Commit ecfd4dfd authored by Peterson, Peter's avatar Peterson, Peter
Browse files

Fixes to equations

parent 2673f194
......@@ -527,14 +527,14 @@ Then, with the assumption that :math:`\Delta < 1`, Eq. :eq:`Im` can be manipulat
Similarly:
.. math::
:label: Im_delta_proof
:label: Im_delta_proof
I_m \Delta &= I_1 \Delta \sum_{i=2}^n \Delta^{i-1} = I_1 \sum_{i=2}^n \Delta^{i}
I_m \Delta = I_1 \Delta \sum_{i=2}^n \Delta^{i-1} = I_1 \sum_{i=2}^n \Delta^{i}
Subtracting Eq. :eq:`Im_delta_proof` from Eq. :eq:`Im_proof`, we have:
.. math::
:label: Im_delta_difference
:label: Im_delta_difference
I_m - I_m \Delta &= I_1 \sum_{i=2}^n \Delta^{i-1} - I_1 \sum_{i=2}^n \Delta^{i} \\
&= I_1 (\Delta - \Delta^{n})
......@@ -542,7 +542,7 @@ Subtracting Eq. :eq:`Im_delta_proof` from Eq. :eq:`Im_proof`, we have:
Which, solving for :math:`I_m` and based on the assumption :math:`\Delta < 1`, implying :math:`\Delta^n << \Delta`, we arrive at:
.. math::
:label: Im_geometric
:label: Im_geometric
I_m &= I_1 \frac{\Delta - \Delta^{n}}{1 - \Delta} \\
&\approx I_1 \frac{\Delta }{1 - \Delta}
......@@ -553,7 +553,7 @@ However, comparisons of both equations for cylinders show that Eq. :eq:`Im_geome
From Eq. :eq:`Im_geometric`, we are left with calculating :math:`\Delta`:
.. math::
:label: delta_equation
:label: delta_equation
\Delta &= \frac{I_n}{I_{n-1}} = \frac{I_2}{I_1} \\
&= \frac{ J_0 \rho^2 \frac{d\sigma}{d\Omega} \left( \theta_1 \right) \frac{d\sigma}{d\Omega} \left( \theta_2 \right) \int_{V} \int_{V} \frac{exp \left[ -\mu (\lambda_1) l_1 + - \mu (\lambda_{12}) l_{12} + - \mu (\lambda_2) l_2 \right]}{l_{12}^2} dV dV }
......@@ -564,11 +564,11 @@ From Eq. :eq:`Im_geometric`, we are left with calculating :math:`\Delta`:
Using the isotropic approximation, we arrive at:
.. math::
:label: delta_equation_elastic
:label: delta_equation_elastic
\Delta_{elastic} &= \frac{ \rho \left( \frac{\sigma_s}{4 \pi} \right)^2 \int_{V} \int_{V} \frac{exp \left[ -\mu (\lambda_1) l_1 + - \mu (\lambda_{12}) l_{12} + - \mu (\lambda_2) l_2 \right]}{l_{12}^2} dV dV }
{ \frac{\sigma_s}{4 \pi} \int_{V} exp \left[ -\mu (\lambda_1) l_1 + -\mu (\lambda_2) l_2 \right] dV } \\
&= \frac{ \rho \sigma_s A_2 V^2 }{ 4 \pi A_1 V } = \frac{ \rho V \sigma_s A_2 }{ 4 \pi A_1 }
&= \frac{ \rho \sigma_s A_2 }{ 4 \pi A_1 }
where :math:`A_2` is the secondary scattering absorption factor and :math:`A_1` is the single scattering absorption factor, equivalent to :math:`A` in Eq. :eq:`absorption_factor`.
The absorption factors can be further simplified by using the elastic scattering assumption from Eq. :eq:`absorption_factor_elastic`.
......
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