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 .
The full scattering angle is defined by a right-triangle with base d and height :
The total momentum transfer is:
We can also write:
Where we take for granted that q must be positive.
If (and ), then , , and:
The other component can be thought of in terms of the sides of a right-triangle with angle :
If , then , , and:
The components are:
For arbitrary 3D scattering vectors, the momentum transfer components are:
In vector form:
Note that this provides a simple expression for q total:
As a check of these results, consider:
Note that , and so: