Fluorescence correlation spectroscopy is a promising technique for the study of chemical kinetics and diffusion processes in small, well-defined sample volumes. Laser-induced fluorescence is used as a specific highly sensitive probe of concentration, permitting the analysis of the characteristic time dependence of spontaneous concentration fluctuations. We present here a quantitative analysis of the statistical and systematic errors inherent in such an experiment. Theoretical expressions are derived for the signal-to-noise ratio for various operational definitions of the photocount correlation signal. It is shown that the critical parameter governing the statistical accuracy of the experiment is not the total number of photocounts per correlation time, but (when background fluorescence is negligible) the photocounts per correlation time, per fluorescent molecule, a parameter independent of the number of fluorescent molecules in the sample volume. Comparisons are made with parallel results for laser-light scattering experiments derived by other authors. Finally, we consider the effects of background fluorescence, photolysis, and other deviations from ideality.

The desirability of considering "dressed" rather than "bare" states of atoms in strong electromagnetic fields has been noted by others. It is suggested here that such a treatment is conveniently carried out by performing a unitary transformation of the Hamiltonian so that the "dressed" states in the new basis are precisely the "bare" states of the usual basis. A perturbative method for constructing such a transformation, closely related to a procedure described by Heitler, is proposed. The method is applied through second order to a two-level atom model, and it is verified that the transformation makes the dressed ground atomic state stable, and the excited state unstable with the proper decay probability; furthermore, the energies of the two states are renormalized by the correct amounts. The method is then compared with the "nonperturbative" momentum-translation approximation, and shown to contain already in first order the entire operator content of that approximation, but to yield different transition probabilities. Finally, the transformation method is applied through second order to the standard scalar-field model and shown to produce the usual dressing transformation, mass renormalization, and induced nucleon-nucleon potential of that model, a result which tends to substantiate the hypothesis that the transformation produces the correct dressed states of a physical system.

Energies of the coupled system of a spin-1/2 atom in a static magnetic field and an oscillating rf field are analyzed with the resolvent formalism and numerically calculated by quantizing the rf field, for the general case that the strong rf field is oriented in an arbitrary direction to the static magnetic field. It is shown that, if the component of the static field along the direction of the rf field has particular values, level crossings take place and multiphoton transitions by even or odd numbers of rf photons become forbidden, even when the polarization of the rf field satisfies the parity condition for the multiphoton transitions. Particular attention is given to the direction and amount of the shifts of the level crossings by the rf field, together with the shifts of the anticrossings or of the multiphoton transitions. Theoretical results are verified by the optical-pumping experiments with cesium vapor.

The nuclear hexadecapole shielding factor $R_H$ for the $4f$ electrons of the holmium atom ($Z=67\mathrm{}$) has been obtained. This calculation was carried out in connection with a recent atomic-beam experiment of Dankwort, Ferch, and Penselin, which indicates the presence of a nuclear hexadecapole interaction with the $4f$ electron, leading to a value of the nuclear hexadecapole moment $H\left({\mathrm{Ho}}^{165}\right)=0.8\mathrm{}\times {10}^{-48\mathrm{}}$ ${\mathrm{cm}}^4$. This result has to be corrected for shielding or antishielding effects of the holmium atomic core electrons on the $4f$ electron, which experiences the interaction with the nucleus. The present paper gives the procedure and the results of the calculation of $R_H$, which was found to be + 0.24 (shielding). Thus the value of the moment $H\left({\mathrm{Ho}}^{165}\right)$ obtained by Dankwort et al. should be multiplied by the factor $\frac{1}{(1-R_H)}=\frac{1}{0.76}$, giving a corrected value $H{\left({\mathrm{Ho}}^{165}\right)}_{\mathrm{corr}}=1.06\mathrm{}\times {10}^{-48\mathrm{}}$ ${\mathrm{cm}}^4$.

Using Sinanoğlu's method of successive partial orthogonalizations, pair correlation functions have been extracted from the energetically accurate configuration-interaction wave functions of Weiss for a series of two-, three-, and four-electron ions. For selected systems, the pair correlation functions are examined by constructing contour diagrams which explicitly reveal the relative corrections to the Hartree-Fock wave functions brought about by inclusion of pairwise correlation interactions. For each pair function, an associated pair correlation energy is evaluated. The characteristics of both the pair functions and the pair energies are examined together with (i) their $Z$ dependence for a constant number of electrons and (ii) their dependence upon the number of electrons for constant $Z$. The small differences which occur between the $K$ shells for ions of a constant $Z$ have been rationalized in terms of an orthogonality requirement, while the slight but smooth variation of the $K$-shell pair energies with $Z$ has enabled us to comment on the stability of the ion ${\mathrm{He}}^{-}$. Our pictorial representation has highlighted differences between the $L$- and $K$-shell correlation effects within the Be-like ions. In particular, the large degree of angular correlation in the $L$ shell, brought about by the near degeneracy of the $2s$ and $2p$ orbitals, is clearly apparent from the contour diagrams. Also revealed is a nearly linear variation of the $L$-shell pair energies with $Z$. Comparison of the pair energies for the four-electron ions with the results of other workers shows that the wave functions analyzed here provide a good description of the correlation effect within the $K$- and $L$-shells, the description of the latter being superior to that of the former. The intershell correlation behavior, however, is accounted for only very approximately. For the Be-like series as a whole, the $L$-shell pair energies are expected to be highly reliable.

