# Form Factor:Pyramid

## Contents

## Equations

For pyramid of base edge-length 2*R*, and height *H*. The angle of the pyramid walls is . If then the pyramid is truncated (flat top).

- Volume
- Projected (
*xy*) surface area

### Form Factor Amplitude

- where

### Isotropic Form Factor Intensity

This can be computed numerically.

## Derivations

### Form Factor

For a pyramid of base-edge-length 2*R*, side-angle , truncated at *H* (along *z* axis), we note that the in-plane size of the pyramid at height *z* is:

Integrating with Cartesian coordinates:

A recurring integral is (c.f. cube form factor):

Which gives:

This can be simplified automated solving. For a regular pyramid, we obtain:

### Form Factor near *q*=0

#### qy

When :

So:

#### qx

When :

Since sinc is an even function:

And:

#### qz

When :

So:

#### q

When :

So:

And:

#### qx and qy

When :

So:

To analyze the behavior in the limit of small and , we consider the limit of where . We replace the trigonometric functions by their expansions near zero (keeping only the first two terms):

Note that since is symmetric . When and are small (but not zero and not necessarily equal), many of the above arguments still apply. It remains that , and:

### Isotropic Form Factor Intensity

To average over all possible orientations, we note:

and use:

## Regular Pyramid

A regular pyramid (half of an octahedron) has faces that are equilateral triangles (each vertex is 60°). The 'corner-to-edge' distance along each triangular face is then:

This makes the height:

So that the pyramid face angle, is:

The square base of the pyramid has edges of length 2*R*. The distance from the center of the square to any corner is *H*, such that:

### Surface Area

For a non-truncated, regular pyramid, each face is an equilateral triangle (each vertex is 60°). So each face:

The base is simply:

Total:

### Volume

For a regular pyramid, the height and :