A10 - invariant calculation

the integral over all $\vec{q}$ of the appropriately dimensionally-weighted intensity of a multi-phase system is a constant, directly proportional to the mean-square average fluctuation in SLD and the phase composition
- a multi-phase system is a system containing regions with different SLDs
this is independent of the arrangement of the phase domains - ie. it is invariant
- here, a phase is any portion of the material with an SLD distinctly different from the average SLD
this constant is known as the scattering or porod invariant , or simply as the invariant, $Q^{*}$
as a fundamental law of scattering, it can be used for sanity checks, cross-calibration of SAS instruments, and can yield an independent estimate of volume fractions or contrast terms

calculation

Q^{*} = \int_{0}^{\infty} q^{2} I (q) d q

Q^{*} = Δ q_{v} \int_{0}^{\infty} q I (q) d q

where, $Δ q_{v}$ is the slit height

the difficulty arises due to experimental data never being measure over the range $q \in [0, \infty]$
therefore, extrapolation to both low and high $q$ is required
SasView currently allows extrapolation to a user-set range, with the default range: $q \in [10^{- 5}, 10] Å$
since the integrals above are weighted by $q^{2}$ or $q$ , the high- $q$ extrapolation is weighted far more heavily than the low- $q$
low- $q$ region ( $\leq q_{m i n}$ in data):
- guinier function: $I_{o} \exp (- q^{2} R_{g}^{2} / 3)$
- alternatively, but not recommended: power law
high- $q$ region ( $\geq q_{m a x}$ in data):
- power law: $A / q^{m}$
- $m$ is user-fitted, typically between 3 and 4 for pinhole resolution, and $4$ for sharp interfaces

Q^{*} = 2 π^{2} (Δ ρ)^{2} ϕ_{1} ϕ_{2}

where, $Δ ρ = ρ_{1} - ρ_{2}$ is the SLD contrast, and $ϕ_{1}$ and $ϕ_{2}$ are the volume fractions of the two phases $(ϕ_{1} + ϕ_{2} = 1)$

therefore, one can calculate either the SLD contrast or the volume fraction if the other is given
currently, SasView only allows the latter

in some cases for which no good analytical description exists, if the contrast term can be reasonably estimated, the volume fraction can be estimated from the invariant

ϕ_{1} ϕ_{2} = \frac{Q^{*}}{2 π^{2} (Δ ρ)^{2}} \equiv A

\begin{matrix} ϕ_{1} = \frac{1 - \sqrt{1 - 4 A}}{2} \\ ϕ_{2} = \frac{1 + \sqrt{1 - 4 A}}{2} \end{matrix}

the specific surface area, the total surface area per unit volume $(S_{v})$ , is reflected in he scattering of the material
for a two phase system:

\begin{matrix} S_{v} = \frac{C_{p}}{2 π (Δ ρ)^{2}} \\ C_{p} = lim_{q} \to \infty I (q) q^{4} \end{matrix}

where, $C_{p}$ , the porod constant, is given by porod's law

the invariant is a general concept, so the formalism can be extended to more phases

Q^{*} = 2 π^{2} ⟨ η^{2} ⟩

where, $η$ is the SLD fluctuation

⟨ η^{*} ⟩ = ⟨ (ρ^{*})^{2} ⟩ - ⟨ (ρ^{*}) ⟩^{2}