We review in this article the current theoretical understanding of collective and single particle diffusion on surfaces and how it relates to the existing experimental data. We begin with a brief survey of the experimental techniques that have been employed for the measurement of the surface diffusion coefficients. This is followed by a section on the basic concepts involved in this field. In particular, we wish to clarify the relation between jump or exchange motion on microscopic length scales, and the diffusion coefficients which can be defined properly only in the long length and time scales. The central role in this is played by the memory effects. We also discuss the concept of diffusion under nonequilibrium conditions. In the third section, a variety of different theoretical approaches that have been employed in studying surface diffusion such as first principles calculations, transition state theory, the Langevin equation, Monte Carlo and molecular dynamics simulations, and path integral formalism...

Collective and single particle diffusion on surfaces

We review in this article the current theoretical understanding ofcollective and single particle diA usion on surfaces and how it relates to the existing experimental data. We begin with a brief survey of the experimental techniques that have been employed for the measurement of the surface diA usion coeA cients. This is followed by a section on the basic concepts involved in this ®eld. In particular, we wish to clarify the relation between jump or exchange motion on microscopic length scales, and the diA usion coeA cients which can be de®ned properly only in the long length and time scales. The central role in this is played by the memory eA ects. We also discuss the concept of diA usion under nonequilibrium conditions. In the third section, a variety of diA erent theoretical approaches that have been employed in studying surface diA usion such as ®rst principles calculations, transition state theory, the Langevin equation, Monte Carlo and molecular dynamics simulations, and path integral formalism are presented. These ®rst three sections form an introduction to the ®eld of surface diA usion. Section 4 contains subsections that discuss surface diA usion for various systems which have been investigated both experimentally and theoretically. The focus here is not so much on speci®c systems but rather on important issues concerning diA usion meas

Collective and single particle diÄ usion on surfaces

Mode-coupling theory is an approach to the study of complex behavior in the supercooled liquids which developed from the idea of a nonlinear feedback mechanism. From the coupling of slowly decaying correlation functions the theory predicts the existence of a characteristic temperature ${T}_{c}$ above the experimental glass transition temperature ${T}_{g}$ for the liquid. This article discusses the various methods used to obtain the model equations and illustrates the effects of structure on dynamics and scaling behavior over different time scales using a wave-vector-dependent model. It compares the theoretical predictions, experimental observations, and computer simulation results, and also considers phenomenological extensions of mode-coupling theory. Numerical solutions of the model equations to study the dynamics from a nonperturbative approach are also reviewed. The review looks briefly at recent observations from landscape studies of model systems of structural glasses and their relation to the mode-coupling temperature ${T}_{c}$. The equations for the mean-field dynamics driven by the $p$-spin interaction Hamiltonian are similar to those of mode-coupling theory for structural glasses. Related developments in the nonequilibrium dynamics and generalization of the fluctuation-dissipation relation for the structural glasses are briefly touched upon. The review ends with a summary of the open questions and possible future direction of the field.

Mode-coupling theory and the glass transition in supercooled liquids

There are at least three fundamental states of matter, depending upon temperature and pressure: gas, liquid, and solid (crystal). These states are separated by first-order phase transitions between them. In both gas and liquid phases a complete translational and rotational symmetry exist, whereas in a solid phase both symmetries are broken. In intermediate phases between liquid and solid, which include liquid crystal and plastic crystal phases, only one of the two symmetries is preserved. Among the fundamental states of matter, the liquid state is the most poorly understood. We argue that it is crucial for a better understanding of liquids to recognize that a liquid generally has the tendency to have a local structural order and its presence is intrinsic and universal to any liquid. Such structural ordering is a consequence of many-body correlations, more specifically, bond angle correlations, which we believe are crucial for the description of the liquid state. We show that this physical picture may naturally explain difficult unsolved problems associated with the liquid state, such as anomalies of water-type liquids (water, Si, Ge, ...), liquid-liquid transition, liquid-glass transition, crystallization and quasicrystal formation, in a unified manner. In other words, we need a new order parameter representing a low local free-energy configuration, which is a bond orientational order parameter in many cases, in addition to a density order parameter for the physical description of these phenomena. Here we review our two-order-parameter model of liquid and consider how transient local structural ordering is linked to all of the above-mentioned phenomena. The relationship between these phenomena is also discussed.

