An exact scaling relation developed for nonlocal operators is presented. The matrix elements of a general nonlocal operator are related to a set of reduced matrix elements through scaling coefficients which do not explicitly depend on the particular nonlocal operator. The theory reduces to a conventional local scaling formulation in that limit. The guide to developing the nonlocal scaling relation comes from consideration of general symmetry properties of the nonlocal operator in a chosen representation. Operators with high degrees of symmetry will generally lead to scaling relations requiring a small number of scaling coefficients for practical applications. The theory is expected to be useful not only for providing exact nonlocal scaling relations but also as a tool for developing approximate scaling relations which still retain the nonlocal nature of the operator to an appropriate extent.

Results of a differential time-of-flight energy-loss study of ${\mathrm{H}}^{\mathrm{-}}$+Na collisions in the 100–500-eV laboratory collision energy range are discussed. It is shown that the relative importance of electron-detachment processes in the total stripping cross section increases with collision energy. The relative magnitude of Na excitation processes is determined. As in negative-ion–molecule collisions, charge exchange to the ${\mathrm{Na}}^{\mathrm{-}}$${(\mathrm{}}^3$P) shape resonance lying at 80 meV may be important.

Experimental evidence of field-correlation effects on weak-field two-photon absorption in atomic sodium is presented. In the case of a nearly Lorentzian laser power spectrum the absorption profile has four times the spectral width of the exciting field, in agreement with predictions by Mollow. The measurement is carried out with counterpropagating laser beams to cancel Doppler broadening. The width of the two-photon absorption spectrum is decreased by partially decorrelating the counterpropagating laser beams. Increasing the correlation time of the frequency fluctuations, resulting in a nearly Gaussian laser power spectrum, has also been observed to decrease the width of the absorption spectrum. An extension of the time-dependent second-order perturbation theory to these additional cases yields good agreement.

We have studied the Stark effect for the bound Rydberg states of xenon ${(}^2$${\mathrm{P}}_{3/2}$ core) in the vicinity of n=17. These are the first detailed measurements of Rydberg-state Stark structure in a heavy rare-gas atom. The experiment was performed by one-photon laser spectroscopy from the metastable 6s’${\left[\mathrm{}\right(1/2\left)\mathrm{}\right]}_0$ state in an atomic beam. Laser polarization can be chosen to excite pure M=0 or ‖M‖=1 levels. Stark maps are presented for electric fields from 0 to 2000 V/cm. The major low-field features are a prominent linear manifold for states with l≥3 and an isolated group of states arising from mixing within the overlapping p and d terms. A simple model of the p,d complex, showing strong mixing at fields as low as 200 V/cm, is used to correlate observed states with the zero-field states. The spectra become exceedingly complex for fields greater than 1000 V/cm because of mixing within the extensive fine structure of xenon.

Stimulated recombination is examined as a stochastic process. The master equation, its stationary solution, and the fluctuation around the stationary solution are obtained. The existence of multiple solutions and its implication are discussed.

We use a combination of R-matrix and hyperspherical coordinate techniques to study the effects of electron correlations in atomic valence shells. The 3snd-3$p^2$ mixing in Mg ${}_{}{}^1{}_{}{}^{}\mathrm{D}_{}^{\mathrm{e}}{}_{}{}^{}$ is analyzed in detail and its relevance to the $s^2$$p^n$d-${\mathrm{sp}}^{n+2\mathrm{}}$ interactions throughout the row is indicated by a study of the 3$s^2$nd-3s3$p^2$ ${}^2$$D^e$ mixing in Al. Multichannel quantum-defect theory parameters for Mg ${}_{}{}^1{}_{}{}^{}\mathrm{D}_{}^{\mathrm{e}}{}_{}{}^{}$ and Al ${}_{}{}^2{}_{}{}^{}\mathrm{D}_{}^{\mathrm{e}}{}_{}{}^{}$ are calculated ab initio. We obtain excellent agreement with experiment for the quantum defects of the 3snd series in Mg and a significant improvement over previous theoretical studies for the quantum defects of the 3$s^2$nd series in Al. Two-electron correlations are illustrated by mapping their joint probability density for alternative eigenmodes.

