The Math Behind

SAXS

In models.sphere the form factor is defined as:

\[\begin{split}\def\v#1{{\bf #1}} ff_{sph}(q, r) &= { 3 ~ sin(qr) - qr ~ cos(qr) \over (qr)^3 } \\ v_{sph}(r) &= {\tiny {4 \over 3}} ~ \pi ~ r^3 \\ v_{sph,abs}(r, \Delta\rho) &= \Delta\rho^2 ~ v_{sph}(r)\end{split}\]

Where q is

the scattering vector loaded from the data file and possibly preprocessed, respectively filtered by defining min/max q or masking invalid values equal or below zero.

r denotes

the radius of the sphere set in the user interface (UI) or varied during optimization.

\(\Delta\rho\) denotes

the scattering length density difference constant of the model against the solution which is defined in the UI.

Sphere.formfactor(dataset)

Calculates the form factor of a sphere defined by:

\(F(q, r) = { 3 ~ sin(qr) - qr \cdot cos(qr) \over (qr)^3 }\)

Sphere.volume()

Calculates the volume of a sphere defined by:

\(v(r) = {4\pi \over 3} r^3\)

Sphere.absVolume()

Calculates the volume of a sphere taking the scattering length density difference \(\Delta\rho\) into account:

\(v_{abs}(r, \Delta\rho) = v_{sph}(r) \cdot \Delta\rho^2\)

SASModel.weight()

Calculates an intensity weighting used during fitting. It is based on the scatterers volume. It can be modified by a user-defined compensation exponent c. The default value is \(c={2 \over 3}\)

\(w(r) = v(r)^{2c}\)

SASModel.calcIntensity(data, compensationExponent=None)

Returns the intensity I, the volume \(v_{abs}\) and the intensity weights w for a single parameter contribution over all q:

\(I(q,r) = F^2(q,r) \cdot w(r)\)