I=
∫
0∞N a I a da (a.12)where N a( ) is a size distribution function of the sphere. As the simplest example we assumed a Gaussian distribution function
A.3 Scaled Form Factor of Birefringent Sphere
The major purpose of the studying scaled structure factor is to determine the universal structure features in the spatiotemporal evolution of aggregation or decomposition systems.8 The same is true of scaled form factor. For a nucleation-growth system, the volume fraction of growing new phase is the function of time. In order to obtain a universal feature, the time evolution of the scaled form factor should be reduced with the invariant, Q. For depolarized scattering, the scaled form factor,
sphere( )
( )
45 2
0 Hv
Q=
∫
∞I ° q q dq (a.15)On the basis of the Meeten et al.’s theory, Q is ascribed to both mean-square optical anisotropy, δ2
〈 〉 , and the mean-square density fluctuation, 〈 〉 . A problem now arises: due to the excess η2 scattering intensity of which arises from 〈 〉 , η2 IHv45°( )q q2 shows a divergent behavior in high- q regime, and Q does not converge to a constant. To avoid this problem, we may expediently neglect the isotropic term in eqs 10 and 11 to calculate the equivalent invariant, Qan, and9
2 2
an s
Q ∝ δ = ∆ (a.16) φ µ
where φ is the volume fraction of the birefringent sphere. Thus, we can clarify the relative s contribution of the volume fraction of the growing new phase on the scaled form factor.
In previous work we demonstrated PHvsphere( )qa to show the universal feature of the depolarized scattering from a birefringent sphere, but some deviate at qa<4. At that time, we incorrectly related the deviation to the long-range correlation of the dispersed spheres according to the observation that the excess scattering intensity is biased with q2. Later we knew that the observed excess scattering yields important information concerning the elastic background.
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