Attenuation correction for sample shape

In cases of strongly scattering or absorbing samples, the detected scattering intensity is lower than the 'true' scattering. Moreover, for oddly-shaped samples, the extinction of scattering may be anisotropic: some parts of the detector image are more attenuated than others because of the differing path-lengths through the sample.

Formulation

In the following, we assume a transmission-scattering (TSAXS) experiment.

The measured scattering, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_m} , at a particular scattering angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Theta_o, \chi_o)} (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta_o} is the full (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \theta} ) scattering angle between the scattered ray and the incident beam, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_o} is the azimuthal angle: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi=0^{\circ}} corresponds to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_z} axis, whereas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi=90^{\circ}} is along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_r} axis) can be computed by summing the scattering contributions for all the elements along the beam path through the sample.

We define a realspace coordinate system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)} where z points vertically, y points along the beam direction, and x points horizontally with respect to the sample. Let the sample size along the beam direction be L. Defining the point where the beam first enters the sample as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0,0)} we write:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} S_m(\Theta_o,\chi_o) = \int \limits_{l=0}^{l=L} S(l) \mathrm{d}l \end{alignat} }

The scattering from a particular location within the sample is affected by two attenuation effects:

1. The beam flux within the sample decreases due to absorption/scattering, such that the flux at position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} is not the incident flux, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_0} but attenuated to:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(l) = I_0 e^{- \alpha l }}

   where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is (Beer-Lambert like) extinction coefficient. If the 'true' scattering probability is given by $\sigma$ (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_0 \sigma} is the scattering observed in the absence of attenuation), then the scattering at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} is:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(l) = I(l) \sigma = I_0 e^{-\alpha l} \sigma}

2. The scattered radiation is itself attenuated as it passes through the sample. Let this path-length (from scattering location Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,l,0)} until it exits the sample along the direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Theta_o, \chi_o)} ) be denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(l)} . In such a case, the scattering that exits the sample is:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} S(l) & = I(l) \sigma \times \mathrm{attenuation}(l) \\ & = I_0 e^{-\alpha l} \sigma e^{-\alpha p(l)} \\ & = I_0 \sigma e^{-\alpha (l+p(l))}\\ \end{alignat} }

The measured scattering is thus:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} S_m(\Theta_o,\chi_o) & = \int \limits_{l=0}^{L} I_0 \sigma e^{-\alpha (l+p(l))} \mathrm{d}l \\ & = I_0 \sigma \int \limits_{0}^{L} e^{-\alpha (l+p(l))} \mathrm{d}l \\ \end{alignat} }

The integral of course depends on the form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(l)} which depends on the sample shape. Note that in the limiting case of weak attenuation (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\approx0} ), we obtain the very simple result:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} S_m(\Theta_o,\chi_o) & = I_0 \sigma \int \limits_{0}^{L} e^{0} \mathrm{d}l \\ & = I_0 \sigma L \\ \end{alignat} }

As expected, scattering intensity scales with the scattering volume.

Normalization

To normalize-out the effect of attenuation, one can simply divide by the integral:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} S_{\mathrm{norm}} (\Theta_o,\chi_o) & = \frac{S_m(\Theta_o,\chi_o)}{\int_{0}^{L} e^{-\alpha (l+p(l))} \mathrm{d}l } \\ & = I_0 \sigma \end{alignat} }

Of course in the case of weak attenuation the integral is simply L, and we are normalizing by the beam-path through the sample.

For a vector that starts at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0,0)} and terminates at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)} , pointing along direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Theta_o,\chi_o)} , the full length is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathbf{v}| = v = \sqrt{ x^2 + y^2 + z^2 }}

We can consider triangles in various planes:

1. xz plane (looking along beam):

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(90^{\circ}-\chi_o) = \frac{z}{x}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(90^{\circ}-\chi_o) = \frac{x}{\sqrt{x^2+z^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(90^{\circ}-\chi_o) = \frac{z}{\sqrt{x^2+z^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\chi_o) = \frac{x}{z}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\chi_o) = \frac{z}{\sqrt{x^2+z^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\chi_o) = \frac{x}{\sqrt{x^2+z^2}}}

2. xy plane (looking from above):

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\omega_{xy}) = \frac{x}{y}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\omega_{xy}) = \frac{y}{\sqrt{x^2+y^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\omega_{xy}) = \frac{x}{\sqrt{x^2+y^2}}}

3. yz plane (looking from side):

