-                      re166cb9
+                      ra5d0d00
 #note - calculation requires double precision
 """
+Calculates the scattering from a **body-centered cubic lattice** with paracrystalline distortion. Thermal vibrations
+are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is
+assumed to be isotropic and characterized by a Gaussian distribution.
+Calculates the scattering from a **body-centered cubic lattice** with
+paracrystalline distortion. Thermal vibrations are considered to be negligible,
+and the size of the paracrystal is infinitely large. Paracrystalline distortion
+is assumed to be isotropic and characterized by a Gaussian distribution.
 The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale.
 …
 ----------
 The scattering intensity *I(q)* is calculated as
+The scattering intensity $I(q)$ is calculated as
 .. image:: img/image167.jpg
+.. math:
+where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume
+correction for the crystal structure, *P(q)* is the form factor of the sphere (normalized), and *Z(q)* is the
+paracrystalline structure factor for a body-centered cubic structure.
+    I(q) = \frac{\text{scale}}{V_P} V_\text{lattice} P(q) Z(q)
-Equation (1) of the 1990 reference is used to calculate *Z(q)*, using equations (29)-(31) from the 1987 paper for
-*Z1*\ , *Z2*\ , and *Z3*\ .
+The lattice correction (the occupied volume of the lattice) for a body-centered cubic structure of particles of radius
+*R* and nearest neighbor separation *D* is
+where *scale* is the volume fraction of spheres, *Vp* is the volume of the
+primary particle, *V(lattice)* is a volume correction for the crystal
+structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$
+is the paracrystalline structure factor for a body-centered cubic structure.
+.. image:: img/image159.jpg
+Equation (1) of the 1990 reference is used to calculate $Z(q)$, using
+equations (29)-(31) from the 1987 paper for *Z1*\ , *Z2*\ , and *Z3*\ .
+The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)*
+The lattice correction (the occupied volume of the lattice) for a
+body-centered cubic structure of particles of radius $R$ and nearest neighbor
+separation $D$ is
 .. image:: img/image160.jpg
+.. math:
+where *g* is a fractional distortion based on the nearest neighbor distance.
+    V_\text{lattice} = \frac{16\pi}{3} \frac{R^3}{\left(D\sqrt{2}\right)^3}
+The distortion factor (one standard deviation) of the paracrystal is included
+in the calculation of $Z(q)$
+.. math:
+    \Delta a = g D
+where $g$ is a fractional distortion based on the nearest neighbor distance.
 The body-centered cubic lattice is
 .. image:: img/image168.jpg
+.. image:: img/bcc_lattice.jpg
 For a crystal, diffraction peaks appear at reduced q-values given by
 .. image:: img/image162.jpg
+.. math:
+where for a body-centered cubic lattice, only reflections where (\ *h* + *k* + *l*\ ) = even are allowed and
+reflections where (\ *h* + *k* + *l*\ ) = odd are forbidden. Thus the peak positions correspond to (just the first 5)
+    \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2}
+.. image:: img/image169.jpg
+where for a body-centered cubic lattice, only reflections where
+$(h + k + l) = \text{even}$ are allowed and reflections where
+$(h + k + l) = \text{odd}$ are forbidden. Thus the peak positions
+correspond to (just the first 5)
+**NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of**
+**points to properly capture the sharp peaks of the paracrystalline scattering.** So be warned that the calculation is
+SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This
+makes a triple integral. Very, very slow. Go get lunch!
+.. math:
+This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above
+default values.
+    \begin{eqnarray}
+    &q/q_o&&\quad 1&& \ \sqrt{2} && \ \sqrt{3} && \ \sqrt{4} && \ \sqrt{5} \\
+    &\text{Indices}&& (110) && (200) && (211) && (220) && (310)
+    \end{eqnarray}
+.. image:: img/image170.jpg
+**NB: The calculation of $Z(q)$ is a double numerical integral that must
+be carried out with a high density of points to properly capture the sharp
+peaks of the paracrystalline scattering.** So be warned that the calculation
+is SLOW. Go get some coffee. Fitting of any experimental data must be
+resolution smeared for any meaningful fit. This makes a triple integral.
+Very, very slow. Go get lunch!
+This example dataset is produced using 200 data points,
+*qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above default values.
+.. image:: img/bcc_1d.jpg
 *Figure. 1D plot in the linear scale using the default values (w/200 data point).*
+The 2D (Anisotropic model) is based on the reference below where *I(q)* is approximated for 1d scattering. Thus the
+scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model
+computation.
+The 2D (Anisotropic model) is based on the reference below where $I(q)$ is
+approximated for 1d scattering. Thus the scattering pattern for 2D may not
+be accurate. Note that we are not responsible for any incorrectness of the 2D
+model computation.
 .. image:: img/image165.gif
+.. image:: img/bcc_orientation.gif
 .. image:: img/image171.jpg
+.. image:: img/bcc_2d.jpg
 *Figure. 2D plot using the default values (w/200X200 pixels).*
 REFERENCE
+---------
 Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765
 …
     assumed to be isotropic and characterized by a Gaussian distribution.
     """
+category="shape:paracrystal"
 parameters = [

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