The absolute differential production efficiencies (photons/eV sr electron) for x rays emitted from each of three transition radiators were measured for incident electron-beam energies of 17.2, 25, and 54 MeV. The radiators were made of stacks of 1.0-μm-thick foils: 18 foils of beryllium, 18 foils of carbon, and 30 foils of aluminum. The radiation spectra were most intense between 0.5 and 2.5 keV, peaking at 0.8, 1.3, and 1.3 keV, respectively. The angular distribution of the transition radiation from the beryllium-foil stack was measured for the three electron-beam energies and found to agree well with theoretical predictions. Owing to K-shell absorption, the photon-energy spectra from the carbon and aluminum stacks are narrowed. Theoretical calculations, which include both the two-surface interference and photon attenuation in the foil material, agree well with these data. A method of enhancing output using a split-foil stack is considered; cursory experiments with a split stack of Mylar foils showed enhanced emission. The use of transition radiation as a source of x rays for lithographic purposes may be practical.

The two-photon resonant, three-photon ionization spectrum of atomic iodine in the ${\mathrm{I}}^{\mathrm{*}}$ ${}_{}{}^2{}_{}{}^{}\mathrm{P}_{1/2}^{}{}_{}{}^{}$ state is presented. The ${\mathrm{I}}^{\mathrm{*}}$ ${}_{}{}^2{}_{}{}^{}\mathrm{P}_{1/2}^{}{}_{}{}^{}$ state is produced by photodissociation of methyl iodide, and its two-photon excitation spectrum exhibits ${(}^3$${\mathrm{P}}_2$)np and ${(}^3$${\mathrm{P}}_2$)nf Rydberg series converging to the ${\mathrm{I}}^{+}$ ${}_{}{}^3{}_{}{}^{}\mathrm{P}_2^{}{}_{}{}^{}$ ground state. With use of this technique, the single-photon forbidden np and nf series are readily observed to n>30.

It is shown that the usual length-form approximation to the momentum matrix element (m/iħ)($E_i$-$E_f$)〈f‖R‖i〉, although correct for the usual single-electron single-photon transitions, is off by about a factor of 2 for two-electron single-photon transitions in the x-ray region. A much more accurate result is obtained if the total atomic-energy difference $E_i$-$E_f$ is replaced by the single-particle-energy difference of the electrons involved in the transition (e.g., $E_{2p}$-$E_{1s}$).

The hydrogenic model of Burch et al. for calculating the energy shifts of the K x ray and Auger electrons for a vacancy in the L shell is extended to incorporate the effect of relativity. Numerical results are presented for 18 values of Z (from 10 to 95 in steps of 5). The computations are based on two different choices of the screening parameters. The effective nuclear charges ($Z_{\mathrm{eff})}$ determined by the method of O’Connell and Carroll yield numbers for the energy shifts close to the predictions of Hartree-Fock (HF) and Hartree-Fock-Slater (HFS) calculations. On the other hand, $Z_{\mathrm{eff}}$ obtained from Slater’s rule gives numbers which are considerably larger than the HF and HFS values. However, the choice of screening is crucial only for low-Z elements and relativity plays an increasingly dominant role as one goes to higher atomic numbers. It is found that the relativistic effect is more pronounced for a vacancy in the $L_2$ subshell.

Lifetimes of 3pπ $D^1$${\Pi }_u^{\mathrm{-}}$ vibrational levels of molecular hydrogen, v’=3–15, excited by vuv synchrotron radiation, were measured by time-resolved spectroscopy. The observed lifetimes range from 2.3 to 3.0 ns and are almost constant (about 2.6 ns) and independent of vibrational quantum number within experimental error. This value agrees with calculated radiative lifetimes. The emission corresponding to the D-E transition is observed in the 6000–7500-Å spectral range, and a cascade in the 3000–6000-Å range is identified to I and G to B transitions. The D-F transition could not be detected.

The time-evolution operator is in general gauge dependent. Its gauge transformation property follows from the gauge transformation of the wave function and ensures gauge-invariant matrix elements. The same transformation property is shown here to follow from the formal solution of the Schrödinger equation for the time-evolution operator, which is a time-ordered exponential of the time integral of the Hamiltonian. The gauge transformation property of the time-evolution operator in the interaction picture is also obtained. The perturbation expansion of the time-evolution operator in one gauge can be transformed to give the perturbation expansion for the time-evolution operator in another gauge. The A⋅p versus E⋅r controversy in the electric dipole approximation is resolved by specifying the correct initial and final states.

Total attenuation cross sections in the elements La, Ce, Pr, Nd, Sm, Gd, Dy, Ho, and Er were derived from the measured total cross sections of their simple oxide compounds, by employing the mixture rule at 145.4-keV photon energy. The compound cross sections have been measured by performing transmission experiments in a good geometry setup. From the derived total cross sections of elements, photoeffect cross sections have been obtained by subtracting the theoretical scattering cross sections. A good agreement is observed between the present data of photoeffect cross sections and Scofield’s theoretical data.

