Showing papers in "Physical Review in 1968"

Transition temperature of strong-coupled superconductors.

Abstract: The superconducting transition temperature is calculated as a function of the electron-phonon and electron-electron coupling constants within the framework of the strong-coupling theory. Using this theoretical result, we find empirical values of the coupling constants and the "band-structure" density of states for a number of metals and alloys. It is noted that the electron-phonon coupling constant depends primarily on the phonon frequencies rather than on the electronic properties of the metal. Finally, using these results, one can predict a maximum superconducting transition temperature.

Global structure of the Kerr family of gravitational fields

Abstract: The Kerr family of solutions of the Einstein and Einstein-Maxwell equations is the most general class of solutions known at present which could represent the field of a rotating neutral or electrically charged body in asymptotically flat space. When the charge and specific angular momentum are small compared with the mass, the part of the manifold which is stationary in the strict sense is incomplete at a Killing horizon. Analytically extended manifolds are constructed in order to remove this incompleteness. Some general methods for the analysis of causal behavior are described and applied. It is shown that in all except the spherically symmetric cases there is nontrivial causality violation, i.e., there are closed timelike lines which are not removable by taking a covering space; moreover, when the charge or angular momentum is so large that there are no Killing horizons, this causality violation is of the most flagrant possible kind in that it is possible to connect any event to any other by a future-directed timelike line. Although the symmetries provide only three constants of the motion, a fourth one turns out to be obtainable from the unexpected separability of the Hamilton-Jacobi equation, with the result that the equations, not only of geodesics but also of charged-particle orbits, can be integrated completely in terms of explicit quadratures. This makes it possible to prove that in the extended manifolds all geodesics which do not reach the central ring singularities are complete, and also that those timelike or null geodesics which do reach the singularities are entirely confined to the equator, with the further restriction, in the charged case, that they be null with a certain uniquely determined direction. The physical significance of these results is briefly discussed.

Behavior of Current Divergences under SU 3 × SU 3

Abstract: We investigate the behavior under SU3×SU3 of the hadron energy density and the closely related question of how the divergences of the axial-vector currents and the strangeness-changing vector currents transform under SU3×SU3. We assume that two terms in the energy density break SU3×SU3 symmetry; under SU3 one transforms as a singlet, the other as the member of an octet. The simplest possible behavior of these terms under chiral transformations is proposed: They are assigned to a single (3,3*)+(3*,3) representation of SU3×SU3 and parity together with the current divergences. The commutators of charges and current divergences are derived in terms of a single constant c that describes the strength of the SU3-breaking term relative to the chiral symmetry-breaking term. The constant c is found not to be small, as suggested earlier, but instead close to the value (-sqrt[2]) corresponding to an SU2×SU2 symmetry, realized mainly by massless pions rather than parity doubling. Some applications of the proposed commutation relations are given, mainly to the pseudoscalar mesons, and other applications are indicated.

Exact Solution for an N-Molecule-Radiation-Field Hamiltonian

Abstract: The exact solution for a problem of $N$ identical two-level molecules interacting through a dipole coupling with a single-mode quantized radiation field at resonance is given. Approximate expressions for the eigenvectors and eigenvalues for the ground and low-lying excited states, as well as the most highly excited states, are developed and compared with the exact results.

Crystalline Order in Two Dimensions

Abstract: If $N$ classical particles in two dimensions interacting through a pair potential $\ensuremath{\Phi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are in equilibrium in a parallelogram box, it is proved that every $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{ e}0$ Fourier component of the density must vanish in the thermodynamic limit, provided that $\ensuremath{\Phi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})\ensuremath{-}\ensuremath{\lambda}{r}^{2}|{\ensuremath{ abla}}^{2}\ensuremath{\Phi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})|$ is integrable at $r=\ensuremath{\infty}$ and positive and nonintegrable at $r=0$, both for $\ensuremath{\lambda}=0$ and for some positive $\ensuremath{\lambda}$. This result excludes conventional crystalline long-range order in two dimensions for power-law potentials of the Lennard-Jones type, but is inconclusive for hard-core potentials. The corresponding analysis for the quantum case is outlined. Similar results hold in one dimension.

