# sphere¶

Spheres with uniform scattering length density

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

sld

Layer scattering length density

10-6-2

1

sld_solvent

Solvent scattering length density

10-6-2

6

50

The returned value is scaled to units of cm-1 sr-1, absolute scale.

For information about polarised and magnetic scattering, see the Polarisation/Magnetic Scattering documentation.

Definition

The 1D scattering intensity is calculated in the following way (Guinier, 1955)

$I(q) = \frac{\text{scale}}{V} \cdot \left[ 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} \right]^2 + \text{background}$

where scale is a volume fraction, $$V$$ is the volume of the scatterer, $$r$$ is the radius of the sphere and background is the background level. sld and sld_solvent are the scattering length densities (SLDs) of the scatterer and the solvent respectively, whose difference is $$\Delta\rho$$.

Note that if your data is in absolute scale, the scale should represent the volume fraction (which is unitless) if you have a good fit. If not, it should represent the volume fraction times a factor (by which your data might need to be rescaled).

The 2D scattering intensity is the same as above, regardless of the orientation of $$\vec q$$.

Validation

Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006).

Source

sphere.py $$\ \star\$$ sphere.c $$\ \star\$$ sas_3j1x_x.c

References

1. A Guinier and G. Fournet, Small-Angle Scattering of X-Rays, John Wiley and Sons, New York, (1955)

Authorship and Verification

• Author: