Classical Rate Theory Research Articles

Erratum: Soluble models of rate processes in periodic systems with many degrees of freedom

We show that the parameters of classical many-body rate theory can be evaluated exactly for activated processes in large, chainlike systems. The methods are applied to four simple systems which simulate, in some degree, activated atomic motions in crystals. Each is a linear chain of atoms interacting with near-neighbor forces. One represents evaporation of an adatom from a surface, one an interstitialcy migration, one a vacancy migration, and one the motion of the interface between two phases. Exact analytical calculation of the parameters of many-body rate theory, namely the activation energy and the effective, or attack, frequency are given for each model. The results illustrate the role of cooperative effects in atomic jumps. The commonly made assumption that the attack frequency for such jumps is the characteristic lattice-vibration frequency is shown to be valid in some conditions, but to be a very poor approximation in others. For the evaporating adatom the relative mass of the adatom is allowed to be arbitrary. It is found that the transition rate is proportional to the square root of the relative mass for very light atoms, but otherwise the relative mass enters in a more complicated way, and, in particular, the transition rate remainsmore » finite for an adatom of infinite mass.« less

Soluble models of rate processes in periodic systems with many degrees of freedom.

We show that the parameters of classical many-body rate theory can be evaluated exactly for activated processes in large, chainlike systems. The methods are applied to four simple systems which simulate, in some degree, activated atomic motions in crystals. Each is a linear chain of atoms interacting with near-neighbor forces. One represents evaporation of an adatom from a surface, one an interstitialcy migration, one a vacancy migration, and one the motion of the interface between two phases. Exact analytical calculation of the parameters of many-body rate theory, namely the activation energy and the effective, or attack, frequency are given for each model. The results illustrate the role of cooperative effects in atomic jumps. The commonly made assumption that the attack frequency for such jumps is the characteristic lattice-vibration frequency is shown to be valid in some conditions, but to be a very poor approximation in others. For the evaporating adatom the relative mass of the adatom is allowed to be arbitrary. It is found that the transition rate is proportional to the square root of the relative mass for very light atoms, but otherwise the relative mass enters in a more complicated way, and, in particular, the transition rate remainsmore » finite for an adatom of infinite mass.« less

Measurements of Permeation of Hydrogen Isotopes through α-Iron by Pressure Modulation and Ion Bombarding

Permeation response of hydrogen and deuterium through α-iron under the condition of pressure modulation was measured to determine the diffusion coefficient which contained no surface effects. The diffusion coefficients of hydrogen isotopes are obtained between 500 and 1000 K.Permeation under hydrogen ion bombardment was also studied to extend the temperature range of diffusion coefficient measurements.The ratio of the diffusion coefficients of hydrogen isotopes (DH⁄DD) depends on the temperature; the value of the ratio approaches unity with increasing temperature. This behavior differs from that of common fcc metals in which the value of the ratio approaches \sqrt2 with increasing temperature. The isotope effect of hydrogen diffusivity in α-iron can not be explained by the classical rate theory, and the quantum-mechanical correction is needed.

Recent Developments in Transition State Theory

AbstractA number of recent developments in classical and quantum transition state rate theory are briefly discussed.

Plastic deformation of β-CuZn single crystals at low temperatures

Plastic behavior of single crystals of Cu 52Zn 48 alloy with the CsCl structure has been examined over a temperature range between 2 K and room temperature. Both at 4.2 K and 77 K, the operative slip plane is the twinning {112} plane in most orientations and it is the anti-twinning {112} plane only in orientations with the maximum shear stress plane close to the anti-twinning {112} plane. Orientation dependence of the yield stress τ mssp showed a maximum near the orientation of the slip plane transition. These results agree well with those of computer simulation of superlattice screw dislocation motion [S. Takeuchi, Phil. Mag. A 41, 541 (1980)]. Temperature and strain-rate dependences of yield stress below ~130K are essentially similar to those in bcc metals. The above results together with in situ TEM observation indicate that low temperature deformation in β-CuZn is governed by motion of screw dislocations overcoming the Peierls potential. Activation analyses revealed that the value of exponent in the Arrhenius rate equation, ΔH kT , decreases rapidly below 10 K toward zero at 0 K, showing discrepancy of the classical rate theory in the very low temperature range.

