The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

This paper describes the first synthesis of a new class of topological macromolecules which we refer to as “starburst polymers.” The fundamental building blocks to this new polymer class are referred to as “dendrimers.” These dendrimers differ from classical monomers/oligomers by their extraordinary symmetry, high branching and maximized (telechelic) terminal functionality density. The dendrimers possess “reactive end groups” which allow (a) controlled moelcular weight building (monodispersity), (b) controlled branching (topology), and (c) versatility in design and modification of the terminal end groups. Dendrimer synthesis is accomplished by a variety of strategies involving “time sequenced propagation” techniques. The resulting dendrimers grow in a geometrically progressive fashion as shown: Chemically bridging these dendrimers leads to the new class of macromolecules—”starburst polymers” (e.g., (A)n, (B)n, or (C)n).

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A new class of polymers: Starburst-dendritic macromolecules

In this chapter, we will restrict our attention to the ferrites and a few other closely related materials. The great interest in ferrites stems from their unique combination of a spontaneous magnetization and a high electrical resistivity. The observed magnetization results from the difference in the magnetizations of two non-equivalent sub-lattices of the magnetic ions in the crystal structure. Materials of this type should strictly be designated as “ferrimagnetic” and in some respects are more closely related to anti-ferromagnetic substances than they are to ferromagnetics in which the magnetization results from the parallel alignment of all the magnetic moments present. We shall not adhere to this special nomenclature except to emphasize effects, which are due to the existence of the sub-lattices.

Ferromagnetic materials

We present an overview of the current status of transition-state theory and its generalizations. We emphasize (i) recent improvements in available methodology for calculations on complex systems, including the interface with electronic structure theory, (ii) progress in the theory and application of transition-state theory to condensed-phase reactions, and (iii) insight into the relation of transition-state theory to accurate quantum dynamics and tests of its accuracy via comparisons with both experimental and other theoretical dynamical approximations.

Current status of transition-state theory

Systems with a ferroelectric to paraelectric transition in the vicinity of room temperature are useful for devices. Adjusting the ferroelectric transition temperature (T(c)) is traditionally accomplished by chemical substitution-as in Ba(x)Sr(1-x)TiO(3), the material widely investigated for microwave devices in which the dielectric constant (epsilon(r)) at GHz frequencies is tuned by applying a quasi-static electric field. Heterogeneity associated with chemical substitution in such films, however, can broaden this phase transition by hundreds of degrees, which is detrimental to tunability and microwave device performance. An alternative way to adjust T(c) in ferroelectric films is strain. Here we show that epitaxial strain from a newly developed substrate can be harnessed to increase T(c) by hundreds of degrees and produce room-temperature ferroelectricity in strontium titanate, a material that is not normally ferroelectric at any temperature. This strain-induced enhancement in T(c) is the largest ever reported. Spatially resolved images of the local polarization state reveal a uniformity that far exceeds films tailored by chemical substitution. The high epsilon(r) at room temperature in these films (nearly 7,000 at 10 GHz) and its sharp dependence on electric field are promising for device applications.

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Room-temperature ferroelectricity in strained SrTiO3.

Quantum Theory of Diffusion with Application to Light Interstitials in Metals, C. P. Flynn and A. M. Stoneham [Phys. Rev. B 1, 3966 (19'70)]. A notational error might cause confusion. In Eq. (3.6), X„, should read R„„the potential associated with the pth interstice (the division of K„, into terms associated with different interstices is to some extent arbitrary but does not affect subsequent steps). Equations after (3.6) should all contain &&+X„,rather than X„,alone. The proper terms were used throughout the actual calculations, so the conclusions are not affected. A final point is that the integrand in Eq. (A6) should contain x rather than xa as printed. The related formulas are correct.

Quantum Theory of Diffusion with Application to Light Interstitials in Metals

Electron spin relaxation measurements on low-spin ${\mathrm{Fe}}^{3+}$ in several proteins show that they occupy a space of fractal dimensionality $d=1.65\ifmmode\pm\else\textpm\fi{}0.04$, in conformity with the dimensionality $d=\frac{5}{3}$ of a self-avoiding random walk. Analysis of myoglobin x-ray data independently confirms this fractal dimension.

