A4 - structure factor

image: CJ Gommes, S Jakschb and S Frielinghausb (2021)

structure factor captures the collective contribution by a large number of particles

| a | = ρ V [P (q)]^{1 / 2}

the phase depends on particle position
for identical particles:

I (q) = ρ^{2} V^{2} P (q) {| \sum_{i} \exp (i ϕ_{i}) |}^{2}

where, $ϕ_{i} = ϕ_{θ} (x, y, z)$ at say its centre of mass

randomly positioned particles

considering randomly positioned particles, so the vector orientations are also random, forming a random walk
$N$ successive steps are made with the same length, $a$ and random directions
consequently, average walk length is not proportional to $N$ , but the average squared length is

⟨ | A |^{2} ⟩ = N | a |^{2}

∴ I (q) = c ρ^{2} V^{2} P (q)

where, $c$ is the concentration

$I \propto N$ is typical of incoherent scattering

aggregated particles

in most cases, particles aggregate with $n_{a} = 3$
the trajectories end up being more elongated, and hence larger intensities are obtained
considering $q \to 0$ , $λ^{*} ≫$ aggregates
all particles in an aggregate scatter coherently, so the length of a step is $n_{a} a$
aggregates scatter incoherently, so a random walk with $N / n_{a}$ steps

⟨ | A |^{2} ⟩ ≃ n_{a} N | a |^{2}

∴ I (q) = c ρ^{2} V^{2} P (q) S (q)

where, $S (q)$ is the structure factor

image: CJ Gommes, S Jakschb and S Frielinghausb (2021)