Molecular dynamics of surface diffusion. I. The motion of adatoms and clusters

The motion of isolated adatoms and small clusters on a crystal surface is investigated by a novel and efficient simulation technique. The trajectory of each atom is calculated by molecular dynamics, but the exchange of kinetic energy with the crystal lattice is included through interactions with a ’’ghost’’ atom. This atom represents surface atoms of the lattice and is subjected to random and dissipative forces that are related by the fluctuation–dissipation theorem. The diffusion process is characterized by measurements of the velocity autocorrelation function, mean square displacement, directional correlations between hops, and the mean displacement per hop. In addition, the rate of evaporation of single adatoms and the rate of dissociation of clusters are discussed. The diffusion of an isolated adatom is found to be somewhat faster than that predicted by the classical rate theory for an activated process. This effect is a result of diffusion jumps of several atomic diameters that occur preferentially at high temperatures. But Arrhenius behavior is observed over the entire range of temperatures below the melting point. Dimers and larger clusters are found to diffuse more slowly than individual atoms, but with a smaller apparent activation energy. These results do not exhibit the high-temperature anomalies that have been inferred from some experimental data on surface mass transport. In a subsequent paper the method is extended to treat mass transport in the layer of adatoms and clusters that results from a dynamic equilibrium with the vapor phase.

Off-CenterCu+Ions in Potassium Halides Studied with Ionic Thermocurrents

From previous optical and theoretical work it had been concluded that substitutional ${\mathrm{Cu}}^{+}$ ions may occupy off-center positions in certain alkali halides. The method of ionic thermocurrents (ITC) was applied to verify directly the electric dipole properties of ${\mathrm{Cu}}^{+}$ ions in three potassium halides, and to study the temperature dependence of their relaxation behavior. ITC bands were observed for ${\mathrm{Cu}}^{+}$ in KCl, KBr, and KI, with maxima at 62, 70, and 83 \ifmmode^\circ\else\textdegree\fi{}K, respectively. From the integrated ITC currents of crystals with calibrated ${\mathrm{Cu}}^{+}$ concentrations, the electric dipole moment was determined, yielding values of 1.5, 1.9, and 2.6 $e\mathrm{\AA{}}$ for ${\mathrm{Cu}}^{+}$ in KCl, KBr, and KI, respectively. The shape of the ITC bands was found in quantitative agreement with the calculated behavior for dipole reorientation by classical rate theory with a single relaxation time. The activation energy for reorientation of the ${\mathrm{Cu}}^{+}$ dipole was found to be 177, 196, and 232 meV for KCl, KBr, and KI, respectively. Using additional material on ${\mathrm{Cu}}^{+}$ and ${\mathrm{Ag}}^{+}$ ions, some systematic trends in the appearance of the off-center effect, and in the static and dynamic off-center dipole properties of heavy-metal ions, are demonstrated and discussed.

The diffusion coefficients of hydrogen and deuterium in vanadium, niobium, and tantalum by gorsky‐effect measurements

AbstractGorsky‐effect measurements on vanadium‐, niobium‐, and tantalum‐samples loaded with different concentrations (0.2 to 4.3 at%) of hydrogen or deuterium are reported. From the relaxation strength the trace of the dipole‐moment tensor of hydrogen and deuterium was determined and found to be consistent with the lattice parameter changes. From the relaxation time of the Gorsky effect the diffusion coefficients of hydrogen and deuterium in vanadium, niobium, and tantalum were determined in the temperature range from 600 to about 270° K (in the case of hydrogen in niobium down to 120° K). The diffusion coefficients are very large and show only a weak exponential temperature dependence, which moreover decreases for hydrogen in niobium below 300° K. In contrast to classical rate theory the activation energies are isotope dependent whereas the pre‐exponential factors are isotope independent within the experimental errors.

Classical and Quantum Rate Theories for Solids

A rate formula for defect motion in solids including quantum effects is derived by generalization of the classical equilibrium statistical approach. The development makes use of an interpretation of the canonical density matrix for a harmonic oscillator as an ensemble of oscillators which are in minimum-uncertainty states. The classical limit of the rate formula differs from Vineyard's in the pre-exponential factor. The difference is due to the assumption regarding the degree of mode interaction on the hill of the potential surface. The present derivation assumes no interaction, while Vineyard assumes sufficient interaction to maintain thermal equilibrium on the hill. The derived quantum rate formula includes the effects of quantum statistics but not tunnelling effects. Its low-temperature characteristics are explored for the Debye model.

Nature of “anomalous” fine structure observed in the low temperature recovery of copper and aluminum

Observed recovery maxima less than 2°K apart are shown to be consistent with appropriate classical rate theory, although the theoretical resolution of two processes is about 3°K. The recovery must consist of more processes than the number of maxima observable.

