Results of Clustering Analyses

The development leading to (9.47) is extremely broad, and the notation of (9.48) is correspondingly broad. An important question is just what is meant precisely by notations such as yi and specifically ((e-/3zic/ w =o)) A development 

The quantities averaged on the right here are the Boltzmann factor for the binding energy, as is usual for the PDT, but multiplied by the indicator function for the event that there are zero occupants of the defined inner shell. 

Notice that the quantity of (9.49) is the right-most factor of (9.34), and also appears in the denominators of the equilibrium ratios (9.48). Thus, that factor in (9.34) cancels precisely the same factor in the denominator of contributions from (9.48). This observation gives some perspective on how the basic quasichemical formula (9.34) is built, but does not address the leading term in (9.34) of 1 , which does not have such a denominator. But in multiplying out of (9.34) we see that the term (9.49) is the correct contribution to the PDT formula for the case m = 0. Again, this observation serves to explain how the equilibrium ratios and the outer-shell contribution come together to formulate the general result. 

If we simplify the sum of (9.34) to include only the maximum term, the mth term, we find the formula 

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Figure 9.5. Results of cluster analyses for M. A. Brewer s (1988) experiment...

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