Mean-square displacement surface atoms

Vibrations in the surface plane, however, will be rather similar to those in the bulk because the coordination in this plane is complete, at least for fee (111) and (100), hep (001) and bcc (110) surfaces. Thus the Debye temperature of a surface is lower than that of the bulk, because the perpendicular lattice vibrations are softer at the surface. A rule of thumb is that the surface Debye temperature varies between about 1/3 and 2/3 of the bulk value (see Table A.2). Also included in this table is the displacement ratio, the ratio of the mean squared displacements of surface and bulk atoms due to the lattice vibrations [1]. 

Finally, if the temperature increases, becomes larger until the crystal melts. The Lindemann criterion predicts that melting sets in when becomes about 0.25 a2, where a is the interatomic distance of the metal. Because the mean squared displacements of surface atoms is higher we expect that the surface melts at lower temperatures than the bulk does [2]. Indeed, evidence has been presented that the (110) surface of lead starts to melt at 560 K whereas the bulk melting temperature is about 600 K [13]. 

Surface diffusion can be studied with a wide variety of methods using both macroscopic and microscopic techniques of great diversity.98 Basically three methods can be used. One measures the time dependence of the concentration profile of diffusing atoms, one the time correlation of the concentration fluctuations, or the fluctuations of the number of diffusion atoms within a specified area, and one the mean square displacement, or the second moment, of a diffusing atom. When macroscopic techniques are used to study surface diffusion, diffusion parameters are usually derived from the rate of change of the shape of a sharply structured microscopic object, or from the rate of advancement of a sharply defined boundary of an adsorption layer, produced either by using a shadowed deposition method or by fast pulsed-laser thermal desorption of an area covered with an adsorbed species. The derived diffusion parameters really describe the overall effect of many different atomic steps, such as the formation of adatoms from kink sites, ledge sites... 

Adatom diffusion, at least under the low temperature of field ion microscope measurements, almost always follows the direction of the surface channels. Thus adatoms on the W (112) and Rh (110) surfaces diffuse in one direction along the closely packed atomic rows of the surface channels. Such one-dimensional surface channel structures and random walks can be directly seen in the field ion images, and thus the diffusion anisotropy is observed directly through FIM images. Unfortunately, for smoother surfaces such as the W (110) and the fee (111), no atomic or surface channel structures can be seen in field ion images. But even in such cases, diffusion anisotropy can be established through a measurement of the two-dimensional displacement distributions, as discussed in the last section. Because of the anisotropy of a surface channel structure, the mean square displacements along any two directions will be different. In fact this is how diffusion anisotropy on the W (110) surface was initially found in an FIM observation.120... 

Atomic jumps in random walk diffusion of closely bound atomic clusters on the W (110) surface cannot be seen. A diatomic cluster always lines up in either one of the two (111) surface channel directions. But even in such cases, theoretical models of the atomic jumps can be proposed and can be compared with experimental results. For diffusion of diatomic clusters on the W (110) surface, a two-jump mechanism has been proposed by Bassett151 and by Cowan.152 Experimental studies are reported by Bassett and by Tsong Casanova.153 Bassett measured the probability of cluster orientation changes as a function of the mean square displacement, and compared the data with those derived with a Monte Carlo simulation based on the two-jump mechanism. The two results agree well only for very small displacements. Tsong Casanova, on the other hand, measured two-dimensional displacement distributions. They also introduced a correlation factor for these two atomic jumps, which resulted in an excellent agreement between their experimental and simulated results. We now discuss briefly this latter study. 

Arnold and Hobert (231) studied the chemisorption of ferrocene, (C5H5)2Fe, on a silica surface from an alcohol solution. Ferrocene itself shows a symmetric quadrupole splitting. After chemisorption this doublet is no longer symmetric, and the authors explain this in terms of a Gol danskii-Karyagin effect, where the iron atoms in the adsorbed state have a larger mean square displacement perpendicular to the silica surface than parallel to it. 

The mean-square displacement of surface atoms should be sensitive to changes in the number and type of neighboring atoms. The adsorption on the clean surface of gases that chemically interact with the surface atoms (e.g., oxygen on nickel or tungsten) strongly affects the vibrational amplitude of surface atoms. 