Single-Site Approximations in the Electronic Theory of Simple Binary Alloys

Abstract: A single-band model Hamiltonian is used to describe the electronic structure of a three-dimensional disordered binary alloy. Several common theories based on the single-site approximation in a multiple-scattering description are compared with exact results for this Hamiltonian. The coherent-potential theory of Soven and others is shown to be the best of these. Within the appropriate limits, it exhibits dilute-alloy, virtual-crystal, and well separated impurity-band behavior. Hubbard and Onodera's and Toyozawa's simple model density of states is employed in numerical calculations for a wide variety of concentrations and scattering-potential strengths. Explicit results are exhibited for the total density of states, the partial density contributed by each component, and such $k$-dependent properties as the Bloch-wave spectral density and the distribution function. These illustrate the general conclusions as well as the limitations of the quasiparticle description.

Computer "Experiments" on Classical Fluids. II. Equilibrium Correlation Functions

Abstract: : Equilibrium correlation functions for a dense classical fluid are obtained by integrating the equation of motion of a system of 864 particles interacting through a Lennard-Jones potential. The behaviour of the correlation function at large distance, and that of its Fourier transform at large wave number are discussed in detail and shown to be related to the existence of a strong repulsion in the potential. A simple hard sphere model is shown to reproduce very well the Fourier transform of those correlations functions at high density, the only parameter of the model being the diameter a of the hard spheres. (Author)

Coherent Averaging Effects in Magnetic Resonance

Abstract: A theory is developed to describe the slow component of the transient decay of transverse spin magnetization, and the central component of the slow-passage absorption spectrum, of a system of spins which is subjected to a periodic and cyclic perturbation. The theory is used to analyze and compare various schemes for high-resolution NMR of solids, including the spinning of the sample about an axis oriented at the "magic angle," the rotating-frame magic-angle experiment of Lee and Goldburg, pulsed versions of the latter, and a number of new pulsed-NMR experiments recently developed in this laboratory. Attention is focused on the factors, both theoretical and practical, which are important in obtaining optimal suppression of static dipole-dipole interactions and quadrupole splittings, and retention of chemical and Knight shifts and scalar spin-spin interactions. Several new experiments are proposed.

Nonlinear realizations of chiral symmetry.

Abstract: We explore possible realizations of chiral symmetry, based on isotopic multiplets of fields whose transformation rules involve only isotopic-spin matrices and the pion field. The transformation rules are unique, up to possible redefinitions of the pion field. Chiral-invariant Lagrangians can be constructed by forming isotopic-spin-conserving functions of a covariant pion derivative, plus other fields and their covariant derivatives. The resulting models are essentially equivalent to those that have been derived by treating chirality as an ordinary linear symmetry broken by the vacuum, except that we do not have to commit ourselves as to the grouping of hadrons into chiral multiplets; as a result, the unrenormalized value of $\frac{{g}_{A}}{{g}_{V}}$ need not be unity. We classify the possible choices of the chiral-symmetry-breaking term in the Lagrangian according to their chiral transformation properties, and give the values of the pion-pion scattering lengths for each choice. If the symmetry-breaking term has the simplest possible transformation properties, then the scattering lengths are those previously derived from current algebra. An alternative method of constructing chiral-invariant Lagrangians, using $\ensuremath{\rho}$ mesons to form covariant derivatives, is also presented. In this formalism, $\ensuremath{\rho}$ dominance is automatic, and the current-algebra result from the $\ensuremath{\rho}$-meson coupling constant arises from the independent assumption that $\ensuremath{\rho}$ mesons couple universally to pions and other particles. Including $\ensuremath{\rho}$ mesons in the Lagrangian has no effect on the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths, because chiral invariance requires that we also include direct pion self-couplings which cancel the $\ensuremath{\rho}$-exchange diagrams for pion energies near threshold.

Electron Correlations at Metallic Densities

Abstract: The dielectric function of a degenerate electron gas in the random-phase approximation, and the one proposed by Hubbard, which takes exchange effects into account, have been extensively used in the study of metallic properties. However, both dielectric functions lead to an overestimate of the short-range correlations between particles. This is manifest from the fact that the pair-correlation function is negative for small interparticle separations over the whole range of metallic densities, and implies an overestimate of the correlation energy. An improved expression of the dielectric function is given, which includes explicitly, in an approximate way, the short-range correlations arising from both Coulomb and exchange effects by being a functional of the structure factor. The structure factor and the dielectric function can then be determined in a self-consistent manner. The numerical solution of the self-consistent scheme yields a pair-correlation function which is positive for all values of the density up to ${r}_{s}=4$. For ${r}_{s}g4$, it is very slightly negative at small separations, but it is so small that it can be considered to be zero for all practical purposes. New estimates of the correlation energy are given for the entire metallic density range, and are smaller than the earlier estimates. These results are used to recalculate the cohesive energy of the alkali metals. A discussion of the plasmon dispersion relation, the compressibility, and the liquid-solid transition, both for the electron system and for an astrophysically interesting system of protons over a background of electrons, is also given.