Classical Unimolecular Rate Theory. III. Effect of Initial Conditions on Lifetime Distributions

The distribution of lifetimes of classical anharmonic and harmonic triatomic molecules capable of dissociation is found to depend upon the distribution of initial conditions immediately after collisional activation. Two types of collisional-activation mechanisms are treated: one in which the density of initial points in phase space depends only upon the energy (the random mechanism), and the second in which the vibrational energy of the molecule is initially entirely kinetic (the impulsive mechanism). The random mechanism leads to a roughly exponential distribution of lifetimes for both harmonic and anharmonic molecules, in agreement with earlier work. The impulsive mechanism leads to lifetime distributions showing very marked peaks and minima for the harmonic model; this effect is much smaller for the anharmonic model.

Classical Unimolecular Rate Theory. II. Effect of the Distribution of Initial Conditions1

Classical Unimolecular Rate Theory. II. Effect of the Distribution of Initial Conditions¹

ISOTOPE EFFECTS IN METHYL RADICAL ABSTRACTION REACTIONS

The high kH/kD ratio found experimentally for methyl radical abstraction reactions is explained on the basis of classical rate theory. It is shown that if the bending frequencies in the activated complex approach zero, good agreement is found between experimental and calculated kH/kD values. For abstraction of H and D from hydrocarbons, both the tunneling and classical rate theories predict acceptable kH/kD values. Abstraction of H and D from H2 and D2, however, can best be explained on the basis of a classical process.

Phonon induced tunneling of ions in solids

A quantum mechanical analysis is given of the change of position or orientation of an atom, ion or molecule in a crystal as it occurs e.g. in the processes of diffusion or hindered rotation. One-phonon processes, Raman processes and indirect processes are discussed. The limits of applicability of the classical rate theory are given. Specifically, the analysis is applied to the reorientation of the O2- center in alkali halides under the influence of external mechanical stresses at low temperatures, which has been investigated experimentally byKanzig. The measured dependence of the reorientation time upon temperature and applied stress can be explained by one-phonon processes. Random internal strains in the crystal are shown to play an important part. As further application of the theory the proton motions in ice and in iron are elucidated. Finally, an estimate is given of the effect of direct processes involving imperfections on the thermal conductivity of alkali halides.

Isotope Effect in an Electric Field and Jump Frequencies for Diffusion in Ionic Crystals

The simultaneous diffusion of two isotopes of a given ionic impurity in an electric field is considered, and an improved method of analyzing the experimental data to obtain ΔD/D is discussed. (Here D is the diffusion coefficient of one of the isotopes, and ΔD the difference between those of the two isotopes.) As noted by Chemla, an electric field increases the accuracy with which ΔD/D can be determined. Equations are found giving this increase in accuracy as a function of field and diffusion time. If the ratio w2α/w2β of the jump frequencies of the two isotopes is known, relative values of the vacancy or interstitialcy jump frequencies near an impurity can be calculated from ΔD/D and μ/D, where μ is the drift mobility. As an example, data in the literature for Na22 and Na24 diffusing in KCl are considered. For self-diffusion, w2α/w2β may be calculated from ΔD/D and known values of the correlation factor. When data in the literature are used to estimate w2α/w2β for Na22 and Na24 diffusing in NaCl, a value smaller than that predicted by classical rate theory is found.

Single-Pulse Shock Tube Studies of the Kinetics of the Reaction N2+O2⇄2NO between 2000–3000°K

The single temperature pulse technique, developed at the Cornell Aeronautical Laboratory, has been used to study the kinetics of the formation of nitric oxide in the temperature range from 2000° to 3000°K. It has been found that the kinetics of the reaction are consistent with the chain mechanism proposed by Zeldovich. The rate-determining step in the chain is: O+N2→NO+N, with ΔH25000=75.8 kcal/mole. The activation energy for this step is found to be 74±5 kcal/mole. The collision cross-section, as calculated from classical reaction rate theory, is 10—16 cm2, which corresponds to a steric factor of about 0.05, assuming σN2,O=5 A.

Frequency factors and isotope effects in solid state rate processes

The movement of defects in solids on the basis of classical absolute rate theory is reexamined with special attention to many-body aspects. The effective frequency in the Arrhenius expression governing these processes is shown to be, in harmonic approximation, the ratio of the product of the N normal frequencies of the entire crystal at the starting point of a transition to the product of the N−1 normal frequencies of the crystal when it is constrained in a saddle point configuration. The influence of the masses of the various atoms on the effective frequency is investigated. It is shown that an effective mass which depends on the direction of the path through the saddle point in configuration space determines this frequency. In the case of chemical diffusion by the vacancy mechanism the effective mass is approximately the same as the mass of the solute atom, and must always lie between the mass of the solute and the mass of the solvent. It is finally shown that the classical rate theory, even with many-body considerations, is unable to explain the recent observations of L azarus and O kkerse on the isotope effect in the diffusion of iron in silver.