# Geometry:TSAXS 3D

In transmission-SAXS (TSAXS), the x-ray beam hits the sample at normal incidence, and passes directly through without refraction. TSAXS is normally considered in terms of the one-dimensional momentum transfer (*q*); however the full 3D form of the *q*-vector is necessary when considering scattering from anisotropic materials. The *q*-vector in fact has three components:

This vector is always on the surface of the Ewald sphere. Consider that the x-ray beam points along +*y*, so that on the detector, the horizontal is *x*, and the vertical is *z*. We assume that the x-ray beam hits the flat 2D area detector at 90° at detector (pixel) position . The scattering angles are then:

where is the sample-detector distance, is the out-of-plane component (angle w.r.t. to *y*-axis, rotation about x-axis), and is the in-plane component (rotation about *z*-axis). The alternate angle, , is the elevation angle in the plane defined by .

## Contents

## Total scattering

The full scattering angle is defined by a right-triangle with base *d* and height :

The total momentum transfer is:

Given that:

We can also write:

Where we take for granted that *q* must be positive.

## In-plane only

If (and ), then , , and:

The other component can be thought of in terms of the sides of a right-triangle with angle :

Summarizing:

## Out-of-plane only

If , then , , and:

The components are:

Summarizing:

## Components (angular)

For arbitrary 3D scattering vectors, the momentum transfer components are:

In vector form:

### Total magnitude

Note that this provides a simple expression for *q* total:

#### Check

As a check of these results, consider:

And:

## Components (distances)

Note that , and so:

And: