Coincidence lattice correlation function

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

For a perfect crystal, all lie on a perfect lattice, and therefore the correlation function Tz(r) is nonzero only when r coincides with the lattice. However, for crystals with imperfections, rjk are subject to statistical fluctuations, and the correlation function rz(r) described by (3.35) is no longer strictly on a lattice but is smeared out, as illustrated in Figure 3.13. Note that in the imperfection of the first kind the autocorrelation function is smeared out equally at every lattice point except at the origin, but in the imperfection of the second kind the degree of smearing of the autocorrelation function becomes more severe as the distance from the origin is increased. 

As pointed out by Koehler (1968), even the solution obtained from these modified force constants is not the most meaningful one, because it is obtained from the harmonic part only of the wavefunction (3.22). He shows that the complete dispersion curves are not obtained from the force constant matrix alone, but first it has to be transformed by a unitary matrix diagonalizing the matrix of second derivatives of the correlation function/ . For q O, however, the frequencies coincide and (3.27) is sufficient for the calculation of lattice frequencies. 

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