Bond orientational order in liquids: Towards a unified description of water-like anomalies, liquid-liquid transition, glass transition, and crystallization: Bond orientational order in liquids.

The α-relaxation behaviour of polymers and glass forming viscous liquids is well described by the Vogel-Fulcher-Tammann equation, introducing a Vogel temperature T0 at which the relaxation time τα diverges (T0 ≈ Tg - 50 K). With the recent development of the mode coupling theory a new critical temperature Tc is introduced located about 50 to 80 K above Tg. The relevance of the various temperatures is discussed on the basis of dynamic light scattering studies and dielectric relaxation data. The dynamic and static light scattering experiments revealed some unexpected features, which cannot be explained on the basis of conventional liquid state theories: (1) In static light scattering the intensity I(q→0) is no longer proportional to the isothermal compressibility. (2) This excess scattering Iexc shows a strong q-dependence (q = (4πn/λ)sin(θ/2)) corresponding to a correlation length ξ in the range of 20–200 nm. (3) The Landau-Placzek ratio IRayleigh/2IBrillouin is much too high, compared with the results of light scattering theories. (4) In photon correlation spectroscopy a new ultraslow hydrodynamic mode (Γ∼q2) is detected with relaxation rates Γ about 10-4 to 10-7 lower than those of the α-process at a given temperature.

These effects are caused by long density fluctuations indicating a nonhomogeneous distribution of free volume. The redistribution of free volume in space causes the new ultraslow mode. A tentative model is proposed which describes the long range density fluctuations as a result of the coexistence of molecules with two different dynamic states, which show up in the Fabry-Perot and Raman spectroscopy.

Light scattering and dielectric studies on glass forming liquids

The influence of frozen-in topological defects in a crystal on the long-wavelength quantum states of a particle is considered. In the continuum limit of a conveniently defined tight-binding model one is led to a covariant Schr\"odinger equation on a Riemann-Cartan manifold. When the tight-binding transfer energies are assumed to depend on the local lattice deformations caused by the defects, additional noncovariant terms are generated in the Hamiltonian. These terms generate bound states of the particle to edge dislocations and enhance the scattering of particles on screw dislocations.

Single-Particle Quantum States in a Crystal with Topological Defects

We present a theoretical description of conduction electrons interacting with a domain wall in ferromagnetic metals. The description takes into account the interaction between electrons. Within the semiclassical approximation we calculate the spin and charge distributions, particularly their modification by the domain wall. In the same approximation we calculate the local transport characteristics, including relaxation times as well as charge and spin conductivities. It is shown that these parameters are significantly modified near the wall and this modification depends on the electron-electron interaction. The spatial nonuniformity of the transport characteristics may give rise to new phenomena.

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Electrons in a ferromagnetic metal with a domain wall

A previously established general framework for the description of long-wavelength quantum states of electrons in a crystal with topological defects is used to discuss the scattering of electrons on a screw dislocation. The corresponding Schr\"odinger equation contains contributions of the type of a vector potential as well as of a repulsive scalar potential. Together they give rise to modified Aharonov-Bohm interferences in the scattering amplitude, for which the far-field expression is calculated exactly.

Scattering of electrons on screw dislocations

For the description of the long-wavelength quantum states of electrons in crystals with topological defects we establish a general framework on the basis of the continuum theory of defects. The resulting one-particle Schrodinger equation lives on a Riemann-Cartan manifold, representing the distorted crystal, and consists of two parts. All effects due to the topology of the defects are contained in the first part which has a covariant form and describes the purely geometric particle motion. The second part is non-covariant and derives from deformation-dependent tunneling rates of the particle which e.g. are responsible for the existence of bound states to edge dislocations.

Łukasz A. Turski

Papers

Single-Particle Quantum States in a Crystal with Topological Defects

Electrons in a ferromagnetic metal with a domain wall

Scattering of electrons on screw dislocations

Quantum motion of electrons in topologically distorted crystals

Collective diffusion on hexagonal lattices: repulsive interactions