We present a cluster growth model for trees (random aggregates without loops). The intrinsic dimension $d_l$ and fractal dimension $d_f$ are adjustable. We study the ‘‘skeletons’’ of trees embedded in d=2, and find that the intrinsic dimension of a skeleton is $d_l^s$=1 for $d_l$≤$d_l^c$≊1.65, and $d_l^s$≊1+$d_l$-$d_l^c$ for $d_l$≥$d_l^c$. Thus, for 1≤$d_l$≤$d_l^c$ these trees are finitely ramified, and for $d_l^c$<$d_l$≤2 infinitely ramified. The possibility that structures are fractals in l space and compact in r space also is discussed.

A new derivation of the diffusion equation and the radiation boundary condition (at a partially absorbing boundary) from the Fokker-Planck equation is presented. The range of validity of the diffusion equation is discussed.

We report observation of oscillatory instabilities leading to chaos in a single longitudinal but multi-transverse-mode cw ${\mathrm{CO}}_2$ laser. Particularly interesting is the observation of stable oscillation involving three transverse modes en route to chaos. Furthermore, these results are obtained under predominantly homogeneously broadened lasing transition.

A weighted random walk is considered on regular two-dimensional lattices. The weight 1 corresponds to the free random walk and the weight 0 to the walk performed in the combinatorial solution of the Ising model. It is argued that the walk with weight 1/2 might have the same connective constant as the self-avoiding walk. This is shown for the honeycomb lattice. For the square lattice the connective constant is in good agreement with recent estimates. For the triangular lattice, where the construction of the walk is not unique, a ‘‘natural’’ choice results in fair agreement with these estimates.

We show that in cholesteric liquid crystals subject to a vertical external temperature gradient a new type of pattern competition exists: Either a dynamic instability (convection) as in nonequilibrium systems or a static instability (thermoundulations) as in constrained pattern-forming equilibrium systems can arise first—depending on the pitch—as the temperature gradient is increased. Numerical estimates are given for both instabilities: Convection occurs first for large pitches and thermoundulations dominate for small pitches.

In this investigation term values for high-angular-momentum states in sodium are calculated by numerical integration of Schrödinger’s wave equation. It is assumed that a single valence electron (Rydberg electron) interacts with the atom’s nucleus and a core of electrons. Polarization and penetration are taken into account and effective polarizabilities determined by fitting calculations to observed splittings between l=3, 4, and 5 states of sodium. The effective dipole and quadrupole polarizabilities are found to be 0.9965(100) and 0.376(150) a.u., respectively, and an improvement over a previous comparison of ab initio theory and experiment is made for the quadrupole polarizability.

Probabilities have been computed for transitions induced by a β-decay process from the ground rovibronic state of the ${\mathrm{T}}_2$ molecule to various bound rovibrational and scattering states connected with the first six ${}_{}{}^1{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$ electronic states of the ${}_{}{}^3{}_{}{}^{}$${\mathrm{HeT}}^{+}$ ion. Analogous calculations were performed for the TH source but only the ground electronic state of the daughter system was considered. For transitions from the ${\mathrm{T}}_2$ source to the ground electronic state of the daughter ion a probability equal to 0.4083 is associated with bound rovibrational states while 0.1659 is associated with scattering states. The same values for the TH source are 0.5572 and 0.0136, respectively. It is shown that the use of a TH source is not advantageous in a neutrino-mass experiment. Numerical accuracies of our calculations have been checked by making extensive use of sum rules. The total error in the final result is discussed.

Atomic Coulomb capture ratios for pions have been measured for beryllium borides and mixtures of the constituents. The ratios for the mixtures are independent of the atomic fraction and are reproduced well by a quantum-mechanical calculation. However, the ratios for the compounds are less than half of those of the mixtures and decrease rapidly with increase in the atomic fraction of boron. These results for the borides are not predicted by any model, except when the ionicity parameter in the mesomolecular model is used as an adjustable parameter.

