Geometry:TSAXS 3D

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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 .

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 :


Out-of-plane only

If , then , , and:

The components are:


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:


As a check of these results, consider:


Components (distances)

Note that , and so:


Total magnitude