. Energy eigenfunctions in an operator form are obtained by factorization. Previously derived eigenfunctions in coordinate space are obtained and are shown to be eigenfunctions for the operators for the center of the orbit as well as for the energy. Corresponding relativistic eigenfunctions are derived by a simple device which enables one to construct solutions of the Dirac equation from solutions of the Schr\"odinger equation.

Quantum Electrodynamics. III. The Electromagnetic Properties of the Electron--Radiative Corrections to Scattering

Abstract: The discussion of vacuum polarization in the previous paper of this series was confined to that produced by the field of a prescribed current distribution. We now consider the induction of current in the vacuum by an electron, which is a dynamical system and an entity indistinguishable from the particles associated with vacuum fluctuations. The additional current thus attributed to an electron implies an alteration in its electromagnetic properties which will be revealed by scattering in a Coulomb field and by energy level displacements. This paper is concerned with the computation of the second-order corrections to the current operator and the application to electron scattering. Radiative corrections to energy levels will be treated in the next paper of the series. Following a canonical transformation which effectively renormalizes the electron mass, the correction to the current operator produced by the coupling with the electromagnetic field is developed in a power series, of which first- and second-order terms are retained. One thus obtains second-order modifications in the current operator which are of the same general nature as the previously treated vacuum polarization current, save for a contribution that has the form of a dipole current. The latter implies a fractional increase of $\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}$ in the spin magnetic moment of the electron. The only flaw in the second-order current correction is a logarithmic divergence attributable to an infra-red catastrophe. It is remarked that, in the presence of an external field, the first-order current correction will introduce a compensating divergence. Thus, the second-order corrections to particle electromagnetic properties cannot be completely stated without regard for the manner of exhibiting them by an external field. Accordingly, we consider in the second section the interaction of three systems, the matter field, the electromagnetic field, and a given current distribution. It is shown that this situation can be described in terms of an external potential coupled to the current operator, as modified by the interaction with the vacuum electromagnetic field. Application is made to the scattering of an electron by an external field, in which the latter is regarded as a small perturbation. It is found convenient to calculate the total rate at which collisions occur and then identify the cross sections for individual events. The correction to the cross section for radiationless scattering is determined by the second-order correction to the current operator, while scattering that is accompanied by single quantum emission is a consequence of the first-order current correction. The final object of calculation is the differential cross section for scattering through a given angle with a prescribed maximum energy loss, which is completely free of divergences. Detailed evaluations are given in two situations, the essentially elastic scattering of an electron, in which only a small fraction of the kinetic energy is radiated, and the scattering of a slowly moving electron with unrestricted energy loss. The Appendix is devoted to an alternative treatment of the polarization of the vacuum by an external field. The conditions imposed on the induced current by the charge conservation and gauge invariance requirements are examined. It is found that the fulfillment of these formal properties requires the vanishing of an integral that is not absolutely convergent, but naturally vanishes for reasons of symmetry. This null integral is then used to simplify the expression for the induced current in such a manner that direct calculation yields a gauge invariant result. The induced current contains a logarithmically divergent multiple of the external current, which implies that a non-vanishing total charge, proportional to the external charge, is induced in the vacuum. The apparent contradiction with charge conservation is resolved by showing that a compensating charge escapes to infinity. Finally, the expression for the electromagnetic mass of the electron is treated with the methods developed in this paper.

Three photon annihilation of an electron - positron pair

Abstract: Annihilation of an electron-positron pair accompanied by the emission of three photons is discussed for the case of small relative velocity of the two particles. The energy spectrum of the photons is derived and the cross section for the process is calculated. The ratio of this cross section to that of ordinary two-photon annihilation is found to be 1:370. When the result is applied to the $^{3}S$ ground state of the positronium atom, for which two-photon annihilation is forbidden, one finds a lifetime 1.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}7}$ sec.

