Easy derivation of the formula relating the fluctuations of a binary system to the X-ray scattering intensity extrapolated to s = 0

Abstract

Bhatia & Thornton [Phys. Rev. B (1970), 2, 3004-3012] have derived the formula that relates the scattering inten sity of a binary system to the Fourier transform of the local number density and concentration. For the intensity at the limit of zero scattering angle, the same formula can be derived by a much simpler process.

Two different regimes in alcohol-induced coil–helix transition: effects of 2,2,2-trifluoroethanol on proteins being either independent of or enhanced by solvent structural fluctuations

“…The expression ∆ of ∆𝑓 and 999 of 𝑓 ̅ indicate the difference in f between the components and the mean value of f in the system, respectively. For the X-ray scattering in s → 0, 28,58 f is approximated to be Z, the total number of electrons:…”

“…where Z is the number of electrons in a particle. According to the Bhatia-Thornton theory, the particle number fluctuations for s → 0 are evaluated using the zero-angle X-ray scattering intensities together with the thermodynamic quantities of vi and κT as: 28,58 𝐼(0…”

Formulating rationally via statistical thermodynamics

“…When divided by the bulk phase concentrations, Δ 1 and Δ 2 become Δ 1 and Δ 2 (Figures 3 and 4) and are called the Kirkwood-Buff integrals (KBI). Introduced originally in 1951 [24], they have been applied to study the structure of solution mixtures via thermodynamic measurements [25][26][27], small angle scattering [28,29] and simulations [30,31]. KBI are the spatial integration of the increment of radial distribution function (RDF) from its bulk value (Figure 4).…”

Statistical thermodynamic foundation for mesoscale aggregation in ternary mixtures

“…The stability function is inversely proportional to the concentration-concentration structure factor at , , via . 25,[29][30][31] 3. From 1 and 2, is inversely proportional to .…”