Fractal Form of Proteins

The role of electron-phonon interactions in energetic displacement cascades in metals is evaluated by considering the diffusion of electrons in a localized heat spike. It is found that the electronic system in this region always falls out of equilibrium with the lattice at long times and that whether electronic equilibrium is achieved at short times depends almost catastrophically on the parameters that describe the electron-phonon interaction. Neglect of the electron-phonon coupling may lead to overestimates of thermal spike temperatures in many metals.

Electron-phonon interactions in energetic displacement cascades

The collisional dynamics between clusters of Cu, Ni, or Al atoms, with energies of 92 eV to 1.0 keV and sizes of 4 to 92 atoms, and substrates of these same metals were studied using molecular-dynamics computer simulations. A diverse behavior was observed, depending sensitively on the size and energy of the cluster, the elastic and chemical properties of the cluster-substrate combination, and the relative mass of the cluster and substrate atoms. For the 92-atom Cu clusters impacting a Cu substrate, the cluster can form a ``glob'' on the surface at low energy, while penetrating the substrate and heavily deforming it at high energies. When the cluster energy exceeds \ensuremath{\approxeq}25 eV/atom, the substrate suffers radiation damage. The 92-atom Al clusters do not much deform Ni substrates, but rather tend to spread epitaxially over the surface, despite the 15% lattice mismatch. For 1-keV collisions, several Al atoms dissociate from the cluster, either reflecting into the vacuum or scattering over the surface. 326-eV Ni clusters embed themselves almost completely within Al substrates and form localized amorphous zones. The potentials for these simulations were derived from the embedded-atom method, although modified to treat the higher-energy events. IAb initioP linear-combination-of-atomic-orbitals--molecular-orbitals calculations were employed to test these potentials over a wide range of energies. A simple model for the expected macroscopic behavior of cluster-solid interactions is included as an appendix for a comparison with the atomistic description offered by the simulations.

Molecular-dynamics simulations of collisions between energetic clusters of atoms and metal substrates.

Two samples of Dy-Y superlattices produced by molecular-beam-epitaxy techniques are shown by neutron diffraction to order magnetically in a helix which is incommensurate with the bilayer thickness. One sample consists of 64 bilayers, each bilayer made up of about 15 growth planes (42 A) of Dy atoms followed by 14 planes (38 A) of Y atoms. The second sample has 90 layers, each layer consisting of 9 Dy atomic planes and 8 Dy/sub 0.5/Y/sub 0.5/ alloy planes. The phase coherence of this ordering extends over several bilayers, and is especially striking in the sample where the layers of localized Dy spins are separated by 14 atomic planes of nonmagnetic Y. The fact that the helix chirality propagates across several bilayers rules out a simple scalar Ruderman-Kittel-Kasuya-Yosida coupling between the Dy planes on either side of an Y layer, but suggests instead that a helical spin density wave is induced in the Y conduction electrons. A simple model for the superlattice structure factor demonstrates that observed asymmetries in the magnetic diffraction-peak intensities can be ascribed to the existence of different magnetic modulation wave vectors in each layer type (Dy and Y). In these superlattices the strain clamping by the intervening non-Dymore » layers and the substrate suppresses the first order ferromagnetic transition found in bulk Dy in both zero and finite fields. Although the planar magnetostriction is clamped, it is observed that the application of a magnetic field in the basal plane produces at low temperatures a second order irreversible transition to a metastable ferromagnetic state. At high temperature the magnetization process is initiated by a reduction of the helical coherence length due to a random-field coupling to the uncompensated Dy layer moment.« less

C. P. Flynn

Papers

Quantum Theory of Diffusion with Application to Light Interstitials in Metals

Fractal Form of Proteins

Electron-phonon interactions in energetic displacement cascades

Molecular-dynamics simulations of collisions between energetic clusters of atoms and metal substrates.

Magnetic structure of Dy-Y superlattices