Multiphonon orbit-lattice relaxation of excited states of rare-earth ions in crystals.

Abstract: Multiphonon orbit-lattice relaxation of excited states of rare earth ions in crystals, measuring fluorescence lifetimes, quantum efficiencies and transition rates

Klein-Gordon Geon

Abstract: A study of the spherically symmetric eigenstates of the Klein-Gordon Einstein equations (Klein-Gordon geons) reveals that these geons have properties that are uniquely different from other gravitating systems that have been studied. The equilibrium states of these geons seem analogous to other gravitating systems; but when the question of stability is considered from a thermodynamical viewpoint, it is shown that, in contrast with other systems, adiabatic perturbations are forbidden. The reason is that the equations of state for the thermodynamical variables are not algebraic equations, but instead are differential equations. Consequently, the usual concept of an equation of state breaks down when Klein-Gordon geons are considered. When the question of stability is reconsidered in terms of infinitesimal perturbations of the basic fields, it is then found that Klein-Gordon geons will not undergo spherically symmetric gravitational collapse. Thus, Klein-Gordon geons are counterexamples to the conjecture that gravitational collapse is inevitable.

Piezo-Electroreflectance in Ge, GaAs, and Si

Abstract: Piezoelectroreflectance in Ge, Si and GaAs studied for uniaxial stress effects on electronic energy bands

Steady-State, ac-Temperature Calorimetry

Abstract: A steady-state technique for measuring heat capacity using ac heating is described. Heat is applied sinusoidally in time to a sample coupled thermally to a reservoir; the resultant equilibrium temperature of the sample contains a term that is both inversely proportional to the heat capacity and measurable with high precision. The effects of various corrections that must be applied to the data are considered in detail. Measurements of the absolute magnitude of the heat capacity of indium and the field dependence of the heat capacity of beryllium have been made and are used to illustrate the power of the method. The observed quantum oscillations in the heat capacity of beryllium are in agreement with predictions based on other measurements.

Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry

Abstract: The radiation at the boundary of an isotropic or cubic medium by a polarization at $2\ensuremath{\omega}$, the amplitude of which is proportional to the product of the incident laser field at $\ensuremath{\omega}$ and a spatial derivative of this field, is examined theoretically. A complete expression for the intensity and polarization of the reflected harmonic radiation as a function of the angle of incidence and state of polarization of the incident laser beam is derived. The angular dependences are in good agreement with observations on Si, Ge, and Ag. Some additional experimental results, not previously reported, are described. The magnitude of the nonlinearity due to bound electrons in these cubic materials is related to the square of the linear susceptibility, and agrees qualitatively with observations in Si, Ge, and alkali halides. This nonlinearity has the same order of magnitude as that caused by conduction electrons in metals, which has been extensively discussed in the literature. The influence of absorbed surface layers is considered.

New Formulation of the Axially Symmetric Gravitational Field Problem. II

Abstract: The field equations governing the gravitational field of a uniformly rotating axially symmetric source are reformulated in terms of a simple variational principle. The new formalism affords a concise unified derivation of the solutions discovered by Weyl and Papapetrou, and permits a simple derivation of the Kerr metric in terms of prolate spheroidal coordinates. More complex solutions are identified by applying perturbation theory.