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\omega_{yz}) = \frac{z}{y}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\omega_{yz}) = \frac{y}{\sqrt{y^2+z^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\omega_{yz}) = \frac{z}{\sqrt{y^2+z^2}}}

4. plane of beam elevation:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\omega_{\mathrm{elevation}}) = \frac{z}{\sqrt{x^2+y^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\omega_{\mathrm{elevation}}) = \frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\omega_{\mathrm{elevation}}) = \frac{z}{\sqrt{x^2+y^2+z^2}}}

5. plane of full scattering angle:

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\Theta_o) = \frac{\sqrt{x^2+z^2}}{y}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\Theta_o) = \frac{y}{\sqrt{x^2+y^2+z^2}}}

   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\Theta_o) = \frac{\sqrt{x^2+z^2}}{\sqrt{x^2+y^2+z^2}}}

Height Z

If the vector's final point is at height Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=Z} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} v_Z & = \sqrt{ x^2 + y^2 +Z^2 } \\ & = \frac{ \sqrt{x^2+Z^2} }{ \sin(\Theta_o) } \\ & = \frac{ 1 }{ \sin(\Theta_o) } \sqrt{\left( Z \tan(\chi_o) \right)^2 + Z^2} \\ & = \frac{ Z }{ \sin(\Theta_o) } \sqrt{\tan^2(\chi_o) + 1} \\ & = \frac{ Z }{ \sin(\Theta_o) } \sec(\chi_o) \\ & = \frac{ Z }{ \sin(\Theta_o) \cos(\chi_o) } \\ \end{alignat} }

This has a minimum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_z=Z} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Theta_o,\chi_o)=(90^{\circ},0^{\circ})} .

Depth L

If the vector's final position is at depth Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=L} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} v_Y & = \sqrt{ x^2 + L^2 +z^2 } \\ & = \frac{ L }{ \cos(\Theta_o) } \\ \end{alignat} }

This has a minimum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_Y=L} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Theta_o=0^{\circ}} .

Width X

If the vector's final position is at width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=X} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} v_X & = \sqrt{ X^2 + y^2 +z^2 } \\ & = \frac{ \sqrt{X^2+z^2} }{ \sin(\Theta_o) } \\ & = \frac{ 1 }{ \sin(\Theta_o) } \sqrt{X^2 + \left( \frac{X}{ \tan(\chi_o) } \right)^2} \\ & = \frac{ |X| }{ \sin(\Theta_o) } \sqrt{1 + \frac{1}{ \tan^2(\chi_o) } } \\ & = \frac{ X }{ \sin(\Theta_o) } \sqrt{\frac{\tan^2(\chi_o)+1}{ \tan^2(\chi_o) } } \\ & = \frac{ X }{ \sin(\Theta_o) } \frac{\sqrt{\tan^2(\chi_o)+1}}{ \sqrt{\tan^2(\chi_o) }} \\ & = \frac{ X }{ \sin(\Theta_o) } \frac{ \sec(\chi_o) }{ \tan(\chi_o) } \\ & = \frac{ X \cos(\chi_o) }{ \sin(\Theta_o) \cos(\chi_o) \sin(\chi_o) } \\ & = \frac{ X }{ \sin(\Theta_o) \sin(\chi_o) } \\ \end{alignat} }

This has a minimum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_X=X} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Theta_o,\chi_o)=(90^{\circ},90^{\circ})} .

Rectangular prism

If the sample is a rectangular prism with dimensions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2X, 2Y, 2Z) = (2X, L, 2Z)} and the beam falls upon the center of the xz front-face, then the beam travels a distance L through the sample, and the scattered radiation in any quadrant passes through the rectangular prism of size Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X,L,Z)} . The distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,l,0)} to the exit-point from the sample is the distance to the closest sample face:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} p\left(l\right) & = \mathrm{min}( d_{\mathrm{top}}(l) , d_{\mathrm{back}}(l) , d_{\mathrm{side}}(l) ) \\ \end{alignat} }

The distances are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} d_{\mathrm{top}}(l) & = v_Z \\ & = \frac{ Z }{ \sin(\Theta_o) \cos(\chi_o) } \\ \end{alignat} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} d_{\mathrm{back}}(l) & = v_{L-l} \\ & = \frac{ L-l }{ \cos(\Theta_o) } \\ \end{alignat} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} d_{\mathrm{side}}(l) & = v_X \\ & = \frac{ X }{ \sin(\Theta_o) \sin(\chi_o) } \\ \end{alignat} }