Standard Rate Theory Research Articles

Competing quantum effects in heavy-atom tunnelling through conical intersections.

Thermally activated chemical reactions are typically understood in terms of overcoming potential-energy barriers. However, standard rate theories break down in the presence of a conical intersection (CI) because these processes are inherently nonadiabatic, invalidating the Born-Oppenheimer approximation. Moreover, CIs give rise to intricate nuclear quantum effects such as tunnelling and the geometric phase, which are neglected by standard trajectory-based simulations and remain largely unexplored in complex molecular systems. We present new semiclassical transition-state theories based on an extension of golden-rule instanton theory to describe nonadiabatic tunnelling through CIs and thus provide an intuitive picture for the reaction mechanism. We apply the method in conjunction with first-principles electronic-structure calculations to the electron transfer in the bis(methylene)-adamantyl cation. Our study reveals a strong competition between heavy-atom tunnelling and geometric-phase effects.

A Rate Theory Model of Radiation-Induced Swelling in an Austenitic Stainless Steel

Many rate theory models of cavity (void) swelling have been published over the past 50 years, all having the same, or similar, structures. A rigorous validation of the models has not been possible because of the dearth of information concerning the microstructures that correspond with the swelling data. Whereas the lack of microstructure information is still an issue for historical swelling data, in the past 10–20 years data have been published on the evolution of the microstructure (point defect yields from collision cascades, cavity number densities, and dislocation densities/yield strengths) allowing certain gaps in information to be filled when considering historic swelling data. With reasonable estimates of key microstructure parameters, a standard rate theory model can be applied, and the model parameter space explored, in connection with historical swelling data. By using published data on: (i) yield strength as a function of dose and temperature (to establish an empirical expression for dislocation density evolution); (ii) cavity number densities as a function of temperature; and (iii) freely migrating defect (FMD) production as a function of primary knock-on atom (PKA) spectrum, the necessary parameter and microstructure inputs that were previously unknown can be used in model development. This paper describes a rate-theory model for void swelling of 316 stainless steel irradiated in the EBR-2 reactor as a function of irradiation temperature and neutron dose.

Point defect evolution under irradiation: Finite size effects and spatio-temporal correlations

The evolution of point defect concentrations under irradiation is controlled by their diffusion properties, and by their formation and elimination mechanisms. The latter includes the mutual recombination of vacancies and interstitials, and the elimination of point defects at sinks. Two models are traditionally used to predict the defect concentration evolution, the standard rate theory (SRT) and the atomistic kinetic Monte Carlo (AKMC). We show in this work a large discrepancy in the defect concentrations between both methods when the average number of defects in the AKMC simulation box is close or lower than one. The reason is that AKMC naturally captures strong space and time correlations between vacancies and interstitials generated by the finite size of the periodic simulation box. These correlations strongly affect the recombination rate and the point defect concentrations. SRT fails to predict such correlations, and the corresponding solution deviates from the more accurate solution given by the AKMC simulation under similar conditions. These finite size correlation effects are strong when the elimination of point defects occurs by recombination only, but can still be significant in the presence of sinks. In order to account for the spacio-temporal correlation in a continuum framework, we introduce a Correlated Pair Theory (CPT). This theory fully takes into account the correlations between vacancy and interstitial pairs and predicts point defect concentrations in good agreement with AKMC simulations. Inversely, if the goal is for the AKMC to reproduce defect concentrations in bulk using small simulation boxes, the naturally occurring correlations need to be corrected. We show here that the CPT can be used to modify the elimination rates in the AKMC simulations, so as to yield point defect concentrations in agreement with SRT.

Influence of dislocation and dislocation loop biases on microstructures simulated by rate equation cluster dynamics

Sensitivity of microstructures under irradiation to absorption biases of dislocations and dislocation loops is investigated by rate equation cluster dynamics (RECD). Contrary to standard rate theory which is often used to estimate sensitivity of cavity growth to biases, it is possible with RECD to quantify the effect of biases on nucleation, growth and coarsening of loops and cavities. For this purpose simple and accurate expressions for dislocation and dislocation loop biases are derived, based on finite element calculations. They are used in a RECD code to quantify the effect of biases on microstructures of austenitic steels irradiated with electrons. It is shown that biases have a strong influence not only on the loop and dislocation network evolutions, but also on cavity nucleation. Sensitivity to some other parameters of the model is also discussed.

