Method

The Rayleigh differential cross section as a function of $q2$ is given by []: $\{d\; \sigma$_R( q²) d Ω}= { πr₀² k²}( {1 + μ² 2 }) | F_T( q ) | ²

where:

Under the assumption that the atoms of a molecule are completely independent, $|\; F$_T( q ) | ² is given by: $|\; F$_T( q ) | ²= ∑_i=1^N{W_iA_i}| F_i( q_i, Z_i) | ²σ_ci( Z_i, E )

where the index i runs on the N atoms in the molecule and:

$W$i	$proportion\; by\; weight$
$Z$i, Ai	$atomic\; number\; and\; weight$
$F$i	$form\; factor$
$\sigma$ci	$total\; atomic\; cross\; section\; for\; coherent\; scattering$

Using the combined composition and rejection sampling method described in GPAIRG ( [PHYS211]) we may set: $f\; (\; q\; )\; =\; \sum$_i=1^Nα_if_i( q )g_i( q ) = ∑_i=1^NA ( q_i²){ | F_T( q ) | ² A ( q_n²) }( {1+μ²2})

where:

Therefore, for given values of the random numbers $r$₁ and $r$₂ , GRAYL samples the momentum of the scattered photon and the scattering angle $\theta$ via the following steps:

1. sample $A\; (\; q2)\; =\; r$₁A ( q_n²)

2. find the $(\; q$_i-1, q_i] interval which gives $A\; (\; q$_i-1²) ≤A ( q²)≤A ( q_i²)

3. calculate the linear extrapolation: $q\; =\; q$_i-1+ ( A ( q²) - A ( q_i-1²) ){q_i- q_i-1 A ( q_i²) - A ( q_i-1²)}

4. calculate $\mu =\; cos\; \theta =\; 1\; -\; q2/(2\; k2)$

5. calculate $g$_i( q ) = (1 + μ²)/2

6. if $g$_i( q ) > r₂ the event is accepted, otherwise go back to 1.

F. Carminati

PHYS260