Optical Properties of V O 2 between 0.25 and 5 eV

Abstract: The optical constants of V${\mathrm{O}}_{2}$ have been determined between 0.25 and 5 eV both below and above the semiconductor-metal transition temperature ${T}_{t}=340\ifmmode^\circ\else\textdegree\fi{}$K. Reflectivity and transmission spectra have been measured on both single crystals and than films. The reflectivity spectra of the bulk crystals were measured with E \ensuremath{\perp} ($c$ axis) in the tetragonal phase [or \ensuremath{\perp} ($a$ axis) in the monoclinic phase], and with E parallel to these axes. While there are some differences in magnitude between the dielectric constants obtained from thin-film and single-crystal measurements, the structural features are in good agreement. Below ${T}_{t}$ there are four prominent absorption peaks centered near photon energies of 0.85, 1.3, 2.8, and 3.6 eV. Above ${T}_{t}$, metallic free-carrier absorption is observed below 2.0 eV, but the same two absorption peaks near 3 and 4 eV are present. The energy location and polarization dependence of these two higher energy peaks can be related to similar absorption peaks in rutile, and are interpreted using the rutile band structure. The results are consistent with a picture in which filled bands arising primarily from oxygen $2p$ orbitals are separated by approximately 2.5 eV from partially filled bands arising primarily from vanadium $3d$ orbitals. Transitions from the filled $2p$ bands are responsible for the high-energy peaks in the optical absorption in both the high- and low-temperature phases. In the high-temperature metallic phase, there is evidence that there is overlap among the $3d$ bands such that at least two bands are partially occupied by the extra $d$ electron per vanadium ion. In the low-temperature semiconductor phase, a band gap of approximately 0.6 eV opens up within the $3d$ bands, separating two filled bands from higher-lying empty bands. The two absorption peaks at 0.85 and 1.3 eV are due to transitions from these two filled bands.

Quantum electromagnetic zero-point energy of a conducting spherical shell and the casimir model for a charged particle.

Abstract: The quantum electromagnetic zero-point energy of a conducting spherical shell of radius $r$ has been computed to be $\ensuremath{\Delta}E(r)\ensuremath{\cong}\frac{0.09\ensuremath{\hbar}c}{2r}$. The physical reasoning is analogous to that used by Casimir to obtain the force between two uncharged conducting parallel plates, a force confirmed experimentally by Sparnaay and van Silfhout. However, while parallel plates are attracted together because of the zero-point energy, a conducting sphere tends to be expanded. Thus although relevant for the understanding of the quantum-mechanical zero-point energy, the result invalidates Casimir's intriguing model for a charged particle as a charged conducting shell with Poincar\'e stresses provided by the zero-point energy and a unique ratio for $\frac{{e}^{2}}{\ensuremath{\hbar}c}$ independent of the radius.

Radiative and Multiphonon Relaxation of Rare-Earth Ions in Y 2 O 3

Abstract: Spontaneous-emission probabilities for electric- and magnetic-dipole transitions from excited electronic states of ${\mathrm{Pr}}^{3+}$, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Er}}^{3+}$, and ${\mathrm{Tm}}^{3+}$ impurities in ${\mathrm{Y}}_{2}$${\mathrm{O}}_{3}$ are calculated using eigenstates in intermediate coupling and experimental intensity parameters. The predicted radiative lifetimes are in good agreement with experimental lifetimes measured using pulsed-selective-excitation techniques for those levels where unit quantum efficiency is expected. Discrepancies do exist, however, which indicate the importance of $J$-state mixing and possible breakdown of the closure approximation used in Judd and Ofelt's treatment. Nonradiative decay rates were determined by subtracting the radiative contribution from the observed lifetime. The rates of multiphonon emission in ${\mathrm{Y}}_{2}$${\mathrm{O}}_{3}$ exhibit a systematic variation with proximity of the next-lowest level, and contribute significantly to the decay for levels having energy gaps to the next-lowest level of less than 3000 ${\mathrm{cm}}^{\ensuremath{-}1}$. Selection-rule restrictions are evident from the reduced relaxation rates for the $^{5}D_{J}$ levels of ${\mathrm{Eu}}^{3+}$. The concentration dependence of the excitation spectra and excited-state lifetimes of ${\mathrm{Tm}}^{3+}$ were investigated to establish the importance of ion-pair relaxation; however, these processes were generally avoided.

Gravitational Radiation in the Limit of High Frequency. I. The Linear Approximation and Geometrical Optics

Abstract: A formalism is developed for obtaining approximate gravitational wave solutions to the vacuum Einstein equations of general relativity in situations where the gravitational fields of interest are quite strong. To accomplish this we assume the wave to be of high frequency and expand the vacuum field equations in powers of the correspondingly small wavelength, getting an approximation scheme valid for all orders of $\frac{1}{r}$, for arbitrary velocities up to that of light, and for all intensities of the gravitational field. To lowest order in the wavelength, we obtain a gauge-invariant linearized equation for gravitational waves which is just a covariant generalization of that for massless spin-2 fields in a flat background space. This wave equation is solved in the WKB approximation to show that gravitational waves travel on null geodesics of the curved background geometry with their amplitude, frequency, and polarization modified by the curvature of space-time in exact analogy to light waves.

