source: sasmodels/sasmodels/models/bcc.py @ b89f519

#bcc paracrystal model
#note model title and parameter table are automatically inserted
#note - calculation requires double precision
r"""
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.

Definition
----------

The scattering intensity $I(q)$ is calculated as

.. math:

    I(q) = \frac{\text{scale}}{V_P} V_\text{lattice} P(q) Z(q)


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.

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

.. math:

    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/bcc_lattice.jpg

For a crystal, diffraction peaks appear at reduced q-values given by

.. math:

    \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2}

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)

.. math:

    \begin{eqnarray}
    &q/q_o&&\quad 1&& \ \sqrt{2} && \ \sqrt{3} && \ \sqrt{4} && \ \sqrt{5} \\
    &\text{Indices}&& (110) && (200) && (211) && (220) && (310)
    \end{eqnarray}

**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.

.. image:: img/bcc_orientation.gif

.. 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
(Original Paper)

Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856
(Corrections to FCC and BCC lattice structure calculation)
"""

from numpy import pi, inf

name = "bcc_paracrystal"
title = "Body-centred cubic lattic with paracrystalline distortion"
description = """
    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.
    """
category="shape:paracrystal"

parameters = [
#   [ "name", "units", default, [lower, upper], "type","description" ],
    [ "dnn", "Ang", 220, [-inf,inf],"","Nearest neighbour distance"],
    [ "d_factor", "", 0.06,[-inf,inf],"","Paracrystal distortion factor" ],
    [ "radius", "Ang",  40, [0, inf], "volume","Particle radius" ],
    [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "", "Particle scattering length density" ],
    [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "","Solvent scattering length density" ],
    [ "theta", "degrees", 60, [-inf, inf], "orientation","In plane angle" ],
    [ "phi", "degrees", 60, [-inf, inf], "orientation","Out of plane angle" ],
    [ "psi", "degrees", 60, [-inf,inf], "orientation","Out of plane angle"]
    ]

source = [ "lib/J1.c", "lib/gauss150.c", "bcc.c" ]

# parameters for demo
demo = dict(
    scale=1, background=0,
    dnn=220, d_factor=0.06, sld=4, solvent_sld=1,
    radius=40,
    theta=60, phi=60, psi=60,
    radius_pd=.2, radius_pd_n=0.2,
    theta_pd=15, theta_pd_n=0,
    phi_pd=15, phi_pd_n=0,
    psi_pd=15, psi_pd_n=0,
    )

# For testing against the old sasview models, include the converted parameter
# names and the target sasview model name.
oldname='BCCrystalModel'
oldpars=dict(sld='sldSph',
             solvent_sld='sldSolv')