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Notes: Diffusion on bcc (110) of dimers made up of two chemically different atoms is examined for a model in which motion occurs by transitions from a ground state, in which two atoms are in nearest-neighbor sites along 〈111〉, to an intermediate state with the atoms in adjacent sites along either 〈100〉 (horizontal) or 〈110〉 (vertical transition). In this representation, diffusion can be viewed as a random walk of the center of positions (COP) of the dimer atoms over a plane grid of stable and intermediate sites. Starting from the Kolmogoroff equation, a detailed stochastic analysis is carried out for horizontal transitions. This yields the mean-square displacement in terms of four elementary jump processes of the dimer; in addition, the moment generatingfunction and higher moments of the dimer displacements along the Cartesian coordinates x and y are also obtained. These quantities are found to differ for the two axes quite apart from differences in length scales: relaxation to the long-time limits occurs at different rates. For systems in which the population of dimers in the intermediate state is negligible, only the rates a and c at which the two different dimer atoms jump to the intermediate state affect the moments significantly; however, these rates are not separately accessible from observations of the mean-square displacement. The probability distribution for dimer displacements over (110) is therefore worked out explicitly. Displacements are referenced to coordinate axes χ and η along the close-packed 〈111〉 direction, along which COP motion is assumed tooccur. On this χ–η grid, the dimer executes a novel random walk, in which transitions from one axis to another are allowed only at intermediate-state positions; at ground-state positions, motion is restricted to one dimension. For short diffusion intervals, the probability distribution is found to be sensitively dependent on the jump rates a and c. The asymmetry of the distribution of displacements about 〈100〉 provides an immediate qualitative indication of any difference in the rates at which the chemically different dimer atoms jump out of the stable configuration. The actual value of these rates can be obtained from a quantitative comparison with experimentally measured distributions. Dimer diffusion involving vertical transitions, through an intermediate oriented along 〈110〉, is also considered, but more briefly. It is shown that the sum of the two rate constants for jumps into thisintermediate can be derived from observations of the rate at which flips occur from one orientation to another. The mean-square displacement under steady-state conditions is demonstrated to have the same general form as for horizontal transitions alone, but with an additional term for the effective jump rate out of the ground state via this new channel. Finally, the general formalism is adapted to the diffusion of simple dimers, of chemically identical atoms, both on bcc (110) and (100). For simple dimers which are not found in the intermediate state to any perceptible extent, a single jump rate suffices to define the mean-square displacement as well as the distance distribution function.

Notes: Diffusion on surfaces has in the past been modeled as a random walk in continuous time between nearest-neighbor sites on a lattice. In order to allow tests for the possible participation of long jumps in actual diffusion processes, we examine the properties of random walks in which transitions are not limited to jumps between nearest-neighbor sites. Two features of such walks are of special interest: (a) the moments of the displacements, which are directly related to the diffusivity and the statistical uncertainties in its determination; (b) the distribution function governing the probability of displacements, which is an important indicator of the contributions from long jumps. The techniques used to develop expressions for these quantities are illustrated for random walks in one dimension, with transitions allowed between neighbors up to three spacings removed. The appropriate probability generating function is derived starting from the Kolmogoroff equation. This is then manipulated to yield both the moments and the distribution of displacements in terms of the jump rates to first-, second-, and higher-nearest neighbors. Inasmuch as this approach to the distance distribution is not universally feasible, the latter is also evaluated directly as a combinatorial problem for diffusion by single and double jumps. Generating function techniques are then used to describe two-dimensional diffusion with contributions from long jumps on the (110) plane of the bcc lattice. Atomic jumps of different lengths are allowed along the close-packed directions, and jumps along the Cartesian coordinates are considered as well. The most challenging problem is to describe diffusion on the (111) plane of the fcc lattice, on which atoms can be bound at two different types of positions, one surrounded by the other, so that double jumps are no longer uncorrelated with single jumps. Lower moments of the displacements are developed from rather complicated generating functions, but combinatorial methods have to be used to derive the complete distribution function governing displacements.