Tunneling Model of the Superconducting Proximity Effect

Abstract: We study a simple theoretical model of the proximity effect between superposed normal and superconducting metal films. We imagine a potential barrier between the films and treat electron transmission through the barrier as a tunneling process. For thin films the model is soluble, and we can treat the BCS potential self-consistently and calculate the excitation spectrum. We compute the transition temperature, the energy gap, and the electronic density of states for comparison with tunneling experiments.

Scattering of Light by One- and Two-Magnon Excitations

Abstract: We present details of the theory of light scattering by one- and two-magnon excitations, and compare predictions of the theory with our experimental results in the tetragonal antiferromagnets Mn${\mathrm{F}}_{2}$ and Fe${\mathrm{F}}_{2}$. Two mechanisms are considered for first-order (one-magnon) light scattering: one involving a direct magnetic-dipole coupling and the other involving an indirect electric-dipole coupling which proceeds through a spin-orbit interaction. Experimental results on the intensity and polarization selection rules of the first-order scattering show that the spin-orbit mechanism is the important one. On the other hand, second-order (two-magnon) scattering is observed to be even stronger than first-order scattering in these antiferromagnets, implying that the process is not due to the spin-orbit mechanism taken to a higher order in perturbation theory. A theory of second-order scattering based on an excited-state exchange interaction between opposite sublattices is given. When coupled with group-theoretical requirements for the ${{D}_{2h}}^{12}$ crystals, the mechanism predicts the intensity, the polarization selection rules, and the magnetic field dependence of the second-order spectrum. Features of the second-order spectra are related quantitatively to magnons at specific points in the Brillouin zone. Analysis of both first- and second-order magnon scattering has thus enabled determination of the complete magnon dispersion relation for Fe${\mathrm{F}}_{2}$.

Atomic Displacement Relationship to Curie Temperature and Spontaneous Polarization in Displacive Ferroelectrics

Abstract: The phase transition from nonpolar to polar in displacive ferroelectric crystals is accompanied by the displacement $\ensuremath{\Delta}z$ of certain atoms from their higher-temperature symmetry positions. A study of all displacive ferroelectrics in which atomic positions have been determined has shown that a fundamental relationship exists between $\ensuremath{\Delta}z$ and the Curie temperature ${T}_{C}$. This relation has been form ${T}_{C}=(\frac{\mathcal{K}}{2k}){(\ensuremath{\Delta}z)}^{2}$, where $\mathcal{K}$ has the dimensions of a force constant, $k$ is Boltzmann's constant, and ${T}_{C}$ is in absolute units. A least-squares fit, based on $\ensuremath{\Delta}z$ and ${T}_{C}$ for ten different ferroelectrics, gives $\frac{\mathcal{K}}{2k}=(2.00\ifmmode\pm\else\textpm\fi{}0.09)\ifmmode\times\else\texttimes\fi{}{10}^{4}\ifmmode^\circ\else\textdegree\fi{}$K ${\mathrm{\AA{}}}^{\ensuremath{-}2}$. In addition, the spontaneous polarization ${P}_{s}$ is found to be related to $\ensuremath{\Delta}z$ by the equation ${P}_{s}=(258\ifmmode\pm\else\textpm\fi{}9)\ensuremath{\Delta}z$ \ensuremath{\mu}C ${\mathrm{cm}}^{\ensuremath{-}2}$. $\mathcal{K}$ is discussed in terms of the interatomic force constant along the polar axis.

Mössbauer Spectra in a Fluctuating Environment

Abstract: The M\"ossbauer line shape in the presence of time-dependent electric field gradients and magnetic fields is considered. Two specific soluble stochastic models are treated: (1) a static electric field gradient with a randomly fluctuating magnetic field which takes on values $+h$ and $\ensuremath{-}h$, each directed along the axis of the field gradient, and (2) as in (1), but with the fluctuating magnetic field perpendicular to the axis of the field gradient. Example (2) is more complex than (1), since the fluctuating field is in this case capable of inducing transitions between the nuclear levels, while in (1) this is not possible. Specific calculations for the two cases illustrate the differences between them.

