Calculation¶

The Debye sum¶

calculate() sums over all points:

\[ \begin{aligned} w_j &= (\rho_j - \rho_s)\, V_j \\ D(q) &= \sum_j \sum_k w_j w_k \, \frac{\sin(q r_{jk})}{q r_{jk}} \\ I(q) &= \mathrm{scale} \cdot 10^8 \, \frac{D(q)}{V_\mathrm{norm}} + \mathrm{background} \end{aligned} \]

with \(\rho_j\) the SLD of point \(j\) in Å⁻², \(\rho_s\) = sld_solvent × 10⁻⁶ Å⁻², \(V_j\) its volume in Å³ and \(r_{jk}\) in Å. The factor \(10^8\) turns Å⁻¹ into cm⁻¹, the unit of \(I\).

sld_solvent is in 10⁻⁶ Å⁻², like every sasmodels SLD parameter. SasView's calculator takes Å⁻² instead; a value that looks like Å⁻² triggers a warning. The structure's own SLDs stay in Å⁻², as the files store them.

Normalisation¶

\(V_\mathrm{norm}\) is the volume of the points that scatter, those whose SLD differs from the solvent's. For a particle of pixels or cells in a box of solvent, the curve at scale=1 then equals sasmodels' form factor at scale=1, so scale is a volume fraction.

When the solvent exactly matches a population of points, those points stop counting and the curve jumps by the volume ratio; for contrast-variation series set set_params(normalisation_volume=...) explicitly ('auto' restores the default). SasView calls the override total_volume and, when some points have zero contrast, also multiplies by the ratio of all points' volume to the scattering volume. This calculator does not.

Size polydispersity¶

set_params(size_pd=0.1) sums blown-up and shrunk copies of the whole structure, as sasmodels does for radius_pd: size_pd_n size factors (35) evenly over ±size_pd_nsigma (3) standard deviations, factors at or below 0 dropped, each weighted by the Gaussian's height there. A factor \(f\) stretches every distance by \(f\) and every amplitude by \(f^3\), so

\[ I(q) = \mathrm{scale} \cdot 10^8 \, \frac{\sum_f n_f f^6 D(f q)}{V_\mathrm{norm}} + \mathrm{background}, \qquad V_\mathrm{norm} = \sum_f n_f f^3 V \]

with \(V\) the volume of the points that scatter and the shares \(n_f\) adding up to 1, so scale stays a volume fraction. The lattice, histogram and average kernels evaluate their one distance spectrum at \(f q\) for every size, which costs only the sums over q; 'pairs' repeats the whole pair sum per size and warns. The automatic histogram bin and the cell-size warning use the largest factor, 1 + size_pd_nsigma * size_pd.

Deuteration¶

set_params(deuterated=['CTA']) gives the hydrogens of these residue names the SLD of D at calculate(), as Structure.deuterated does.

Kernels¶

set_params(kernel=...) picks how D(q) is summed; the profile's method records it.

`kernel`	How	Exact?	Time
`'lattice'`	FFT autocorrelation of the pixel grid	yes	well under a second for 10⁵ pixels
`'histogram'`	pairs counted by distance, then one sum per q over the occupied distances	to about 1e-4	about 2 s for 11 000 atoms, N²
`'average'`	⟨A⟩²: each point's distance from the bounding-box centre	only for spherically symmetric structures	linear in N
`'pairs'`	every pair at every q	to the last digit	2–3 s for 4 000 points, 25 s for 8 000

'auto' (the default) is 'lattice' for pixels on a lattice and 'histogram' for everything else.

The histogram bin is 'auto': \(\min\left(0.01~\text{Å},\ 10^{-3} / (q_\mathrm{max} f_\mathrm{max})\right)\), with \(f_\mathrm{max}\) the largest size factor (1 without polydispersity), which keeps the rounding phase error below \(5 \cdot 10^{-4}\) rad; histogram_bin= sets it, and profile.histogram_bin records it. A cached spectrum with a finer bin is reused.

'average' reproduces SasView's is_avg=True and suits round aggregates above about 10⁵ atoms. calculate(progress=...) can report progress or stop the pair passes of 'histogram' and 'pairs'.

Things to know¶

PDB structures in a solvent are only qualitative

The natural atomic volumes use bulk elemental densities, liquid H₂, N₂ and O₂ for H, N and O. A typical residue gets about 310 Å³ instead of about 133 Å³, so in D₂O a protein comes out about 8× too intense at q → 0. There is no hydration shell and no H/D exchange either. calculate() warns with the actual volume ratio. In vacuum (sld_solvent=0) the result is right. For atoms, scale is not a volume fraction.

Points stand in for cells. Pixels and VTK cells are treated as points, which degrades as q × cell size approaches 1. calculate() warns above that, with the cell size taken as the lattice step or the cell diameter divided by √3.
The curve is theory. profile.to_data1d() returns a Data1D with no dI and no dQ, with qmin, qmax and an empty mask set. A fit of it should state its weighting and resolution explicitly; pass to_data1d(dI=...) to attach uncertainties.
Console output. Progress lines go through the sinccity logger; sinccity.set_verbosity('quiet') silences them.