Notes: Information about pair interactions between adatoms at specified sites on a surface can be readily obtained from the probability P(R) of finding two atoms at a separation R at equilibrium on a surface at temperature T. Observations of two adatoms in such a system at different separations, using the field ion microscope, yield a map of the free energy of interaction F(R), inasmuch as the probability P(R) is proportional to exp[−F(R)/kT]. Only pairs of chemically different metal adatoms have so far been examined in this way, and measurements have therefore been extended to chemically similar atoms. Observations with two iridium adatoms in many different configurations on W(110) reveal behavior similar to that previously found with Re–Pd and W–Pd: interactions extending over long distances, ≥10 A(ring), showing a pronounced orientational anisotropy. With the iridium pair oriented along the close packed [11¯1] direction, interactions are generally attractive at close distances; along [001] and [11¯0], repulsion is the rule. However, the strength of the attractions is greater than in the heteropairs previously studied. The behavior of rhenium atoms is quite different, in that pair interactions are repulsive at short distances. However, at larger ranges, interactions become attractive and are again dependent upon the orientation of the pair on the surface. The extent to which three-atom effects are important in cohesion has been tested for both iridium and rhenium by comparing the experimentally measured dissociation energy of trimers with values of the cohesion estimated from pair interactions. Many-atom contributions turn out to play a significant role in cohesion, not just for rhenium but also for iridium trimers.

Notes: Quantitative information about the range and strength of interactions between adatoms on surfaces is still scarce. However, through the use of the field ion microscope it is possible to pinpoint the location of individual metal atoms on a crystal, and therefore to establish by direct observation the probability P(R) of finding two atoms at sites separated from each other by the vector R. If equilibrium is established at the surface, this probability is related to F(R), the free energy of interaction, by P(R)∝exp[−F(R)/kT]; measurements of the probability therefore provide a quantitative estimate of the interaction between adatoms at specified sites on the surface. Such measurements have been carried out with two metal atoms on the most densely packed plane of tungsten, the (110). To minimize the amount of data taking, initial studies have been made on the distribution of an adatom (Pd) mobile at the temperature of equilibration around a chemically different metal atom (either Re or W) stationary at the center of the W(110) plane. These observations reveal a remarkable anisotropy. Palladium atoms are most frequently found at the nearest-neighbor position to the central atom, but the close by second and third-nearest neighbor sites are not populated. It is shown that this is due to a strong orientational dependence of interactions. Quantitative values for the free energy of interaction with the central adatom have been determined for Pd atoms at different sites on the surface. Along the close-packed direction [11¯1], interactions are attractive out to ≈10 A(ring), and show an oscillatory dependence upon distance; along [001] and [11¯0] they tend to be repulsive for both Re and W as the central atom. These effects are so far not predicted by theory.

Notes: The distribution of condensing metal atoms over the two types of sites present on an atomically smooth Ir(111) has been measured in a field ion microscope. For Ir, Re, W, and Pd from a thermal source, condensing on Ir(111) at ≈20 K, the atoms are randomly distributed, as expected if they condense at the first site struck.

Notes: Measurements with the field ion microscope have been made of the distribution function governing the displacements of a single metal atom diffusing on W(211). It is shown that by comparing such measurements with the predictions of random walk models, the contributions of jumps of different lengths to the overall diffusion can be ascertained, even if diffusion is dominated by transitions between adjacent sites. Extensive observations for rhenium, as well as for molybdenum atoms on W(211) are best represented by a symmetric random walk between nearest-neighbor sites, without any contribution from longer jumps. For iridium and rhodium atoms, however, double jumps do appear to participate in the diffusion, but amount to only ≈3% of the total.

Notes: Diffusivities have for the first time been quantitatively measured for two-dimensional clusters of iridium atoms on a metal surface, Ir(111). As cluster size increases from two to seven atoms, the diffusivity generally diminishes, but tetramers are an exception. It is shown that their high mobility comes about because of weak interatomic binding, and a low activation energy for diffusion, rather than from a change in the mechanism of motion, which involves displacements of individual adatoms independent of cluster size.