Density-functional calculations have been performed for the energy surface of the ${}_{}{}^1{}_{}{}^{}\mathrm{A}_1^{}{}_{}{}^{}$ state of ${\mathrm{Si}}_3$ and for vertical excitation energies of low-lying excited states. In contrast to the results for ${\mathrm{C}}_3$ and the predictions of Walsh’s rules for triatomic molecules with twelve valence electrons, the equilibrium geometry is found to be nonlinear (${\alpha }_{\mathrm{Si}\mathrm{?}\left(\mathrm{h}\mathrm{y}\mathrm{S}\mathrm{i}\mathrm{?}\right(\mathrm{h}\mathrm{y}\mathrm{S}\mathrm{i}\mathrm{\sim }85}$°). The relative importance of d functions in silicon contributes to this and other differences between the energy surfaces of ${\mathrm{C}}_3$ and ${\mathrm{Si}}_3$. The trends in bond lengths and bond angles are discussed for dimers, trimers, and bulk systems of carbon and silicon. The energy surface for the lowest ${}_{}{}^3{}_{}{}^{}\mathrm{B}_2^{}{}_{}{}^{}$ state shows a minimum for α=60° which is almost degenerate with the ${}_{}{}^1{}_{}{}^{}\mathrm{A}_1^{}{}_{}{}^{}$ minimum.

The hyperfine structure has been measured in several rotational states of the X${}^1$${\Sigma }_g^{+}$, v’’=0 ground state of ${\mathrm{Na}}_2$ with the aid of the laser-rf double-resonance technique on a molecular beam. The rotational states observed were especially chosen to avoid distortion of the line shapes by laser-induced coherence effects. The electric quadrupole, nuclear spin-rotation, tensor spin-spin, and scalar spin-spin hyperfine coupling constants are, respectively, eqQ=[-458.98(4)-0.007 28(25)J(J+1)] kHz, c=242.9(1.5) Hz, d=302.6(5.0) Hz, and δ=1066.7(6.5) Hz.

Near–Hartree-Fock information entropies $S_{\rho }$=-Fρ(r)lnρ(r)dr and $S_{\gamma }$=-F γ(p)lnγ(p)dp, where ρ(r) and γ(p) are one-electron densities in coordinate and momentum space, respectively, have been computed for atoms in their ground and excited states. The information entropies for the harmonic oscillator and hydrogen atom as prototype systems are also discussed. The result of this exercise is a numerical discovery that for the ground state, $S_{\rho }$+$S_{\gamma }$ shows its minimum value. This study for atoms in their ground states, using wave functions of single-zeta, double-zeta, and near–Hartree-Fock quality, also unearths a startling feature that the entropy sum, $S_{\rho }$+$S_{\gamma }$, increases with an enhancement in the ground-state wave-function quality.

The perturbed Hamiltonian for magnetic dipole transitions is rewritten in terms of the torque operator, instead of the angular momentum operator, which, owing to its nondifferential form, permits tactical advantages in actual calculations of magnetic susceptibility. The translational gauge invariance of the magnetic properties is used to obtain a large series of sum rules involving linear and angular momenta and torque, force, and position operators. These are found to be very general quantum-mechanical relations, restating in a synthetic and unitary form the Thomas-Reiche-Kuhn sum rule, the basic operator commutation properties, the hypervirial theorem, and the conservation of the current-density vector, which are reduced to the same theoretical framework. Accurate calculations of the magnetic properties of the HF molecule, based on the equation-of-motion approach, reveal that the gauge-invariant sum rules can be used for rigorous tests of the quality of approximate molecular wave functions.

For the first time, a theory has been developed for the structure of the $2^3$P state of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ which is consistent with all experimental data. Using our effective Hamiltonian, containing both spin-symmetric and -antisymmetric operators, we have obtained new values for several important hyperfine parameters. The three most accurate of these are the spin-symmetric constants C, D, and E corresponding, respectively, to the contact, orbital, and tensor interactions. We find C${=4283.84\mathrm{}}_{\mathrm{-}0.01}^{+0.02\mathrm{}}$ MHz, D=-28.145(21) MHz, and D/E=-3.9445(21). We have also obtained improved values for other hyperfine parameters and for the orbital g factor. Our results are shown to be in agreement with ab initio calculations after making some plausible corrections for isotope and relativistic shifts. We believe that this state is now well understood at the level of 20 kHz in the interaction constants.

A lengthy series of experiments has resulted in the production of a high-purity source of ${}_{}{}^{86}{}_{}{}^{}\mathrm{Rb}$ which was used in the internal-source method of pair-production cross-section measurements. The 1.077-MeV γ ray in the decay of ${}_{}{}^{86}{}_{}{}^{}\mathrm{Rb}$ was used to measure the absolute pair-production cross section ${\sigma }_{\mathrm{PP}=1.92\mathrm{}\pm 0.11}$ mb in a target of Z=82. This is the lowest-energy measurement for a target with Z≥32 and shows a significant departure from the theoretical value of 1.34 mb. This is 6% lower than our result published earlier because of a far more detailed correction for pair production by inner bremsstrahlung radiation.