Molecular Vibration Spectra by Inelastic Electron Tunneling

Abstract: An experimental and theoretical study has been carried out on the process of inelastic tunneling between metal films. The experimental results show that this process gives rise to the observation of vibrational spectra for molecules contained in the junction region. The phenomena have been studied for a number of molecular species and at temperatures ranging from 1 to 300\ifmmode^\circ\else\textdegree\fi{}K. The principal effect of temperature is to broaden the observed spectral lines which can be accounted for quantitatively by considering the thermal smearing of the electron distribution in metals. Two detailed interaction mechanisms are discussed which can give rise to infrared dipole or Raman selection rules. None of the experiments carried out thus far can distinguish between these two processes, because of the low symmetry of the molecules used. In addition to the vibrational spectra, it appears that it should be possible to observe electronic transitions as well.

Phonon Dispersion Curves by Raman Scattering in SiC, Polytypes 3 C , 4 H , 6 H , 1 5 R , and 2 1 R

Abstract: Phonon dispersion curves for SiC have been constructed from first-order Raman scattering data. The method used is a new one that exploits the existence of polytypes. Excitation by an argon ion laser made possible the observation of nine one-phonon lines in $4H \mathrm{SiC}$, 16 lines in $15R$, and 14 lines in $21R$. The symmetry type of each phonon mode was determined by polarization analysis, and the modes were further classified by the use of a standard large zone. The three groups of one-phonon lines, together with 15 lines previously reported for $6H \mathrm{SiC}$, were then all assigned to their positions in a single large-zone plot, to yield a set of SiC phonon dispersion curves comparable with those obtained for other materials by neutron diffraction. The results verify the existence of a common phonon spectrum for all SiC polytypes in the axial direction. Longitudinal and transverse acoustic velocities are obtained from the dispersion curves, and are in good agreement with experimental values. Thus, optic modes in polytypes give information on acoustic properties. All SiC polytypes have in common a set of strong modes in which the Si and C sublattices vibrate against each other. The anisotropy of one of these modes varies with polytype in the same way as the $\frac{c}{a}$ axial ratios. Both the anisotropy and the $\frac{c}{a}$ ratio are related empirically to the percentage of hexagonal planes in the polytype stacking arrangement.

Finite-Energy Sum Rules and Their Application to π N Charge Exchange

Abstract: We derive and discuss the finite-energy sum rules, which form consistency conditions imposed by analyticity on the Regge analysis of a scattering amplitude Their finite form makes them particularly useful in practical applications We discuss the various applications, emphasizing a new kind of bootstrap predicting the Regge parameters from low-energy data alone We apply our methods to $\ensuremath{\pi}N$ charge exchange and are able to derive many interesting features of the high-energy amplitudes at various $t$ In particular, we establish the existence of zeros of the amplitudes and of additional $\ensuremath{\rho}$ poles On the basis of the finiteenergy sum rules and the analysis of the $\ensuremath{\pi}N$ amplitudes, we present theoretical and experimental evidence that double counting is involved in the interference model, which adds direct-channel resonances to the exchanged Regge terms

Gravitational radiation in the limit of high frequency. II - Nonlinear terms and the effective stress tensor.

Abstract: The high-frequency expansion of a vacuum gravitational field in powers of its small wavelength is continued. We go beyond the previously discussed linearization of the field equations to consider the lowest-order nonlinearities. These are shown to provide a natural, gauge-invariant, averaged stress tensor for the effective energy localized in the high-frequency gravitational waves. Under the assumption of the WKB form for the field, this stress tensor is found to have the same algebraic structure as that for an electromagnetic null field. A Poynting vector is used to investigate the flow of energy and momentum by gravitational waves, and it is seen that high-frequency waves propagate along null hypersurfaces and are not backscattered by the lowest-order nonlinearities. Expressions for the total energy and momentum carried by the field to flat null infinity are given in terms of coordinate-independent hypersurface integrals valid within regions of high field strength. The formalism is applied to the case of spherical gravitational waves where a news function is obtained and where the source is found to lose exactly the energy and momentum contained in the radiation field. Second-order terms in the metric are found to be finite and free of divergences of the $\frac{(\mathrm{ln}r)}{r}$ variety.