(2.5) represents a static state and gives us only the statistical distribution of configuration as a function of composition. In other words, the configuration probability as given in Eq. However, the configuration probability thus derived is based solely on the Gibb's canonical ensemble concept in which the “system-average” is equated to the “time-average.” Therefore, strictly speaking, the configuration probability we have derived does not truly represent the liquid state in terms of local atomic configurations that are subject to a continual fluctuation. We have shown that certain atomic configurations are more probable than others for a given composition through the configuration probability, P AB nn′(β). Prior to a discussion on the correlations, (d) and (e) relating crystal structure and its mode of formation-whether congruently or incongruently melting, we must examine the following facts. The effect of atomic size difference on P AB nn'(β) is to shift β m toward the side with the larger atomic radius and the magnitude of shift is proportional to the atomic radius ratio. Since the first peak in the ρ(r) of having B (atom with larger radius) at the origin occurs at d AB (which is the limit of our interest), no such complication exists and no modification is required of W n′(α).įig. With this correction, the probability of A being surrounded by n B's is modified to read W n ∗(β′) = (β′) n − 1 ⋅ α′. As a first approximation, we shall correct the probability of finding one B in a melt of composition β by setting β′ = β (d AA/d AB) and α’ = 1 − β′. Therefore, W n(β), describing the probability of A being surrounded by n B's requires modification. This is to say that A will “see” fewer B's than are actually present in the system of composition β. Inasmuch as d AA is shorter than d AB, the integrated area up to d AB in the case of having A at the origin will contain more area due to peak, d AA, than due to peak, d AB. This difference in the ρ(r) can be related to the configuration probability by the following reasoning. On the other hand, when R A/R B ≠ 1, the ρ(r) will be different ( Fig. It is clear that when R A/R B = 1, the ρ(r) for having A or B at the origin are essentially the same ( Fig. This effect can best be seen by comparing the ρ(r)'s-the radial distribution functions (RDFs)-in the range 0 < r < d AB (where d AB = R A + R B), for both R A/R B = 1 and R A/R B ≠ 1. We shall now examine the effect of atomic size difference between A and B on the P AB nn ′( β) curves.
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