Self-correlation triangle

Certain features of the self-correlation triangle centered at the origin are of interest and can be interpreted directly in terms of the structural parameters. In view of the definition of the correlation function, Equation (5.159), we find the following ... 

The height of the self-correlation triangle, representing fee maximum overlap area in the autocorrelation, is ... 

The one-dimensional correlation function K z) can be calculated directly from I(s) [2,5,7,8]. Figure 9.4a shows K z) for an ideal lamellar stack. The self-correlation triangle, centered at the origin, reflects the electron density correlation within a lamella. For a two-phase system, the maximum Q at z = 0 is... 

Fig. 1.3 Relaxation map of polyisoprene results from dielectric spectroscopy (inverse of maximum loss frequency/w// symbols), rheological shift factors (solid line) [7], and neutron scattering pair correlation ((r(Q=1.44 A )) empty square) [8] and self correlation ((t(Q=0.88 A" )) empty circle) [9],methyl group rotation (empty triangle) [10]. The shadowed area indicates the time scales corresponding to the so-called fast dynamics [11]...

Fig. 4.14 Results on fully protonated PIB by means of NSE [147]. a Time evolution of the self-correlation function at the Q-values indicated and 390 K. Lines are the resulting KWW fit curves (Eq. 4.9). b Momentum transfer dependence of the characteristic time of the KWW functions describing Sseif(Q,t) at 335 K (circles), 365 K (squares) and 390 K (triangles). In the scaling representation (lower part) the 335 K and 390 K data have been shifted to the reference temperature 365 K applying a shift factor corresponding to an activation energy of 0.43 eV. Solid (dotted) lines through the points represent (q-2 power laws. Full...

Fig. 5.24 Q-dependence of the average times at 390 K corresponding to SseiKQ.t) empty triangle), Spair(Q.t) (filled triangle), and Sd,am(Q,t) (filled circle). The deduced values for the self-correlation obtained from the description of Schain(Q>f) in terms of the model proposed by Allegra et al. are also shown (empty circle). The solid line represents a (j3=0.55) power...

Fig. 5.25 Result of applying shift factors corresponding to an activation energy of 0.43 eV to the relaxation times observed for the collective dynamics (empty symbol) and the self-correlation (full symbol) of PIB 335 K (circle), 365 K (square), and 390 K (triangles) (reference temperature 365 K). The dotted line through the self-correlation data shows the dependence implying Gaussian behaviour (Reprinted with permission from [147]. Copyright 2002 The American Physical Society)...

Fig. 5.23 Time evolution of the three functions investigated for PIB at 390 K and Q=0.3 A"h pair correlation function (empty circle) single chain dynamic structure factor (empty diamond) and self-motion of the protons (filled triangle). Solid lines show KWW fitting curves. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

Fig. 6 Temperature dependence of the self-diffusion coefficient (top) and the thermal conductivity (bottom) of liquid ammonia on the basis of the force field by Eckl et al. [97]. Simulation results at 10 MPa (filled circles) and 200 MPa (filled triangles) are compared to experimental data (open symbols) [250] and to a correlation of experimental data (solid line) [251]...

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