Quantum Electrodynamics. II. Vacuum Polarization and Self-Energy

Abstract: The covariant formulation of quantum electrodynamics, developed in a previous paper, is here applied to two elementary problems---the polarization of the vacuum and the self-energies of the electron and photon. In the first section the vacuum of the non-interacting electromagnetic and matter fields is covariantly defined as that state for which the eigenvalue of an arbitrary time-like component of the energy-momentum four-vector is an absolute minimum. It is remarked that this definition must be compatible with the requirement that the vacuum expectation values of a physical quantity in various coordinate systems should be, not only covariantly related, but identical, since the vacuum has a significance that is independent of the coordinate system. In order to construct a suitable characterization of the vacuum state vector, a covariant decomposition of the field operators into positive and negative frequency components is introduced, and the properties of these associated fields developed. It is shown that the state vector for the electromagnetic vacuum is annihilated by the positive frequency part of the transverse four-vector potential, while that for the matter vacuum is annihilated by the positive frequency part of the Dirac spinor and of its charge conjugate. These defining properties of the vacuum state vector are employed in the calculation of the vacuum expectation values of quadratic field quantities, specifically the energy-momentum tensors of the independent electromagnetic and matter fields, and the current four-vector. It is inferred that the electromagnetic energy-momentum tensor, and the current vector must vanish in the vacuum, while the matter field energy-momentum tensor vanishes in the vacuum only by the addition of a suitable multiple of the unit tensor. The second section treats the induction of a current in the vacuum by an external electromagnetic field. It is supposed that the latter does not produce actual electron-positron pairs; that is, we consider only the phenomenon of virtual pair creation. This restriction is introduced by requiring that the establishment and subsequent removal of the external field produce no net change in state for the matter field. It is demonstrated, in a general manner, that the induced current at a given space-time point involves the external current in the vicinity of that point, and not the electromagnetic potentials. This gauge invariant result shows that a light wave, propagating at remote distances from its source, induces no current in the vacuum and is therefore undisturbed in its passage through space. The absence of a light quantum self-energy effect is thus indicated. The current induced at a point consists, more precisely, of two parts: a logarithmically divergent multiple of the external current at that point, which produces an unobservable renormalization of charge, and a more involved finite contribution, which is the physically significant induced current. The latter agrees with the results of previous investigations. The modification of the matter field properties arising from interaction with the vacuum fluctuations of the electromagnetic field is considered in the third section. The analysis is carried out with two alternative formulations, one employing the complete electromagnetic potential together with a supplementary condition, the other using the transverse potential, with the variables of the supplementary condition eliminated. It is noted that no real processes are produced by the first order coupling between the fields. Accordingly, alternative equations of motion for the state vector are constructed, from which the first order interaction term has been eliminated and replaced by the second order coupling which it generates. The latter includes the self action of individual particles and light quanta, the interaction of different particles, and a coupling between particles and light quanta which produces such effects as Compton scattering and two quantum pair annihilation. It is concluded from a comparison of the alternative procedures that, for the treatment of virtual light quantum processes, the separate consideration of longitudinal and transverse fields is an inadvisable complication. The light quantum self-energy term is shown to vanish, while that for a particle has the anticipated form for a change in proper mass, although the latter is logarithmically divergent, in agreement with previous calculations. To confirm the identification of the self-energy effect with a change in proper mass, it is shown that the result of removing this term from the state vector equation of motion is to alter the matter field equations of motion in the expected manner. It is verified, finally, that the energy and momentum modifications produced by self-interaction effects are entirely accounted for by the addition of the electromagnetic proper mass to the mechanical proper mass---an unobservable mass renormalization. An appendix is devoted to the construction of several invariant functions associated with the electromagnetic and matter fields.

On the interpretation of neutron-proton scattering data by the Schwinger variational method

Abstract: Variational methods developed by Schwinger are applied to neutron-proton scattering at energies below 10 Mev. $S$-wave scattering alone is considered, and the tensor force is not taken into account. An expansion is obtained for the phase shift in powers of the energy. The coefficients can be evaluated explicity from the wave function. The first term of the series is related to Fermi's scattering length, the second term involves an "effective range." The third and higher terms turn out to be negligible.The results are used to define an "intrinsic range" for a potential well of arbitrary shape. Thus a reasonable comparison of potential wells of different shapes is made possible. The relation between intrinsic range and effective range is discussed.The experimental data on coherent and incoherent neutron-proton scattering are discussed in terms of a "shape-independent" approximation. The best value for the effective range in the triplet state is ${r}_{t}=(1.56\ifmmode\pm\else\textpm\fi{}0.13)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm. The effective range in the singlet state is not well determined by the present data.The effect of higher, shape-dependent terms in the expansion of the phase shift is considered. These terms become more important as the well shape becomes more long tailed, but they are found to be negligible within experimental uncertainties for the four well-shapes considered here (square, Gaussian, exponential, and Yukawa).The results for the scattering phase shifts can be extrapolated to negative energy to give an approximate algebraic equation for the energy of the bound state of the deuteron.All the numerical results are shown in graphical form; interpolation formulas are provided where higher accuracy may be needed.

