A9 - guinier approximation

source: I W Hamley, small-angle scattering, John Wiley & Sons (2021)

I (q) = 4 π \int_{0}^{D_{m a x}} p (r) \frac{\sin (q r)}{q r} d r

where, $p (r) = Γ (r) r^{2}$ is the pair distance distribution function, and $Γ (r)$ is the debye correlation function

substituting with the truncated expansion of $sinc (q r) :$

I (q) = 4 π \int_{0}^{D_{m a x}} p (r) (1 - \frac{(q r)^{2}}{3!}) d r

the radius of gyration:

R_{g}^{2} = \frac{\int_{0}^{D_{m a x}} p (r) r^{2} d r}{2 \int_{0}^{D_{m a x}} p (r) d r}

⟹ I (q) = I (0) (1 - \frac{q^{2} R_{g}^{2}}{3} + \dots)

using the expansion for $e^{x}$ , truncating at the second term (small $q$ ):

I (q) = I (0) \exp (- \frac{q^{2} R_{g}^{2}}{3})

this is the guinier equation
the guinier approximation is useful for systems with non-interacting particles $(S (q) = 1)$ , and typically valid for $q < 1 / R_{g}$