The effects of inhomogeneous broadening, spatial hole burning, and Gaussian beams are calculated for the four coefficients derived in previous papers in this series [Phys. Rev. A 31, 3112 (1985); 31, 3124 (1985)] on the quantum theory of multiwave mixing. These coefficients uniformly describe how one or two strong classical-field modes and one or two weak quantum-field modes interact in a two-level medium. The phenomena for which this theory is applicable includes resonance fluorescence, Rayleigh scattering, laser and/or optical bistability instabilities, and phase conjugation. The averaged formulas are illustrated by resonance-fluorescence spectra. We find the sidepeaks can be ‘‘washed out’’ and that asymmetries can occur.

The concept of Nth-order squeezing and intrinsic Nth-order squeezing is introduced and applied to several physical situations that are known to exhibit second-order squeezing. It is shown that the field produced in degenerate parametric down-conversion is squeezed to all orders, as well as being intrinsically squeezed to orders 2,6,10,. . . . The same is true for the fundamental mode in second-harmonic generation within the short-time approximation, but in this case there is only intrinsic second-order squeezing. In resonance fluorescence from a two-level atom squeezing is again found to be intrinsically a second-order phenomenon, although an intrinsic second-order effect can give rise to a weak form of higher-order squeezing.

The semiclassical equations of motion for a system of coupled lasers are developed and the frequency locking of the lasers comprising the system is analyzed. It is shown that the frequency-coupling range, in terms of the coupled cavities’ mismatch, is proportional to the coupling coefficient. For a system where the cavities are uniformally filled with the active medium, the coupling vanishes regardless of the transmittance of the coupling mirrors. Our theory is valid for all values of coupling and for any number of lasers in the array. It may also be adapted to study different types of coupling arrangements.

We derive the effects of random telegraph noise on the equations of optical resonance. The random-telegraph-signal model allows some results to be found exactly, independent of the strength of the noise or the strength of the field that induces the resonant transitions. Our optical resonance equations depart from the well-known empirical optical Bloch equations in several ways. We obtain excellent agreement at high, low, and intermediate Rabi frequencies with the data of DeVoe and Brewer [Phys. Rev. Lett. 50, 1269 (1983)]. This is the first such agreement that has been reported with a value of the noise coherence time smaller than about 20 μsec.

It is shown in the limit ħ→0 that an approximate solution of the time-dependent Schrödinger equation can be obtained by the use of generalized Fourier integrals. These are constructed by the continuous superposition of eikonals, each generated by solving a problem in classical mechanics. The solution thus generated is valid even in the presence of caustics, provided that they are not too dense. The method applies to localized wave functions and provides a short-time solution. The nature of the solution and its relation to an exact solution is demonstrated in the case of the one-dimensional problem of the surface-state electron. The Wigner distribution f arises in a natural way, leading to the interpretation of the semiclassical limit as classical kinetic theory.

We have studied the dependence of the charge-state distributions of heavy-ion fragments resulting from the foil-induced dissociation of 4.2-MeV ${\mathrm{N}}_2$${\mathrm{}}^{+}$ ions on the thickness of the carbon target foil. The results are compared to those distributions measured for impact of 2.1-MeV ${\mathrm{N}}^{+}$ projectiles. Whereas the mean charge state for atomic-ion impact is already equilibrated in moderately thin targets (2 μg/${\mathrm{cm}}^2$), those measured for molecular-ion impact are strongly dependent on the target thickness, even for the thickest targets (20 μg/${\mathrm{cm}}^2$). The distributions for molecular-ion impact show a marked shift towards lower charge states, evidencing an enhanced electron-capture probability over the case of monatomic-ion impact. A quantitative model is described which explains this phenomenon by reducing the two-center potential of the cluster fragments to a separation-dependent one-center potential.

The eikonal theory of electron capture by relativistic projectiles is developed. It is shown that as long as the electron spin is not measured, a density-matrix formalism can greatly simplify the calculation. The exact eikonal cross section is expressed in terms of two-dimensional integrals. For 1s-1s capture an approximate closed-form expression valid for not too high nuclear charges is also given. This form explicitly displays contributions from magnetic capture and from relativistic modifications of the electron orbits. At each stage of the development the corresponding Oppenheimer-Brinkman-Kramers (OBK) result can be retrieved as a special case. It is concluded that the multiple-scattering contributions described by the eikonal approach are crucial in reducing the calculated cross section significantly below the OBK and second-Born-approximation values. In this way, good agreement with experimental data is obtained.