A detailed theoretical investigation of the helium atomic Rydberg states is carried out. Results for level transition energies and doublet level splittings are found to agree extremely well with the recent high-precision microwave-optical experiments. The physical effects contributing to the electrostatic fine structures are systematically studied.

According to early experiments and calculations, doubly excited states of helium are grouped into channels (Rydberg series and continua) with very different excitation and decay probability. Macek has identified channels by using hyperspherical coordinates [$R={(r_1^2+r_2^2)}^{\frac{1}{2}},tan\alpha =\frac{r_2}{r_1},\Omega \equiv \{\alpha ,{\hat{r}}_1,{\hat{r}}_2\}$] and solving the Schrödinger equation initially at fixed $R$; each eigenvalue $U_{\mu }\mathrm{}\left(R\right)\mathrm{}$ and eigen-function ${\Phi }_{\mu }(R;\Omega )$ characterizes one channel and the approximate solution of the whole equation has the form $F_{\mu }\mathrm{}\left(R\right)\mathrm{}{\Phi }_{\mu }(R;\Omega )$. We study the electron correlations in each channel by obtaining explicit expressions for the ${\Phi }_{\mu }(R;\Omega )$, and extend the range of Macek's investigation to higher channels. Pairs of orbital quantum numbers ($l_1$, $l_2$) are found to be quasiconstants for each channel; channels with the same ($l_1$, $l_2$) and ($L$, $S$, $\pi$) differ mainly in the degree of excitation of the radial correlation. Channel functions ${\Phi }_{\mu }(R;\Omega )$ of ${}_{}{}^1{}_{}{}^{}S$, ${}_{}{}^1{}_{}{}^{}P+$, and ${}_{}{}^3{}_{}{}^{}P+$ channels have an antinode at or near $\alpha ={45}^{\circ }$, while those of ${}_{}{}^3{}_{}{}^{}S$, ${}_{}{}^1{}_{}{}^{}P-$, and ${}_{}{}^3{}_{}{}^{}P-$ channels have a node at or near $\alpha ={45}^{\circ }$. An antinode or a node appears to occur generally at or near $\alpha ={45}^{\circ }$. This location of nodes in the minus and ${}_{}{}^3{}_{}{}^{}S$ channels is shown to be related to the weakness of coupling both among these channels and to plus channels with the same ($L$, $S$, $\pi$) symmetry. Examples for ${\mathrm{H}}^{-}$ are discussed. The interactions between channels with different ($l_1$, $l_2$) are important where their potential curves are nearly degenerate. Examples of the effect of this degeneracy in the spectra of alkaline-earth atoms are indicated.

We denote as a "primary minimum principle" one in which a quantity $B$ of physical interest is represented as the minimum value with respect to variations in a trial function ${\psi }_t$ of a functional $F\left({\psi }_t\right)$; $F$ then provides a variational upper bound on $B$. (The Rayleigh-Ritz principle for the ground-state energy of a system is a familiar example.) If $F$ is quadratic in ${\psi }_t$, the variational property of $F$ enables one to determine the linear parameters relatively easily, but the minimum property is required if the nonlinear parameters are to be determined in a way which allows for systematic improvement of ${\psi }_t$. We show here that for a wide class of problems for which primary minimum principles do not exist, useful and rigorous secondary or "subsidiary minimum principles" are available. That is, we construct a functional $F^{\prime }\left({\psi }_t\right)$ whose minimum value is reached for ${\psi }_t$ equal to some function $\psi$ of dynamical interest. (The Rayleigh-Ritz method provides a subsidiary minimum principle for the approximate determination of the ground-state wave function of a system.) If $B=B\left(\psi \right)$, then a study of $F^{\prime }\left({\psi }_t\right)$ provides a powerful tool for the estimation of $\psi$ and therefore $B$, though $B\left({\psi }_t\right)$ is not normally a variational bound on $B\left(\psi \right)$. Subsidiary minimum principles have recently been obtained for the approximation of the auxiliary functions that appear in the variational principle for the matrix element (${\chi }_n$, $W{\chi }_m$), where ${\chi }_n$ and ${\chi }_m$ are bound-state wave functions and $W$ is an arbitrary operator. Here we extend the method to the estimation of matrix elements of the Green's function $g\left(\epsilon \right)$ of a bound system with $\epsilon$ below the continuum threshold energy. The response of the system to an external perturbation can be represented by matrix elements of this type. While no new results on the bound-state problem are obtained, our formulation is a convenient starting point for the further extension of the method to continuum problems. The new result obtained here is the derivation of a subsidiary minimum principle for the problem of scattering of a projectile by a target whose bound-state wave function is only imprecisely known. The subsidiary minimum principle allows for systematic improvement of the closed-channel component of the trial scattering wave function that appears in a Kohn-type variational calculation of the scattering amplitude.