Surfaces are heterogeneous on the atomic scale. Atoms appear in flat terraces, at steps, and at kinks. There are also surface point defects, vacancies, and adatoms. These various surface sites achieve their equilibrium surface concentrations through an atom-transport process along the surface that we call surface diffusion. Adsorbed atoms and molecules reach their equilibrium distribution on the surface in the same way. This view of surface diffusion as a site-to-site hopping process leads to the random-walk picture, in which the mean-square displacement of the adsorbed particle along the. r-component of the coordinate is given by... 

The harmonic-oscillator and elastic-continuum models can be used to explain the presence of surface phonons (Rayleigh waves and localized surface modes of vibration) and the larger mean-square displacement of surface atoms compared to that of atoms in the bulk. 

Let us consider the surface contact on an atomic scale. A surface atom is bound to its neighbors by strong chemical bonds that add up to 150 kcal/mole (600 kJ/mole). We would like to estimate the mechanical force necessary to break these bonds. If the atom behaves as a harmonic oscillator, stretching its bonds by 0.1 A (10 " nm) about its equilibrium position—five times its root-mean square displacement of =0.02 A (2 X 10 nm)—would certainly break the bonds. Thus, the force... 

From a conceptual standpoint, it is useful to have an understanding of the time scales for motions of particles near metal-water interfaces, to be able to better understand their nature, as well as how molecules and atoms near these interfaces differ from those of the bulk. The two most commonly calculated dynamic properties for metal surfaces are the mean-square displacement and the velocity autocorrelation functions, because these can be used to calculate diffusion constants and spectra. 

The latter technique was also applied for demonstrating the equality of Dx and D (in the absence of adsorbate-adsorbate interactions) with N atoms diffusing on a Ru(0 001) surface [40]. Exclusive dissociation of NO at 300K at monoatomic steps creates a quasi 6-function of Nad concentration at f = 0. Determination of their mean square displacement away from the step as a function of time leads to a straight line whose slope yields D = (3.4 0.4) x lO cm /s (Fig. 1.10). The solution of Pick s second law for the indicated initial concentration profile (where the N atoms do not cross the step) yields a Gaussian of the form n x,t) = NAx/y/nD t where n(x, t) is the number of... 

FIGURE 1.10. Diffusion of N atomsonaRu(0 001) surface at 300K mean square displacement of N atoms from a monoatomic step where they were created by dissociation of NO as a function of time. The inset shows the measured distribution of distances from the step after 2 h (points) and a Gaussian with adjusted parameters (full line) [40]. 

Surface phonon bands along symmetry lines of the SBZ are given for fee metals in Figs. 5.2-49-5.2-55 and in Table 5.2-20. In all figures the horizontal axis is the reduced wave vector, expressed as the ratio to its value at the zone boundary. Table 5.2-21 gives the surface Debye temperatures for some fee and bcc metals, as well as the amplitudes of thermal vibrations of atoms in the first layer p as compared with those of the bulk pb-In the harmonic approximation, the root mean square displacement of the atoms is proportional to the inverse of the Debye temperature. 

The lattice-vibration instability model [40, 55-59] extends Lindemann s vibrational-lattice instability criterion [60]. The melting behavior of a nanosolid is related to the ratio (/ ) of the root-mean-square displacement (RMSD, of an atom at the surface to the RMSD of an atom inside a spherical dot. p is a size-independent parameter ... 

Surface diffusion establishes mass transfer along concentration gradients, and it also refers to the random walk of a constant concentration of diffusing species without any net flux of mass. The first case is called mass transfer, and the second is called intrinsic diffusion [7]. For mass transfer diffusion, the concentration of random walkers n changes with temperature, location, and time, as particles are suppHed from sources and consumed by sinks. The sources and sinks most often are kinks at atomic steps, but they may also be screw dislocations and even flat terraces where adatoms or vacancies can be created. The atomic processes associated with the sources and sinks and the mean square displacement between equivalent sites are all thermally activated, and therefore their respective rate is given by a Boltzmann term with an energy barrier and a preexponential factor. One defines the diffusion coefiicient as the area traveled per time... 

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