It is proposed that the addition of a matrix optical potential to a close-coupling calculation should lead to improved results in studies of electron-atom scattering. This procedure is described with use of a pseudostate expansion to evaluate the optical potential. The integro-differential equations are solved by a linear-algebraic method. As a test case, applications are made to electron-hydrogen scattering, and the results are compared with those obtained by other calculational procedures, and with experiment.

The photon-attenuation cross sections for beryllium have been measured with use of a Si(Li) detector spectrometer in the energy range 1.5–8.0 keV. The Kα and Lα x-ray lines produced by the heavy-ion interaction with the thick targets of different materials have been used as the low-energy photon sources. High-purity (99.8%) beryllium foils are used. The experimental errors are less than 2.5%, except when stacked foils were used for the Cu Kα line, in which case the error was 2.9%. The results are compared with the available measured values in this energy region and also with widely used tabulated values of Storm and Israel, of Veigele, and with recent tabulations of Hubbell. Furthermore, the photoeffect cross sections are derived from the total attenuation cross sections by subtracting the scattering cross sections and are compared with Scofield’s theoretical calculations.

Classical-trajectory Monte Carlo (CTMC) method is applied to the pickup process, A+(BC)→(AB)+C, of a bound light particle in a heavy particle by an incident light particle as well as the breakup process A+(BC)→A+B+C. Calculation has been made for the cross sections of the processes in the case of A=$e^{+}$,${\mu }^{+}$, B=$e^{\mathrm{-}}$,${\mu }^{\mathrm{-}}$, and C=p. The cross sections for positronium formation agree with those of recent quantum-mechanical calculations except for low impact velocities. The cross sections for (${\mu }^{+}$${\mu }^{\mathrm{-}}$) formation are smaller in magnitude than those of the quantum-mechanical calculations using the first-order Born approximation and the first-order distorted-wave Born approximation.

As one of the possible ways to produce a (${\mu }^{+}$${\mu }^{\mathrm{-}}$) atom, we have theoretically investigated the cross section of the process ${\mu }^{+}$+(${\mu }^{\mathrm{-}}$p)→(${\mu }^{+}$${\mu }^{\mathrm{-}}$)+p. The general behaviors of the cross sections obtained by the first-order Born approximation (FBA) and the first-order distorted-wave Born approximation (DWBA) are discussed. From the comparison of the present results with those of classical-trajectory Monte Carlo (CTMC) calculation and from the reliability of the present method which was discussed in the case of $e^{+}$+H→($e^{+}$$e^{\mathrm{-}}$)+p, we can conclude that the maximum cross section is of the order of ${10}^{\mathrm{-}20}$ ${\mathrm{cm}}^2$ at about 2.8 keV impact energy, which is of the order of the geometrical cross section. The cross sections by FBA are of the same order with those by DWBA, agreeing to within 10–20 %. The DWBA cross section is larger than that by CTMC by 30–50 % in the velocity range of 5–10 keV. Scaling relationships between the cross section and impact velocity are discussed in the cases of ${\mu }^{+}$+(${\mu }^{\mathrm{-}}$p) and $e^{+}$+H.

Measurements of the relative magnetic-substate population of differentially scattered H(2p) atoms formed in 4-, 5-, and 8-keV proton-helium collisions have been made using polarized-photon–scattered-particle coincidence techniques. The relative phases of the scattering amplitudes were also determined. It was found that the differential cross sections for the magnetic substates $m_l$=0 and $m_l$=±1 were of comparable magnitude indicating that translational and rotational coupling play equally important roles in the excitation process. The average measured phase difference was found to be ‖45?(de‖.

Two-center molecular-structure calculations using an l-dependent pseudopotential technique have been performed for alkali-metal-atom–${\mathrm{H}}_2$ systems, and the adiabatic potential energies for the ground states and numerous excited states of these systems have been obtained for the $C_{\infty v}$ and $C_{2v}$ symmetries. The ${\mathrm{H}}_2$ molecule was assumed to lie in its ground state $X^1$${\Sigma }_g^{+}$(v=0) and its bond length fixed to the equilibrium value $r_e$=1.4 a.u. The interaction between the valence electron of the alkali-metal atom and ${\mathrm{H}}_2$ is described by a one-center effective interaction which is modeled to reproduce differential elastic-scattering experimental data at low energies. The results are generally in good agreement with available ab initio calculations indicating the reliability and the usefulness of such an approach. The present calculations fill in the lack of information concerning most of these systems.