Computationally-efficient stochastic cluster dynamics method for modeling damage accumulation in irradiated materials

An improved version of a recently developed stochastic cluster dynamics (SCD) method (Marian and Bulatov, 2012) [6] is introduced as an alternative to rate theory (RT) methods for solving coupled ordinary differential equation (ODE) systems for irradiation damage simulations. SCD circumvents by design the curse of dimensionality of the variable space that renders traditional ODE-based RT approaches inefficient when handling complex defect population comprised of multiple (more than two) defect species. Several improvements introduced here enable efficient and accurate simulations of irradiated materials up to realistic (high) damage doses characteristic of next-generation nuclear systems. The first improvement is a procedure for efficiently updating the defect reaction-network and event selection in the context of a dynamically expanding reaction-network. Next is a novel implementation of the τ-leaping method that speeds up SCD simulations by advancing the state of the reaction network in large time increments when appropriate. Lastly, a volume rescaling procedure is introduced to control the computational complexity of the expanding reaction-network through occasional reductions of the defect population while maintaining accurate statistics. The enhanced SCD method is then applied to model defect cluster accumulation in iron thin films subjected to triple ion-beam (Fe3+, He+ and H+) irradiations, for which standard RT or spatially-resolved kinetic Monte Carlo simulations are prohibitively expensive.

Spectral Rate Theory for Two-State Kinetics.

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality. The reaction coordinate quality can be bounded from below by the directly computable observation quality, thus providing a measure allowing the reaction coordinate quality to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force-probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, focusing here on the case of two-state kinetics.

Stochastic cluster dynamics method for simulations of multispecies irradiation damage accumulation

Accurate modeling of irradiation damage processes is important to predict materials performance in nuclear environments. The mean-field rate theory (RT) approach has been and remains the standard method for calculations of the kinetics of damage accumulation under irradiation. Despite its many advantages, when using RT, very large numbers of ordinary differential equations (ODEs) need to be solved if one is interested in the kinetics of complex defect populations containing defect clusters made up of constituent defect species of different types, e.g. He, H, O, alloying impurities, etc. Here we present a stochastic variant of RT, which we term stochastic cluster dynamics (SCD), intended as an alternative to the standard ODE-based implementation. SCD obviates the need to solve the exceedingly large sets of ODEs and relies instead on sparse stochastic sampling from the underlying kinetic Master Equation. The new method evolves an integer-valued defect population in a finite material volume. In this paper we describe the general ideas behind the SCD method, give essential details of our numerical implementation and present SCD simulations that verify the new method against standard RT implementations on two well characterized material models. We then apply the method to simulate triple beam irradiation experiments of heavy Fe ions, He, and H on model ferritic alloys up to a dose of 1 dpa.

Theory of ultrafast photoinduced heterogeneous electron transfer

Ultrafast heterogeneous electron transfer (HET) between a molecule attached to a semiconductor surface and the conduction band of the semiconductor is discussed theoretically with emphasis on the perylene–TiO2 system. The used description accounts for the specialty of the molecule, i.e. its particular electronic level scheme together with its vibrational degrees of freedom. The band continuum of the semiconductor is included and the approach is ready to describe different optical excitation and detection processes. Using a diabatic-state like separation of the whole system into molecular and semiconductor states, femtosecond photoinduced dynamics are studied. Since the HET is ultrafast standard rate theories cannot be applied. Instead, the respective time-dependent Schrödinger equation governing the electron–vibrational wave function is solved. Based on this approach and using a time-dependent formulation, the steady state linear absorption is calculated. Parameters of perylene attached to nano-structured TiO2 via different bridge–anchor groups are adjusted by a comparison with measured spectra. A direct charge transfer excitation into the conduction band continuum is included into the description. This time-dependent formulation of the absorbance is confronted with a direct formulation in the frequency domain using the molecular Green's function. Then, it is explained how to observe the energetic distribution of the injected electron which carries signatures of the molecular vibrations in a two-photon photon emission spectrum. Some speculations on a laser pulse control of ultrafast HET are finally given.

Absence of saturation of void growth in rate theory with anisotropic diffusion

We present a first attempt at solution the problem of the growth of a single void in the presence of anisotropically diffusing radiation induced self-interstitial atom (SIA) clusters. In order to treat a distribution of voids we perform ensemble averaging over the positions of centres of voids using a mean-field approximation. In this way we are able to model physical situations in between the Standard Rate Theory (SRT) treatment of swelling (isotropic diffusion), and the purely 1-dimensional diffusion of clusters in the Production Bias Model. The background absorption by dislocations is however treated isotropically, with a bias for interstitial cluster absorption assumed similar to that of individual SIAs. We find that for moderate anisotropy, unsaturated void growth is characteristic of this anisotropic diffusion of clusters. In addition we obtain a higher initial void swelling rate than predicted by SRT whenever the diffusion is anisotropic.

