WoodSAS

This model is tailored for fitting the equatorial intensity profile from wood samples (Penttilä et al., 2019). The model consists of three independent contributions:
1) Scattering in the plane perpendicular to the long axis of infinite cylinders packed in a hexagonal lattice with paracrystalline distortion (based on Hashimoto et al., 1994)
2) Gaussian function centered at $q = 0$
3) Power law scattering

The fitted function is
\[ I(q) = A I_{cyl}(q,\bar{R},\Delta R/\bar{R},a,\Delta a /a) + B \exp{-q^2/(2\sigma^2)} + C q^{-\alpha} + background ,\]
where the cylinder radius $R$ has a Gaussian distribution with mean $\bar{R}$ and standard deviation $\Delta R$, and the paracrystalline distortion of the distance $a$ between the cylinders' center points is characterized by $\Delta a$.

The cylinder contribution is
\[ I_{cyl}(q) = \frac{1}{2\pi} \int_{0}^{2\pi} I_{\perp}(q,\psi) d\psi , \]
where $\psi$ is the rotational angle around the cylinder axis and
\[ I_{\perp}(q,\psi) = \left\langle \left| f^2 \right| \right\rangle - \left| \left\langle f \right\rangle \right| ^2 + \left| \left\langle f \right\rangle \right| ^2 Z_1 Z_2 .\]

The form factor of an infinitely long cylinder is
\[ f(q, R) = A_{cyl} \frac{J_1(qR)}{qR} = \pi R \frac{J_1(qR)}{q} ,\]
where $J_1$ is the Bessel function of the first kind and $A_{cyl}$ the cross-sectional area of the cylinder.

The terms with averaging are
\[ \left\langle \left| f^2(q) \right| \right\rangle = \frac{\int_{0}^{\infty} P(R) f^2(q,R) dR }{\int_{0}^{\infty} P(R) dR} \]
and
\[ \left| \left\langle f(q) \right\rangle \right| ^2 = \left( \frac{\int_{0}^{\infty} P(R) f(q,R) dR }{\int_{0}^{\infty} P(R) dR} \right)^2 ,\]
where the Gaussian distribution of the radius is
\[ P(R) \propto \exp \left[ -\frac{(R-\bar{R})^2}{2(\Delta R)^2} \right].\]

The paracrystalline lattice factors $Z_1$ and $Z_2$ for a hexagonal lattice with lattice vectors a$_1$ and a$_2$ are
\[ Z_k(q) = \frac{1- \left| F_k \right|^2}{1 - 2\left| F_k \right| \cos(\mathbf{q \cdot a_k}) + \left| F_k \right|^2} ,\]
where
\[ \left| F_k \right| = \exp \left\{ -\frac{1}{2} \left( \Delta a/a \right)^2 \left[ \left( \mathbf{q \cdot a_1 } \right)^2 + \left( \mathbf{q \cdot a_2 } \right)^2 \right] \right\} ,\]
\[ \mathbf{q \cdot a_1} = -a q \cos{(\psi-\frac{\pi}{6})} ,\]
\[ \mathbf{q \cdot a_2 } = a q \sin{\psi} .\]

The lattice factor $Z_k(q)$ has been modified according to Penttilä et al, 2019:
\[ Z_k =\begin{cases}
Z_k(q_0), & \text{if $q \leq 7.061 \times 10^{-5} a^2 - 0.007413a + 0.2465$} \\
Z_k(q) \text{ as in Hashimoto et al., 1994}, & \text{if $q> 7.061 \times 10^{-5} a^2 - 0.007413a + 0.2465$}
\end{cases}\]

A detailed description of the model is given in reference Penttilä et al., 2019.

For the model to work properly, the scaling parameter of SasView should be fixed to 1.0 and $da/a$ should be larger than 0. The output intensity is given in arbitrary units (not in cm$^{-1}$!).

References
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Hashimoto, T., Kawamura, T., Harada, M., & Tanaka, H. (1994). Macromolecules, 27, 3063-3072. DOI: 10.1021/ma00089a025
Penttilä, P. A., Rautkari, L., Österberg, M., & Schweins, R. (2019). Journal of Applied Crystallography, 52, 369-377. DOI: 10.1107/S1600576719002012

Authorship and Verification
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* **Author:** Paavo Penttilä **Date:** March 15, 2019

Example Data: