A general expression for coherent transients, free induction decay and echoes, of an atomic or spin two-level system excited by resonant laser or rf field in the form of two square pulses is given by using the Fourier-transform method. This expression interprets the main characteristics of the transient signals well. An analytic expression for the shapes of two-pulse echoes is derived by assuming infinite inhomogeneity and the same Rabi frequencies for the two pulses. Two-pulse echo shapes were numerically calculated for various combinations of pulse areas by using the analytic expression. Nuclear-magnetic-resonance experiment was carried out on an inhomogeneously broadened proton spin system. Experimental two-pulse echo shapes are compared with the calculated ones. Agreement between theory and experiment is quite excellent.

Recognizing that the complete initial inversion required in superfluorescence (SF) can only be achieved in a finite time, we use two different master equations for two different stages of SF: one describing the initiation of SF during pumping and another one governing the free evolution of SF after pumping. The pumping process is treated as a Poisson process with varying rate to reflect a pump pulse of arbitrary shape. The exact solution to the first master equation is obtained and used as the initial condition for the second master equation. The statistical properties of the tipping angle and the delay time are then calculated explicity. It is shown that the effects of pumping process on these two quantities can be reduced to a parameter called effective head-start time (EHT). Two different expressions for this EHT have been derived: one with saturation effects and another one without.

We report a novel method for the observation of photon echoes and other coherent transient effects. In this method, the polarization, direction, and frequency of the polarized coherent radiation emitted by a system are switched by optical pulses. We apply this method to detect photon echoes generated on the $6P_{\frac{1}{2}}-7\mathrm{}{S}_{\frac{1}{2}}$ transition in atomic thallium by two collinear and vertically polarized excitation pulses which are separated by 20 nsec. At the time of the echo, two orthogonally polarized probe pulses resonant with the $6P_{\frac{3}{2}}-7\mathrm{}{S}_{\frac{1}{2}}$ transition of atomic thallium are applied to the sample to switch the photon echo from its original polarization and direction. A horizontally polarized photon echo with its propagation vector turned by about 10 mrad and its axis of polarization rotated by 90° is detected. No electro-optical shutter is used in the experiment. The echo signal disappears either when one of the probe pulses is blocked or when both probe pulses are horizontally or vertically polarized. Echo polarization switching is also demonstrated using an unpolarized probe pulse applied prior to the echo formation time.

The two-hole final-state binding energies for the major Auger transitions in CO and ${\mathrm{CO}}_2$ are calculated with the use of the self-consistent-field $X\alpha$ multiple-scattering (SCF-$X\alpha$-MS) method. A simple modification of Slater's transition-state approach is used, and the results give essentially the same degree of agreement with experiment as more first-principles Hartree-Fock methods. The relative transition intensities for the $\mathrm{C}\mathrm{}\left(\mathrm{KVV}\right)\mathrm{}$ and the $\mathrm{O}\mathrm{}\left(\mathrm{KVV}\right)\mathrm{}$ spectra are calculated and compared both with experimental measurements and Hartree-Fock calculations. The utility of the method lies in the ease to which it can be applied to more-complex systems such as large molecules and clusters of atoms simulating molecules absorbed on surfaces.

The influence of quantum noise on the mean beat frequency ${〈{〈\dot{\phi }〉}_F〉}_t$ of a dithered-ring-laser gyroscope is considered. The corresponding Fokker-Planck equation is discussed and approximate analytical expressions and an exact expression for ${〈{〈\dot{\phi }〉}_F〉}_t$ in terms of infinite-matrix continued fractions are obtained. The two approaches are checked against each other.

We present a regorous adiabatic elimination of the atomic variables in the framework of the quantum-statistical single-mode laser model. This procedure, which is systematic and similar to the one used in the semiclassical theory, is based on the hierarchy method proposed years ago by one of us. Using a rigorous definition of the adiabatic limit, we show that the hierarchy is exactly truncated by the limit itself. Thus we recover the Fokker-Planck equation previously derived by us using a nonsystematic procedure. We discuss the advantages of the present adiabatic elimination method over the ones previously used by various authors.