On grain-size-dependent void swelling in pure copper irradiated with fission neutrons

Abstract The effect of grain size on void swelling has its origin in the intrinsic property of grain boundaries as neutral and unsaturable sinks for both vacancies and self-interstitial atoms. The phenomenon had already been investigated in the 1970s and it was demonstrated that the grain-size-dependent void swelling measured under irradiation producing only Frenkel pairs could be satisfactorily explained in terms of the standard rate theory (SRT) and dislocation bias. Experimental results reported in the 1980s demonstrated, on the other hand, that the effect of grain boundaries on void swelling under cascade damage conditions was radically different and could not be explained in terms of the SRT. In an effort to understand the source of this significant difference, the effect of grain size on void swelling under cascade damage conditions has been investigated both experimentally and theoretically in pure copper irradiated with fission neutrons at 623 K to a dose level of about 0.3 displacement per atom. ...

On recoil-energy-dependent defect accumulation in pure copper Part II. Theoretical treatment

Abstract Over the years, an enormous amount of experimental results have been reported on damage accumulation (e.g. void swelling) in metals and alloys irradiated under vastly different recoil energy conditions. Unfortunately, however, very little is known either experimentally or theoretically about the effect of recoil energy on damage accumulation. Recently, dedicated irradiation experiments using 2.5 MeV electrons, 3.0 MeV protons and fission neutrons have been carried out to determine the effect of recoil energy on the damage accumulation behaviour in pure copper and the results have been reported in part I of this paper (Singh et al., 2001, Phil. Mag. A, 80. 2629). The present paper attempts to provide a theoretical framework within which the effect of recoil energy on damage accumulation behaviour can be understood. The damage accumulation under Frenkel pair production (e.g. 2.5 MeV electron) has been treated in terms of the standard rate theory (SRT) model whereas the evolution of the defect micro...

Inhomogeneous Swelling Near Grain Boundaries in Irradiated Materials: Cavity Nucleation and Growth Saturation Effects

ABSTRACTThe effect of inhomogeneous nucleation and growth of cavities near grain boundaries illustrates the failure of the standard rate theory to describe the kinetics of phase transformations in irradiated materials under cascade damage conditions. The enhanced swelling observed near grain boundaries is believed to result from the competition between the diffusional growth of cavities and their shrinkage due to the interaction with mobile interstitial clusters. Swelling rates associated with the two processes behave in a radically different way as a function of the size of growing cavities. For a spatially homogeneous distribution of cavities this gives rise to the saturation of swelling in the limit of large irradiation doses.We investigate the evolution of the population of cavities nucleating and growing near a planar grain boundary. We show that a cavity growing near the boundary is able to reach a size that is substantially larger than the size of a cavity growing in the interior region of the grain. For a planar grain boundary the magnitude of swelling at maximum is found to be up to eight times higher than the magnitude of swelling in the grain interior.

Recent advances in the understanding of damage production and its consequences on void swelling, irradiation creep and growth

The consequences of displacement damage produced by energetic particles on physical and mechanical properties of metals and alloys have been investigated both experimentally and theoretically for several decades. Over the years, a number of theoretical models have been proposed to rationalize the rate and magnitude of defect accumulation under different irradiation conditions. In recent years, significant advances have been made in understanding the nature of the damage produced in this form of multi-displacement cascades. The new knowledge regarding the intra-cascade recombination and clustering of self-interstitial atoms and vacancies during the cooling-down phase of cascades makes it necessary to re-examine the appropriateness of the available models for describing the accumulation of damage under cascade damage conditions. In this paper, recent advances in the understanding of damage production and its consequences are reviewed. A historic perspective is adopted. A comprehensive analysis of the effects of temperature, dose rate and particle type on multi-phenomena (swelling, creep, growth, microstructure evolution, RED, RIS) is presented to discuss the strength and weakness of various models, as they have evolved with the understanding of the damage processes. It has been shown that the irradiation damage modeling has progressed from the standard rate theory model to the BEK model to the production bias model with an increasing degree of sophistication as increasingly more realistic features of the irradiation damage production process were incorporated. It is shown that the newly proposed production bias model uniquely includes the necessary features of cascade damage production in its treatment of the damage accumulation.

Cascade size effects on phase stability under irradiation: A stochastic description

Abstract Cascade size may affect phase stability under irradiation because of two distinct contributions: the replacement to displacement cross-section ratio depends on the deposited energy density; ballistic jumps which tend to disorder ordered compounds occur by bursts, while thermal jumps which restore long range order occur one by one. The latter effect cannot be handled by standard rate theory. A stochastic treatment of the problem, based on a Fokker Planck approximation of the relevant master equation is summarized. It is shown that the possible values of the long range order parameter under irradiation are not affected by the size of the cascade but that the respective stability of the former is cascade size sensitive. As a consequence, the stability diagram of phases under irradiation varies with the size of the cascades. A numerical example of this is given for the B2 structure.

