Atom jumping correlated

Changes in the atomic correlations are enabled by atomic jumps between neighbouring lattice sites. In metals and their substitutional solutions point defects are responsible for these diffusion processes. Ordering kinetics can therefore yield information about properties of the point defects which are involved in the ordering process. 

Carlo-simulations for LI2 superlattice including saddle-point energies for atomic jumps in fact yielded two-process kinetics with the ratio of the two relaxation times being correlated with the difference between the activation barriers of the two sorts of atom. 

Figure 5.17 Correlated motion during vacancy diffusion (a) vacancy can jump to any surrounding position and its motion follows a random walk (b, c) the motion of a tracer atom is correlated, as a jump into a vacancy (b) is most likely to be followed by a jump back again (c).

The high conductivity of (3-alumina is attributed to the correlated diffusion of pairs of ions in the conduction plane. The sodium excess is accommodated by the displacement of pairs of ions onto mO sites, and these can be considered to be associated defects consisting of pairs of Na+ ions on mO sites plus a V N l on a BR site (Fig. 6.12a and 6.12b). A series of atom jumps will then allow the defect to reorient and diffuse through the crystal (Fig. 6.12c and 6.12d). Calculations suggest that this diffusion mechanism has a low activation energy, which would lead to high Na+ ion conductivity. A similar, but not identical, mechanism can be described for (3"-alumina. 

Fig. 4.7 (a) Atomically resolved (1 x l)Pt surface prepared by low temperature field evaporation. (b) and (c) show how atoms jump by pulsed-Iaser heating of the surface to —350 K for 5 ns. Atoms in a small [110] atomic row tend to jump together, or in a highly correlated way. 

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. 

There are two possible directions for both the atoms in both the first and second atomic jumps. If the jumping direction is completely random and the two atoms have the same probability of performing a jump, then these atomic jumps are said to be uncorrelated. A correlation factor, /, has been introduced for the two atomic jumps, which is defined as the extra probability that the atom making the first jump will also make the second jump in the forward direction. The rest of the probability, (1 — /), is then shared equally for either of the two atoms jumping in either of the two directions. Two experimental displacement distributions measured at 299 K and 309 K fit best with a Monte Carlo simulation with / = 0.1 and /=0.36, respectively. The correlation factor increases with diffusion temperature as can be expected. It is interesting to note that when/= 1, only a and steps can occur. 

Correlation diminishes the effectiveness of atomic jumps in diffusional random motion. For example, when an atom has just moved through site exchange with a vacancy, the probability of reversing this jump is much higher than that of making a further vacancy exchange step in one of the other possible jump directions. Indeed, if z is the coordination number of equivalent atoms in the lattice, the fraction of ineffective jumps is approximately 2/z (for sufficiently diluted vacancies as carriers) [C. A. Sholl (1992)]. 

An important assumption in this theory is that there is no correlation between successive jumps and this is generally a good assumption at low H concentrations. However, at larger concentrations, this is not strictly true because correlation effects become significant, i.e. if an atom jumps from a filled site to an empty one, the site vacated is, at this instant, empty whereas the other sites are occupied with a probability c where c is the overall fractional occupation, so that the chance of jumping back is enhanced. This effect was first considered by Ross and Wilson [37] who showed by Monte Carlo simulation that, at finite concentrations, the quasi-elastic peak deviates from the Lorentzian shape. This was the first example of the need to resort to Monte Carlo simulation of the diffusion process to obtain G (r, t) in situations where the diffusion process becomes significantly complicated. This is likely to be important in efforts to understand the diffusive process in complex hydride stores. 

Here Tc is the dipolar correlation time, Av is the observed linewidth, and Avq is the rigid lattice linewidth measured at a temperature low enough that diffusion effects are no longer detectable. For Aptemperature dependence of Ap and Tc is normally set equal to the atomic jump time on the grounds that an atomic jump from one site to another alters the dipolar interaction between neighboring spins substantially. Since the atomic jump frequency is expected to obey an Arrhenius relation [Zener (1952)], Tc should also show such a dependence on temperature. The n values obtained from line-narrowing measurements have usually been fit to the relation... 

There is not sufficient experimental evidence to continue this discussion quantitatively at the present time, but the sparse experimental data suggests that for a given compound, the D0 value is significantly lower than is the case in simple metals. This decrease may be attributed to a low value in the correlation factor which measures the probability that an atom may either move forward or return to its original site in its next diffusive jump. In simple metals this coefficient has a value around 0.8. 

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. 

The correlation factor, for any mechanism, is given by the ratio of the values of the mean square displacement of the atom (often the tracer) moving in a correlated motion to that of the atom (or vacancy) moving by a random-walk process. If the number of jumps considered is large, the correlation factor/can be written as... 

A reasonable approximation for the pair correlation function of the j8-process may be obtained in the following way. We assume that the inelastic scattering is related to imcorrelated jumps of the different atoms. Then all interferences for the inelastic process are destructive and the inelastic form factor should be identical to that of the self-correlation function, given by Eq. 4.24. On... 

Figure 14 Mechanism of oxygen elimination from the structure of BajYC Oy 0. (a) As an effect of temperature increase, atom A may jump into position A (b) As a consequence of this shift, atom B may jump into position B, C into C etc., thus causing a correlated motion of the oxygen atoms terminating with the expulsion of half oxygen atom from the structure (c) Atom C now may jump into positions C or C" generating a second cascade...

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