As laboratory plasmas of increasing atomic number, temperature, size, and/or density are produced, it becomes likely that the details of the doublet opacity profiles of the Lyman series will influence the $K$-shell level populations and spectrum. Accordingly, we have analyzed these effects for a range of plasma parameters, confined to densities low enough for Stark broadening to be unimportant. An analytic model is developed which predicts line power enhancements and level-population changes for $K$-shell plasmas. This model is based upon photon escape probability and collisional quenching concepts and is valid for plasmas of atomic number ∼13-26. Additionally, an extensive set of numerical calculations of line ratios, line profiles, and level populations has been carried out for $K$-shell argon plasmas. Each computation was performed both with detailed fine-structure opacity profiles and with a single-Voigt-profile approximation. The results of these calculations may be scaled for plasmas of atomic number other than 18 using a simple set of rules discussed in the text.

The Metropolis Monte Carlo method is adapted to locate the phase transition for treelike percolation. The relationship between this problem and the $s\to 0$ limit of the Potts model is clarified.

We describe photon autocorrelation measurements of the Rayleigh linewidth in thin binary liquid films near the critical point. Measurements were made on four films of thickness 13.1, 4.1, 2.1, and 0.5 μm. The films were formed by trapping and sealing samples of a 2,6-lutidine + water critical mixture between fused-silica optical flats. Dynamic scaling arguments predict that the Rayleigh linewidth dependence upon $k\xi$ for the films should be indistinguishable from that for otherwise identical large-volume samples, until conditions defined by both $\xi \simeq \frac{s}{2}$ and $\Lambda \simeq s$ are satisfied. Here, $s$ is the film thickness, $\xi$ is the correlation length, and $\Lambda \equiv 2\pi k^{-1\mathrm{}}$ is the sampling length probed at wave number $k$. We find that, for the three thicker films to within 0.1 mK of their phase-separation temperatures and for the 0.5-μm film to within 2 mK of an extrapolated critical temperature, our data agree with the predictions of renormalization-group theory for binary liquid mixtures in three spatial dimensions—with no adjustable parameters. No evidence of finite-size effects was seen, although both conditions are at least marginally satisfied for the 0.5-μm film ($\Lambda \simeq s$, and eight linewidths measured with $\xi >\frac{s}{2}$) and for the 2.1-μm film ($\Lambda \simeq \frac{s}{4}$, and fourteen linewidths measured with $\xi >\frac{s}{2}$). No effects were seen that could be traced to the presence of surface wetting layers. Finally, we observe a time- and film-thickness-dependent drift in the critical temperature which is not a critical phenomenon, and we find evidence for a noncritical phase transition in the thinnest film.

The transient birefringent responses of three simple liquids (Xe, Ar, and ${\mathrm{O}}_2$) have been measured using subpicosecond optical pulses. In addition to molecular reorientational dynamic contributions to the birefringence (in ${\mathrm{O}}_2$), we have directly observed the subpicosecond dynamics arising from electronic and intermolecular interactions (in Xe and Ar). Our results in Xe and Ar constitute the first unambiguous observations of such intermolecular processes in real time. The theoretical connections between such direct observations and the previous indirect, light-scattering studies in the frequency domain are explicated. The importance of intermolecular processes in the subpicosecond dynamics of all dense fluids is noted.

There exist a number of interesting physical problems, such as the ac-driven, dc SQUID (superconducting quantum interference device) or convection in conducting fluids, which can be described by the dynamics of driven coupled oscillators. In order to study their behavior as a function of coupling strength and nonlinearity, we have considered the dynamics of two coupled maps belonging to the same universality class as the oscillators. We have analytically determined some of the parameter values for which they exhibit locked states as well as bifurcations into aperiodic behavior. Furthermore, we found a set of codimension-two bifurcations into quasiperiodic orbits near which the rotation number becomes vanishingly small. These bifurcations are characterized by the existence of periodic regimes interrupted by episodes of phase slippage. Finally, we show the effect of thermal fluctuations on the bifurcation diagram by computing the Lyapunov exponent in the presence of external and parametric noise.

The geometric-optics theory for the Dirac equation is derived using a procedure developed for the description of electromagnetic waves in inhomogeneous plasmas. This semiclassical description of relativistic electrons permits simple estimates of the magnitude and relevance of quantum corrections to the classical particle dynamics which can be used to assess the validity of the classical treatment of relativistic electron beams in a variety of applications. Moreover, these results correct several errors in the literature and are presented in a form which is easily solved, numerically, using straightforward modifications of existing ray-tracing codes for electromagnetic radiation.

The growth of Rayleigh-Taylor instabilities in intertial-confinement-fusion targets can be reduced if smooth density gradients are introduced at the interfaces. Keeping the total mass fixed, we consider spreading a heavy shell either continuously or in a discrete number of subshells. We calculate the rate for the fastest growing mode as a function of the fraction of mass spread and find the minima for the cases of 4, 6, and 8 subshells. The rates are reduced by 1.4-1.6. If all the mass is spread continuously into an exponential profile, we find that the rates are reduced by approximately $\sqrt{2\pi }\approx 2.5$.

The Euler equations for describing the director in the optically induced molecular reorientation of a smectic-$C$ liquid crystal in an external magnetic field are presented. The optical alignment effect is shown to be localized and not to produce point defects. Analytic expressions are given explicitly in the small-distortion regime, which, generally, have the form of zeroth-order Bessel functions. Threshold behavior exists if the polarization of the optical beam is normal to the magnetic field. For a magnetic field of the order of 1 kG, the threshold power varies from 3 to 120 mW for a typical smectic-$C$ sample with a laser spot size of about 1 to 100 μm, but is independent of the thickness of the sample. For polarizations oblique to the magnetic field, there is no threshold and the amplitude of rotation of the director depends on spot size, laser power, and the angle between laser polarization and external magnetic field. The transient response of molecular reorientation to the laser switch on is shown to have exponential time dependence for a normally polarized incident beam with an incident intensity greater than the threshold intensity. The response time of such a reorientation is of the order of milliseconds, depending on incident laser intensity. We propose that, experimentally, the molecular reorientation can be quantitatively measured by the reflectivity or transmissivity of a normally incident probe beam. Optical reflectivity and transmissivity from a typical smectic-$C$ film are also calculated.

Earlier work applying equations of the channel-coupling array theory of many-body scattering to ${\mathrm{H}}_2^{+}$ and ${\mathrm{H}}_2$ has been extended. Primary consideration has been given to the use of the Galerkin-Petrov method, by which the non-Hermitian matrix equations of the theory have been solved approximately with the use of globally defined bases as expansion sets. For ${\mathrm{H}}_2^{+}$, both hydrogenic and Hylleraas-Shull-Löwdin functions were used, but neither set led to minima in the ground-state (gerade) potential-energy curves which lie higher than the exact value nor indicated a convergence to the correct minimum value. Only hydrogenic functions were used for ${\mathrm{H}}_2$ in a two-channel truncation approximation. Slow convergence (from above) to the exact minimum in the singlet ground-state potential-energy curve was indicated. The inner minimum in the ${\mathrm{H}}_2$ $E {}_{}{}^1{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$ curve originally calculated by Davidson was reproduced to within 0.002 a.u. by an extremely simple covalent approximate calculation; there was no indication of Davidson's outer, ionic minimum, but this is not surprising since an ionic component was not built into our approximation. The He${\mathrm{H}}^{+}$ ground state was also determined. The equilibrium separation and energy minimum for the ground state was relatively accurate given the simple (Eckart) wave function used to approximate the He ground state. In addition to these results, the nonphysical ${\mathrm{H}}_2^{+}$ ungerade and ${\mathrm{H}}_2$ triplet potential-energy curves found in the earlier one-state approximate calculations also occurred in the present computations. The failure to achieve accurate convergence and the persistence of nonphysical potential-energy curves remain serious problems that require further investigation. A resolution of both these problems for ${\mathrm{H}}_2^{+}$ has been achieved and is described in the following paper.

