A6 - SANS

a beam of collimated (not necessarily monochromatic) radiation is directed at a sample of a small volume $(V = A t_{s})$
some of the incident radiation is transmitted, some absorbed, some scattered
a detector of dimensions $d x \times d y$ , placed at a distance $L_{s d}$ , and scattering angle $θ$ , records the radiation flux into a solid angle element, $Δ Ω = d x \frac{d y}{L_{s d}^{2}}$
for an incident flux $(I_{0})$ , the flux is:

I (λ, θ) = I_{0} (λ) Δ Ω η (λ) T V \frac{\partial σ}{\partial Ω} (Q)

- where, $η$ is the detector efficiency (response), $T$ is the sample transmission and $\frac{\partial σ}{\partial Ω} (Q)$ is the differential cross-section

the first three terms are instrument dependent, whereas the last one is sample dependent
the objective of a SANS experiment is to determine the differential cross-section
it contains the information on the shape, size and interactions of the scattering bodies in the sample

\frac{\partial σ}{\partial Ω} (Q) = N_{p} V_{p} (Δ δ)^{2} P (Q) S (Q) + B_{i n c}

where,
$N_{p}$ is the number concentration of scattering bodies (particles),
$V_{p}$ is the volume of one body,
$Δ δ$ is the difference in neutron scattering length density (contrast),
$P (Q)$ is the form/shape factor,
$S (Q)$ is the interparticle structure factor,
$Q$ is the modulus of the scattering vector,
and $B_{i n c}$ is the isotropic incoherent background signal

scattering vector

Q = | \vec{Q} | = | {\vec{k}}_{f} - {\vec{k}}_{i} | = \frac{4 π n}{λ} \sin (\frac{θ}{2})

in neutron scattering, $n \sim 1$
combined with bragg's law:

d = \frac{2 π}{Q}

often referred to as the momentum transfer as $h Q = Δ p$

contrast term

for a molecule with $i$ atoms:

δ = \sum_{i} b_{i} \frac{D N_{A}}{M_{w}}

where, $D$ is the bulk density, and $M_{w}$ its molecular weight

the contrast is the difference in $δ$ between the part of sample of interest, and the surrounding medium
if $Δ δ = 0$ , three is no SANS, which is at contrast match

form factor

a function that describes how the differential cross-section is modulated by interference effects between radiation scattered by different parts of the same scattering body
the van de hulst's equation:

P (Q) = \frac{1}{V_{p}^{2}} | \int_{0}^{V_{p}} \exp (i f (Q α)) d V_{p} |

where, $α$ is a 'shape parameter', that might represent a length or a radius of gyration

analytic expressions for common shapes exist

System	Scattering Function (P(Q))
sphere of radius $R_{p}$	$P (Q) = {[\frac{3 (\sin (Q R_{p}) - Q R_{p} \cos (Q R_{p}))}{(Q R_{p})^{3}}]}^{2}$
disc of negligible thickness and radius $R_{p}$	$P (Q) = \frac{2}{(Q R_{p})^{2}} [1 - \frac{J_{1} (2 Q R_{p})}{Q R_{p}}]$
rod of negligible cross-section and length $L$ ( $S_{i}$ is the Sine integral function)	$P (Q) = \frac{2 S_{i} (Q L)}{Q L} - \frac{\sin^{2} (Q L / 2)}{(Q L / 2)}$
gaussian random coil with z-average radius of gyration $R_{g}$ , polydispersity $(Y + 1)$ and $U = \frac{(Q R_{g})^{2}}{(1 + 2 Y)}$	$P (Q) = \frac{2 [(1 + U)^{- \frac{Y}{2}} + U - 1]}{(1 + Y) U^{2}}$
concentrated polymer solution with screening length $ξ$ , where $ξ = R_{g} {(\frac{ϕ}{ϕ^{*}})}^{- \frac{1}{2 ν - 1}}$	$P (Q) = P (0) [\frac{1}{1 + (Q ξ)^{2}}]$

structure factor

a function that describes how the differential cross-section is modulated by interference effects between radiation scattered by different scattering bodies
depends on the degree of local order in the sample