Cluster site approximation

The cluster site approximation, CSA, has the great advantage that the required number of equations to handle the necessary variables is given by the product Cn (Oates and Wentl 1996) as compared to the exponential value C" aassociated with a full CVM treatment (Inden and Pitsch 1991). The CSA is a variant of the quasichemical model proposed by Li as early as 1949, where the number of clusters that are considered to contribute to the entropy are reduced by excluding all clusters that share edges or bonds. Kikuchi (1977) has deduced the consequential changes in entropy for an f.c.c. structure (Eqs 7.32 and 7.33), which places CSA intermediate... 

Clustered sites would be located within a single ecoregion or at least have similar ecological characteristics (e.g., southeastern coastal plain streams), and would consist of a series of geographically proximal locations (approximately 10 to 20 water bodies) where similar atmospheric loads (or load reductions) are anticipated. Cluster sites would be sampled on a routine basis (e.g., quarterly) for a prolonged period of time. Individual sites within a cluster would be specifically chosen to represent a range of site characteristics that are known to affect Hg cycling (e g., pH, DOC,... 

While an exact number of site clusters has not been proposed, the authors consider from approximately 3 to 10 clusters of sites to be appropriate. These clusters should represent different ecoregions with different ecological characteristics as well as different loadings (both in amount and source) of mercury deposition. Care should be taken to monitor different types of water bodies and watersheds (e.g., seepage lakes, drainage lakes, old reservoirs, rivers, and estuaries). Areas that should be considered as potential cluster site locations include lakes in northern New England/the Adirondacks, lakes in the upper Midwest, rivers and streams in the southeastern coastal plain, lakes in south-central and southeastern Canada, western... 

The CPA has proved to be an enormously successful tool in the study of alloys, and has been implemented within various frameworks, such as the TB, linear muffin-tin orbital and Korringa-Kohn-Rostoker (Kumar et al 1992, Turek et al 1996), and is still considered to be the most satisfactory single-site approximation. Efforts to do better than the single-site CPA have focused on multi-site (or cluster) CPA s (see, e.g., Gonis et al 1984, Turek et al 1996), in which a central site and its set of nearest neighbours are embedded in an effective medium. Still, for present purposes, the single-site version of the CPA suffices, and we derive the necessary equations here, within the framework of the TB model. 

It therefore appears that the cluster approximation with AB pairs would improve the model considerably. This was first done by Dickman [60]. His description has been adapted [53] and extended later by several groups, in order to include diffusion [62], unreactive desorption [63,64] and Eley-Rideal steps [64]. (Note that in these papers the cluster approximation is also called a mean-field approximation. They are distinguished by the terminology site-approximation, pair-approximation etc.)... 

Figure 3. Schematic diagram of the aflatoxin gene cluster showing approximate locations of coding regions. The vertical arrows indicate the presence of AflR-binding sites in the promoter regions of genes and the numbers indicate the presence of two sites in the region.

Aconitase was first suggested to have an iron-sulfur cluster by virtue of the presence of acid-labile sulfide, as well as a g = 2.01 EPR signal, in the purified protein. The iron-sulfur cluster giving rise to this g = 2.01 signal was later shown to be a [3Fe-4S]" " cluster, and was characteristic of inactive enzyme. " " Activation by addition of iron and reductant, or by addition of reductant alone, provided the enzymatically active aconitase, although in the latter case only approximately 75% of full activity was obtained. As the active form of aconitase was subsequently shown to be the [4Fe-4S] form, " addition of iron and reductant provided the iron needed to occupy the fourth metal site in the cluster, while addition of reductant alone resulted in rearrangement of the clusters, labilizing iron to allow reassembly of [4Fe-4S] clusters in 3/4 of the cluster sites. 

If additional water is added in a second hydrolysis step, complete hydrolysis quickly occurs, rendering all sites approximately equal in reactivity. Condensation of completely hydrolyzed oligomers occurs by reaction-limited cluster-cluster aggregation. 

To treat the effects of statistical correlations on alloy properties, one invariably needs to go beyond the single-site approximation [3]. A number of non-self-consistent cluster theories were introduced for this purpose that allowed the treatment of a finite cluster of atoms embedded in a medium determined in some way - usually through the self-consistent condition of the CPA [3]. These methodologies, however, do not yield an improved self-energy and are not of further interest to us in this review. 

---