# Form Factor:Cube

## Contents

## Equations

For cubes of edge-length 2*R* (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 2*R*, 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*: