Analytical Calculation of the Scattering Curve I(q) and the Real Space Function p(r) of a Simple Model (Example).
As an example, the scattering curve in reciprocal space and p(r)-function in real space from a model consisting of differently sized spheres are calculated analytically
from the Debye-formula in reciprocal space (Debye, P. J. W. (1915), Ann. Phys. 351, 809–823) and its analogue in real-space (Glatter, O. (1980), Acta Phys. Austriaca 52, 243–256).
We build a model consisting of 19 spheres (with different radii and constant electron densities) as shown below.
The x-y-z coordinates, their respective radii and (optionally different) electron densities of the model are entered as input in
the I(q)-p(r)-Debye calculator.
The results - I(q) scattering curve and p(r) real-space function of the model (maximum dimension is 44 units) - are shown below:
Next, we calculate I(q) and p(r) of 2 different models (each composed of 5 spheres) which have the same histogram of
intraparticular distances despite of the different 3D-shape. This results in an identical reciprocal and real space function
demonstrating the ambiguity of these one-dimensional functions with respect to the 3D real-space model. Note, both models have 2 mirror planes,
one being the same (x-y), the other one being different (y-z in Model1 and x-z in Model2).
Model1 (left) and Model2 (right)
Model1 and Model2 are composed of 5 spheres each. The coordinates, radius and electron densities of the two different models
are entered in the I(q)-p(r)-Debye-calculator.
Model1:
Model2:
Their (identical) scattering curve (left: log(I) vs q) and real space function (right: p(r) vs r) are displayed below:
