Structure factor, integral equations, pair

Figure 5.2 exhibits the site-site static structure factors, Eq. (5.90), calculated from the extended version of the RISM integral-equation theory [58, 59]. The peak positions are fcmax = 1-69 and 1.65 A for A-A and B-B pairs, respectively. Note that in the k 0 limit all the site-site structure factors coincide [40], and we define... 

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

This expression can be integrated with respect to all the non-external coordinates. The result must be multiplied by two factors the first one originates from the factorial 1/n in Equation 53, the second one reflects the internal symmetry of the structure and is represented by the number of ways of pairing of the same type as in the given graph. In the case considered, these factors are equal to 3 (the graph a) and 12 (the graph b). 

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