Light scattering experiments reveal a strong suppression of phase separation near the critical point of a vigorously stirred binary liquid mixture. For stirring Reynolds numbers $R$ ranging from 6.0 × ${10}^3$ to 4.5 × ${10}^3$, the apparent critical temperature is depressed by ∼1 mK to ∼50 mK. This temperature depression $\Delta T_c^{\prime }$ can be fitted to a power law $\Delta T_c^{\prime }\sim R^{\lambda }$ where $\lambda \sim 2$. The magnitude of $\Delta T_c^{\prime }$ is consistent with simple models which attribute the effect to the suppression of composition fluctuations by shear; however, these models predict $\lambda \simeq 0.80$ in contrast to the observed value of ∼2. Below the apparent critical temperature the turbidity $\tau$ changes significantly throughout a temperature range of tens of millikelvin following a power law $\tau \propto {(T_c-T)}^{\zeta }$ where $\zeta$ increases from ∼1 to ∼6 as $R$ is increased.

The triplet correlation function in the strongly coupled classical one-component plasmas in three and two dimensions is obtained by Monte Carlo numerical experiments and compared with the values given by the Kirkwood superposition approximation and the convolution approximation. The superposition approximation works well when two particles are closer than the first peak distance of the pair distribution function, and the convolution approximation works well when three particles are nearly aligned with separations larger than the first peak distance. The equilateral triangle configuration of particles mutually separated by the first peak distance is overestimated by the superposition approximation, while the convolution approximation shows no trace of this structure.

In systems with a complex order parameter, $e^{i\varphi \mathrm{}\left(x\right)\mathrm{}}$, the correlation function $C\left(x\right)=〈e^{i\left[\varphi \mathrm{}\right(x)-\varphi \mathrm{}(0\left)\right]}〉$ can be expanded in cumulants $g^{\mathrm{}\left(2n\right)\mathrm{}}\mathrm{}\left(x\right)={〈{\left[\varphi \mathrm{}\right(x)-\varphi \mathrm{}(0\left)\right]}^{2n}〉}_c$. For large $x$, $g^{\mathrm{}\left(2n\right)\mathrm{}}\mathrm{}\left(x\right)\mathrm{}\sim x^{{\omega }_n}$ where ${\omega }_n$ depends on dimensionality, $d$, and the Hamiltonian. We introduce two important dimensions, $d_L^{*}$ and $d_L$, associated with ${\omega }_n$. For $d>\mathrm{}{d}_L^{*}$, ${\omega }_1>{\omega }_2>\cdots >{\omega }_n$, and for $d_L>d>\mathrm{}{d}_L^{*}$, ${\omega }_1>0\mathrm{}$. ${\omega }_1$ becomes zero when $d=d_L$ and thus $d_L$ corresponds to the usual lower critical dimension at which long-range order in $e^{i\varphi \mathrm{}\left(x\right)\mathrm{}}$ disappears. For $d_L\ge d>\mathrm{}{d}_L^{*}$, the large $x$ behavior of $C\left(x\right)\mathrm{}$ is therefore determined by $g^{\mathrm{}\left(2\right)\mathrm{}}\mathrm{}\left(x\right)\mathrm{}$. At $d=d_L^{*}$ all ${\omega }_n$ are equal, and all cumulants are needed. After introducing these concepts with a nonlinear spin-wave model, we consider applications to correlations in liquid crystals where the physical order parameter $\psi$ is related to the order parameter ${\psi }_{\mathrm{SC}}$ is a gauge where phase fluctuations are a minimum via $\psi ={\psi }_{\mathrm{SC}}{e}^{{-iq}_{0}L}$ where $q_{0}L$ is the phase associated with a gauge transformation. We show that for $q_{0}L$, $d_L^{*}\le 2$ in all cases. Thus the large $x$ behavior of $R\left(x\right)=-\ln 〈e^{{\mathrm{iq}}_0\mathrm{}\left[L\right(x)-L(0\left)\right]}〉$ is determined by the second cumulant of [$L\mathrm{}\left(x\right)-L\mathrm{}\left(0\right)\mathrm{}$]. We evaluate $R\mathrm{}\left(x\right)\mathrm{}$ for $2<d<\mathrm{}4\mathrm{}$. At the anisotropic critical point [${\nu }_{\parallel }=(5-d)\mathrm{}{\nu }_{\perp }$] in three dimensions, $R\mathrm{}\left(x\right)\mathrm{}\sim {\left(\ln x\right)}^2$ rather than Inx as previosuly reported. In addition, we show that the decoupling approximation, $G\mathrm{}\left(x\right)=〈\psi \mathrm{}\left(x\right)\mathrm{}{\psi }^{*}\mathrm{}\left(0\right)\mathrm{}〉\sim 〈{\psi }_{\mathrm{SC}}\mathrm{}\left(x\right)\mathrm{}{\psi }_{\mathrm{SC}}^{*}\mathrm{}\left(0\right)\mathrm{}〉e^{-R\mathrm{}\left(x\right)\mathrm{}}$, is valid in the smectic-$A$ and nematic phases and at the critical point when there is isotropic scaling (${\nu }_{\parallel }={\nu }_{\perp }$) for $2<d<\mathrm{}4\mathrm{}$ and when there is anisotropic scaling for $3<d<\mathrm{}4\mathrm{}$. The decoupling approximation breaks down except in the smectic-$A$ phase when there is anisotropic scaling for $d\le 3$.