Evaluation of irradiation-induced creep rate: application to the vacancy dislocation loop contribution

Irradiation (as in a nuclear reactor) drastically affects the defect structure and its time evolution in a material, and induces new creep mechanisms in it. We present a formalism to evaluate the contribution to creep owing to such mechanisms. Beginning with the phenomenological constitutive relation for the strain appropriate to a given mechanism, we put in simple statistical considerations to derive an expression for the corresponding creep rate. This formal expression is in terms of the defect production rate and a non-equilibrium probability distribution function involving the pertinent properties of the defect type concerned. A convenient approximation scheme for practical calculations is employed, that also makes contact with standard rate theory and provides a proper interpretation for the variables occurring there. As an illustration, we evaluate the contribution to irradiation-induced creep from the orientation-dependent shrinkage of vacancy dislocation loops in an applied stress field. The circumstances inducing transient and non-transient creep are clarified and a numerical estimate is given for the latter component.

Diffusion to a random array of identical spherical sinks

The effect of the insertion of one additional spherical sink on the steady state diffusion field of particles migrating to a random array of identical sinks of the same type is determined. The change in particle concentration as a function of the distance from the inserted sink, when averaged over all permissible configurations of the original array, is shown to be of the same form as that arising from insertion of a sink of the same type in a homogeneous lossy medium. The loss rate per atom in this effective medium is found to be of the form DkM2(c̃ (r) − c̄), where D is the particle diffusivity, c̃(r) an ensemble-averaged concentration (atomic fraction) defined in the text, and c̄ is the prescribed concentration at each sink surface. The sink strength, kM2, is calculated to be kM2 = λ2 {1 + λa − λ2a2 (16 + γ + ln 6λa)}, with λ2 = 4πCa and γ Euler's constant, C being the average volume concentration of sinks, a their common radius. These results are shown to substantiate the standard rate theory treatment of simultaneous diffusion to many sinks and confirm the existence of correction terms in λa to the sink strength normally employed in this method. L'effet de l'insertion d'un piège additionnel de forme sphérique sur le champ de diffusion à l'état stationnaire de particules en migration à travers un arrangement aléatoire de pièges identiques de même type est discuté. Il est montré que la variation de concentration des particules en fonction de la distance au piège inséré, quand on fait la moyenne de toutes les configurations possibles de l'arrangement initial, est de la même forme que celle provenant de l'insertion d'un piege de même type dans un milieu homogène. Le taux de perte par atome dans ce milieu effectif a été trouvé égal àDkM2(c̃ (r) − c̄), oú D est la diffusivité des particules, c̃(r) une concentration moyenne (fraction atomique) définie dans le texte et c̄ est la concentration prescrite pour chaque surface de piège. La force du piège, kM2, est calculée comme étant égale à λ2 [1 + λa − λ2a2 (16 + γ + ln 6λa)avec λ2 = 4πCa et λ la constante d'Euler, C étant la concentration en volume moyenne des pièges, a leur rayon commun. Ces résultats sont présentés comme confirmant le traitement théorique classique de diffusion simultanée vers de nombreux pièges et l'existence de termes correctifs dans λa pour la force des pièges normalement utilisée dans cette méthode. Es wurde der Einfluss des Einbaus einer zusätzlichen sphärischen Senke auf das stationäre Diffusionsfeld von Teilchen untersucht, die zu statistisch angeordneten identischen Senken desselben Typs wandern. Die Änderung der Teilchenkonzentration als Funktion des Abstandes von der eingebauten Senke hat bei einer Mittelung über alle erlaubten Konfigurationen der ursprünglichen Anordnung dieselbe Form wie jene, die vom Einbau einer Senke derselben Art in einem Medium mit homogen verschmierten Senken herrührt. Die Verlustrate pro Atom in diesem effektiven Medium hat die Form DkM2(c̃ (r) − c̄), wobei D der Teilchendiffusionskoeffizient, c̃(r) eine im Text definierte gemittelte Konzentration einer Einheit als Atombruch und c̄ die vorgeschriebene Konzentration an jeder Oberfläche der Senken ist. Die Stärke kM2 der Senke wird angegeben als kM2 = λ2 {1 + λa − λ2a2 (16 + γ + ln 6λa)}, wobei λ2 = 4πCa und γ die Eulersche Konstante ist; C ist die mittlere Volumenkonzentration und a der gemeinsame Radius der Senken. Die Ergebnisse erhärten die klassische reaktionskinetische Behandlung einer gleichzeitigen Diffusion zu vielen Senken und bestätigen die Existenz von λa-Korrekturtermen für die Stärke der Senke, die bei dieser Methode normalerweise verwendet wird.