We suggest a general many-body perturbation theory for the ground state as well as for low-lying excited and charge-transfer intermolecular interactions. In this theory the interactions energies are expressed by the Rayleigh-Schrödinger perturbation formulas, where each individual contribution can be interpreted by a slightly modified Hugenholtz diagrammatic technique.

The $K {\alpha }_1$ and $K {\alpha }_2$ x-ray lines of the transition elements $23\le Z\le 32$ were measured using an improved version of a single-crystal high-angle goniometer in conjunction with a NaI scintillation detector. The x-ray spectrum of these elements was produced by bombarding them with an electron beam of constant energy and flux. A computer program was devised and used to unfold the physical widths and the asymmetries of these lines. Previous experimental results on both linewidths and asymmetries are included for comparison. The interpretation of the asymmetries as the result of a simple exchange interaction is only partially satisfactory. Values of $L_2$ and $L_3$ fluorescence yields were extracted from the measured linewidths and compared with theory.

An improved version of a single-crystal high-angle goniometer and a NaI scintillation detector were used to measure the widths of the $L {\alpha }_{1.2}$, $L {\beta }_{1-4}$, and $L {\gamma }_1$ x-ray lines of the elements $58\le Z\le 74$ and the widths of the less prominent lines $L {\beta }_{15}$, $\mathrm{Ll}$, $L\eta$, and $L{\gamma }_1^{\prime }$ of several elements in the same group. The x-ray spectrum of these elements was produced by bombarding them with an electron beam of constant energy and flux, and a computer program was used to unfold the physical widths of their emission lines. These results, as well as the previously reported values of linewidths of the higher-$Z$ elements, are compared with the most recent theoretical predictions.

In a previous paper, we demonstrated that a set of new resonances can accompany the conventional infrared Lamb-dip or double-resonance transitions in C${\mathrm{H}}_3$F gas. These satellite lines have a different origin from the primary resonances even though they are similar in intensity and linewidth. The traditional double resonance, for example, requires that a molecule interact simultaneously with two radiation fields, causing a transition from an initial to a final state through an intermediate level. Here, the double-resonance concept is extended to the situation of two coherently driven optical transitions that do not share a common level but are coupled by molecular collisions that tip the angular momentum vector while preserving the molecular velocity and rotational energy. Thus, velocity-selective population changes are communicated from one transition to another through collisions. Collision-induced double resonance is observed with Stark tuning as a series of sharp lines, free of Doppler broadening and can be explained in the same order of perturbation theory as the ordinary double-resonance experiment. Each satellite corresponds to a specific level structure involving one or more collision-induced transitions among the space-quantized $M$ states. Virtually all characteristics of these satellite resonances, either in Lamb-dip or double-resonance experiments, are in agreement with the theory presented. While purely optical coupling mechanisms can yield double-resonance behavior also, such effects are estimated to be too weak. For the symmetric top molecule C${\mathrm{H}}_3$F, the dipole-dipole interaction of collision pairs dominates and induces the observed reorienting transitions. The corresponding cross section for low-angular-momentum states ($J,K=4,3\mathrm{} or 5,3$) is found to be ∼ 100 ${\mathrm{A}}^2$ whereas for high-angular-momentum states ($J,K=12,2\mathrm{}$), the cross section is calculated to be about 100 times smaller, and satellite resonances are not even detected. A treatment is presented also for the case where the collisionally coupled $M$ states require a sequential transfer of population over intermediate levels. The problem of velocity smearing is discussed in this context where the energy exchange in a collision is small and also for more energetic collisions that may involve rotational quantum jumps.

A relationship between semiclassical radiation theory and the inverse method of solution for nonlinear dispersive waves is developed through two physical examples. The Josephson transmission line is modeled by Maxwell's equations coupled to a phenomenological quantum mechanics. It is shown that this quantum mechanics contains the same linear problem used in the inverse method to solve the sine-Gordon equation, the equation which governs the evolution of the electromagnetic wave. This (nonlinear) wave equation and the linear quantum equations are of equal importance in the physical description of this system. This same relationship exists among the self-induced transparency (SIT) equations of nonlinear optics. This second example, due to Lamb, is discussed in a manner which again displays the precise relationship of the linear problem of the inverse method to the quantum physics. In addition, analogies between SIT and the Josephson transmission line are discussed.

