Form Factor:Cube

From GISAXS
Jump to: navigation, search
Cube.png

Equations

For cubes of edge-length 2R (volume ):

Form Factor Amplitude

Isotropic Form Factor Intensity

Sources

Byeongdu Lee (APS)

From Supplementary Information of: Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin "DNA-nanoparticle superlattices formed from anisotropic building blocks" Nature Materials 9, 913-917, 2010. doi: 10.1038/nmat2870

Where 2R is the edge length of the cube, such that the volume is:

and sinc is the unnormalized sinc function:

Pedersen

From Pedersen review, Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting Jan Skov Pedersen, Advances in Colloid and Interface Science 1997, 70, 171. doi: 10.1016/S0001-8686(97)00312-6 For a rectangular parallelepipedon with edges a, b, and c:

For a cube of edge length a this would be:

Derivations

Form Factor

For a cube of edge-length 2R, the volume is:

We integrate over the interior of the cube, using Cartesian coordinates:

Such that:

Each integral is of the same form:

Which gives:

Form Factor at q=0

At small q:

Isotropic Form Factor

To average over all possible orientations, we note:

and use:

From symmetry, it is sufficient to integrate over only one of the eight octants:

Isotropic Form Factor Intensity

To average over all possible orientations, we note:

and use:

Solving integrals that involve nested trigonometric functions is not generally possible. However we can simplify in preparation for performing the integrals numerically:

From symmetry, it is sufficient to integrate over only one of the eight octants:

Isotropic Form Factor Intensity contribution when =0

The integrand of the -integral becomes:

For small , the various can be replaced by , and the various can be replaced by :

Which is a constant (with respect to ). The part of the -integral near has the contribution:

Isotropic Form Factor Intensity at q=0

At very small q: