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
\[ \left\langle \left| f^2(q) \right| \right\rangle = \frac{\int_{0}^{\infty} P(R) f^2(q,R) dR }{\int_{0}^{\infty} P(R) dR} \]
\[ \left| \left\langle f(q) \right\rangle \right| ^2 = \left( \frac{\int_{0}^{\infty} P(R) f(q,R) dR }{\int_{0}^{\infty} P(R) dR} \right)^2 ,\]
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
\[ Z_k(q) = \frac{1- \left| F_k \right|^2}{1 - 2\left| F_k \right| \cos(\mathbf{q \cdot a_k}) + \left| F_k \right|^2} ,\]
\[ \left| F_k \right| = \exp \left\{ -\frac{1}{2} \left( \Delta a/a \right)^2 \left[ \left( \mathbf{q \cdot a_1 } \right)^2 + \left( \mathbf{q \cdot a_2 } \right)^2 \right] \right\} ,\]
\[ \mathbf{q \cdot a_1} = -a q \cos{(\psi-\frac{\pi}{6})} ,\]
\[ \mathbf{q \cdot a_2 } = a q \sin{\psi} .\]

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 $q> 7.061 \times 10^{-5} a^2 - 0.007413a + 0.2465$}

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$^{-1}$!).


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

* **Author:** Paavo Penttilä **Date:** March 15, 2019

Example Data:


Created By penttila
Uploaded March 15, 2019, 2:12 p.m.
Category Cylinder
Score 0
Verified This model has not been verified by a member of the SasView team
In Library This model is not currently included in the SasView library. You must download the files and install it yourself.
Files woodsas.py


No comments yet.

Please log in to add a comment.