Impact of 20-30-keV ${\mathrm{He}}^{+}$ ions on polycrystalline niobium and copper surfaces causes some projectiles to be backscattered in a neutral excited state. These projectiles subsequently decay, radiatively emitting Doppler-broadened spectral lines; the broadening is characteristic of the distribution in speeds and direction of the scattered excited projectiles. Analysis of the line shape shows that slow reflected particles have a high probability of losing the excited electron by a radiationless transition while they are close to the surface. The spectral line shape has been predicted using the backscattering theory of McCracken and Freeman with the inclusion of a radiationless deexcitation term. By suitable choice of the radiationless deexcitation coefficients the theory and experiment may be brought into acceptable agreement. The coefficients so derived are in surprising agreement with calculated values of Cobas and Lamb for an Auger deexcitation mechanism.

I have obtained a general analytic result for calculating any higher-order correction to the transition amplitude for $N$-photon absorption by writing the time-dependent perturbation theory in the $S$-matrix form. In obtaining this result, I have made use of the concept of average energy as has been defined by Bebb and Gold. A formal expression has been obtained for the upper bound of the higher-order correction and the results are discussed for the case of the hydrogen atom.

In this paper are reported $\mathrm{ab}$ initio computed cross sections for the two lowest charge-transfer channels and the lowest excitation channel for the system ${\mathrm{Li}}^{+}$ scattered from He. The computations are carried out in completely $\mathrm{ab}$ initio fashion in the adiabatic molecular basis. The molecular potential curves and wave functions were computed by use of standard variational procedures. The coupling matrix elements between the molecular states are used in a classical trajectory formulation of the nuclear motion problem to obtain cross sections for the inelastic processes. The differential cross sections agree well with the experimental results of Lorents and Conklin and of Francois, Dhuicq, and Barat. The role of radial coupling in determining the charge-exchange cross sections is stressed.

Calculations of cross sections for the excitation of fine-structure levels of the ground ${}_{}{}^3{}_{}{}^{}P$ state of atomic oxygen by electron impact in the energy range of 500-15 000 K are obtained by a close-coupling method. Below 3000 K, it is necessary to allow for energy differences between the fine-structure levels. Above 3000 K, polarization effects have been incorporated by use of pseudostates and are found to decrease the cross sections slightly. Collision strengths are a factor of 3-5 lower than those obtained by Breig and Lin. In view of recent multiconfiguration results of Saraph, our single-configuration calculations probably represent a lower limit to the fine-structure collision strengths.

Emission cross sections for the resonance lines of Ne I (744 and 736 Å) and He I (584 Å) have been investigated during collisions of ${\mathrm{He}}^{+}$ + Ne at energies from threshold to 9 keV. The threshold of excitation for the channel which dissociates into the upper state of the 736-Å line is 21.7 ± 2.0 eV, and the large modulation depth in the total cross section shows this channel to be strongly coupled at an internuclear separation of 5.3 Å to another excited channel which is presumed to be ${\mathrm{He}}^{*}\left(1s2s\right)+{\mathrm{Ne}}^{+}\left(2\mathrm{}{p}^5\right)$. The onset of excitation of the 744-Å line occurs at 16.3 ± 0.5 eV, and the emission cross section exhibits oscillatory structure which is less pronounced than that of the 736-Å line. Excitation of the 584-Å line appears to occur through several channels.

Electron-detachment cross sections have been measured for collisions of a negative-ion beam with a target gas. The systems studied are ${\mathrm{O}}^{-}$ on CO, ${\mathrm{H}}_2$, ${\mathrm{O}}_2$, ${\mathrm{N}}_2$, C${\mathrm{O}}_2$, NO, He, ${\mathrm{C}}_2$${\mathrm{H}}_2$, ${\mathrm{C}}_2$${\mathrm{H}}_4$, and C${\mathrm{H}}_4$ along with ${\mathrm{S}}^{-}$ on ${\mathrm{H}}_2$, ${\mathrm{O}}_2$, and CO. These cross sections have been measured in the range of incident ion energies from 0.5 to 20 eV. Some cross sections exhibit an onset near the threshold for direct detachment, while others have a large cross section below the threshold for direct detachment. The latter cross sections can be attributed to an associative detachment process.

The energy scattered from three intense, circularly polarized laser beams into a fourth beam due to vacuum-polarization effects is calculated for general angles between the beams. A semiclassical WKB technique is used which allows an accurate treatment of the geometry of the interaction region. The effects of incoherence and optical phase errors are estimated. Both the center-of-momentum and tetrahedrally symmetric cases are examined explicitly, and the results indicate that light-light scattering may be observable using laser systems of the size needed for laser-fusion studies.