We propose and investigate the properties of a model of pattern formation in crystal growth. The principal dynamical variables in this model are the curvature of the solidification front and the thickness (or heat content) of a thermal boundary layer, both taken to be functions of position along the interface. This model is mathematically much more tractable than the realistic, fully nonlocal version of the free-boundary problem, and still recaptures many of the features that seem essential for studying dendritic behavior, for example. In this paper we describe analytic properties of the model. Preliminary numerical solutions produce snowflakelike patterns similar to those seen in nature.

Nucleation theory is considered in $d$-dimensional systems which undergo a nearly mean-field-like transition, such as Ising magnets or mixtures with a large but finite range $r$ of the interaction, or polymer mixtures with chain lengths $N_A=N_B=N\gg 1$. Near two-phase coexistence the nucleation free-energy barriers are $\Delta F^{*} \propto r^d{(1-\frac{T}{T_c})}^{2-\frac{d}{2}}{\left(\frac{\delta \psi }{{\psi }_{\mathrm{coex}}}\right)}^{-(d-1\mathrm{})\mathrm{}}$ and $\Delta F^{*} \propto N^{(\frac{d}{2}-1)}{(1-\frac{T}{T_c})}^{2-\frac{d}{2}}{\left(\frac{\delta \psi }{{\psi }_{\mathrm{coex}}}\right)}^{-(d-1\mathrm{})\mathrm{}}$, respectively, where ${\psi }_{\mathrm{coex}}$ is the order parameter, $\delta \psi$ the deviation of the order parameter in the metastable state from that at coexistence, and $T_c$ the critical temperatures where the transition is second order. The crossover to nucleation near $T_c$ where $\frac{\Delta F^{*}}{k_B{T}_c}=c{\left(\frac{\delta \psi }{{\psi }_{\mathrm{coex}}}\right)}^{-(d-1\mathrm{})\mathrm{}}$, $c$ for $d<\mathrm{}4\mathrm{}$ being a universal constant, is controlled by the same Ginzburg criterion as for the static critical properties, i.e., crossover occurs at $r^d{(1-\frac{T}{T_c})}^{2-\frac{d}{2}}\approx 1$, $N^{(\frac{d}{2}-1)}{(1-\frac{T}{T_c})}^{2-d}\approx 1$. In polymer mixtures, mean-field-like behavior occurs only for $d>\mathrm{}2\mathrm{}$. In the mean-field region, metastable states are well defined up to a narrow region near the spinodal curve ${\psi }_{\mathrm{sp}}$; the width of this region is given by ${\left[\frac{(\psi -{\psi }_{\mathrm{sp}})}{{\psi }_{\mathrm{coex}}}\right]}^{(3-\frac{d}{2})}\approx {\left[r^d{(1-\frac{T}{T_c})}^{2-\frac{d}{2}}\right]}^{-1\mathrm{}}$ or ${\left[N^{(\frac{d}{2}-1)}{(1-\frac{T}{T_c})}^{2-\frac{d}{2}}\right]}^{-1\mathrm{}}$, respectively. This rounding of the spinodal curve can again be understood by the Ginzburg criterion for the metastable state at $\psi$ near ${\psi }_{\mathrm{sp}}$. At the unstable side of the mean-field region, the linear theory of spinodal decomposition holds outside of correspondingly narrow regions close to the spinodal curve, too.