Nearest- neighbor concurrence

Figure 2. The nearest-neighbor concurrence C(l,2) for different values of the anisotropy parameter y = 1, 0.7, 0.3, 0 with an impurity located at = 3 as a function of the reduced coupling constant A = 7/2/i, where J is the exchange interaction constant and h is the strength of the external magnetic field. The curves correspond to different values of the impurity strength a = 0,0.5,1,1.5 with system size iV = 201.

Figure 2 also shows the variation of nearest-neighbor concurrence as the anisotropy parameter y decreases. For the XY model (y = 0), the concurrence for a = 0 is zero up to the critical point Xc = I and different from zero above Xc = 1. However, as a increases the concurrence develops steps and the... 

Figure 3. Nearest-neighbor concurrence C at nonzero temperature for the transverse Ising model.

Figure 8. Nearest-neighbor concurrence C at zero temperature as a function of the initial magnetic field a for the step function case with final field b.

Figure 9. The nearest-neighbor concurrence C(i, i + I) (upper panel) and the periodic external magnetic field hin(t) = a(l - cos[fi]) see Eq. (14) in the text (lower panel) for K = 0.05 with different values of a as a function of time t.

Relative reaction rates of hydrolysis, condensation, reesterification, and dissolution must be understood and controlled to dictate structural evolution. However, accurate values for rate constants are difficult to obtain because of the enormous number of distinguishable reactions as next nearest neighbors are considered, and to the concurrency of these reactions. Assink and Kay [45] use a simplified statistical model assuming that the local silicon environment does not affect reaction rates, and the reactions for a particular silicon species are the product of a statistical factor and rate constant. These assumptions ignore steric and inductive effects. For example, this model predicts that the relative rate constants for the four sequential hydrolysis steps leading from TMOS to Si(OH)4 would be 4 3 2 1. This model was applied to acid-catalyzed TMOS sols with W values ranging from 0.5 to 2.0. Si NMR spectra on the temporal evolution of various silicon species show the model is in excellent agreement with experimental results. A lower limit for fen was calculated as 0.2 L/mol-min. Values for few and feA are 0.006 and 0.001 L/mol-min, respectively. 

---