Magnetic Domain Patterns on Single Crystals of Silicon Iron

Abstract: Magnetic powder patterns have been obtained on electrolytically polished surfaces of single crystals of iron containing 3.8 weight percent silicon. Domains are easily visible, outlined by accumulations of colloidal magnetic particles. Several techniques have been developed that enable the direction of magnetization in each domain to be determined. Many types of domain patterns are observed, depending on the orientation of the surface with respect to the crystal axes. The simpler patterns can now be interpreted in some detail, and support the idea that the internal domain structure is relatively simple and is usually composed of a series of plates or slabs magnetized at 45\ifmmode^\circ\else\textdegree\fi{} or 90\ifmmode^\circ\else\textdegree\fi{} to the plate length. In one case it is verified that the plate thickness depends on plate length in approximate accordance with theory; and, for the more complicated "tree" patterns, comparison of theory with experiment shows that good agreement can be obtained using theoretical values of the wall energy. Further verification of the theory of Bloch walls is obtained by determining from experiment the change in spin orientation on traversing the wall.

Non-Linear Field Theories

Abstract: This is the first paper in a program concerned with the quantization of field theories which are covariant with respect to general coordinate transformations, like the general theory of relativity. All these theories share the property that the existence and form of the equations of motion is a direct consequence of the covariant character of the equations. It is hoped that in the quantization of theories of this type some of the divergences which are ordinarily encountered in quantum field theories can be avoided. The present paper lays the classical foundation for this program: It examines the formal properties of covariant field equations, derives the form of the conservation laws, the form of the equations of motion, and the properties of the canonical momentum components which can be introduced.

Domain Structures and Phase Transitions in Barium Titanate

Abstract: The arrangements of domains that arise in single crystals of barium titanate in the ferroelectric tetragonal phase have been studied in detail. The domains are the results of tetragonal (101) twinning, and appear by the formation of wedge-shaped laminar domains between two converging (101) twin planes. The distance of penetration of thin wedge-shaped laminae into the crystal follows changes in an applied electric field reversibly. Often thin laminae extend through the thickness of a crystal plate at an angle of 45\ifmmode^\circ\else\textdegree\fi{} to the surface. These laminae frequently advance in groups in the two perpendicular directions parallel to the edges of the rectangular plate. As a result of the intersections of these groups, the domain pattern becomes an array of laminated and unlaminated pyramids and tetrahedra, the birefringence properties of which give rise to net-like patterns of multicolored squares in polarized light. The evidence indicates that the squarenet pattern is an arrangement of twinning reducing as much as possible the total energy of lattice strains. These strains are probably due to an inhomogeneous distribution of impurities causing a bending effect, as in a bimetallic disk.A preliminary investigation of the phase transitions near 5\ifmmode^\circ\else\textdegree\fi{}C and -70\ifmmode^\circ\else\textdegree\fi{}C has been completed. Using clearly defined optical observations, we conclude that the crystal is orthorhombic $\mathrm{Cmm}$ between 5\ifmmode^\circ\else\textdegree\fi{}C and -70\ifmmode^\circ\else\textdegree\fi{}C, and trigonal $R3m$ below -70\ifmmode^\circ\else\textdegree\fi{}C, with the lattice stretched along the polar axis. The twinning is identified in terms of these lattices. It is maintained that the transitions near 5\ifmmode^\circ\else\textdegree\fi{}C and -70\ifmmode^\circ\else\textdegree\fi{}C can only be of the first order, in contrast with the $\ensuremath{\lambda}$-transition at 120\ifmmode^\circ\else\textdegree\fi{}C. While a statistical mechanical treatment of the properties of barium titanate would be exceedingly difficult, a definite qualitative thermodynamic correlation of its properties has been made. Due to the piezoelectric "inter-action," the free energy can be visualized in the ferroelectric states in terms of a simultaneous polarization and lattice deformation. The overall situation may be regarded as one in which a ferroelectric $\ensuremath{\lambda}$-transition from the (ordered) cubic phase can, in principle, take place along any direction of the highly symmetric cubic lattice, but subject to anisotropy effects which favor the [100] directions down to 5\ifmmode^\circ\else\textdegree\fi{}C, the [110] directions between 5\ifmmode^\circ\else\textdegree\fi{}C and -70\ifmmode^\circ\else\textdegree\fi{}C, and the [111] directions below -70\ifmmode^\circ\else\textdegree\fi{}C.