Synthetic visualization of fibre orientation and the corresponding
Orientation Distribution Function (ODF) for prescribed
Hermans Orientation Parameter (HOP) values.
This repository provides a controlled, synthetic visualization of fibre orientation statistics and their relationship to the Hermans Orientation Parameter (HOP).
It generates synthetic fibre cross-sections (line segments within a disk) together with their corresponding Orientation Distribution Functions (ODFs), for prescribed HOP values spanning:
- transverse alignment (HOP < 0),
- isotropic orientation (HOP = 0),
- axial alignment (HOP > 0).
The figure layout is:
- Left column: Real-space fibre configuration in a 2D cross-section (reference direction indicated by an arrow)
- Right column: Orientation Distribution Function (ODF) shown in polar coordinates
(using the standard undirected-fibre mapping α = 2θ to represent θ ∈ [0°, 180°) on a full 0–360° polar axis)
The purpose of this project is to build physical intuition for how a scalar orientational order parameter (HOP) relates to the full angular distribution of fibre orientations.
The Hermans Orientation Parameter,
HOP = 0.5 * (3 ⟨cos²φ⟩ − 1)
represents the second Legendre moment of the orientation distribution relative to a reference axis. While HOP provides a compact scalar measure of orientational order, it does not uniquely define the full ODF. Different angular distributions may yield identical HOP values.
This repository therefore illustrates:
- how increasing |HOP| sharpens the angular distribution,
- how the sign of HOP distinguishes axial vs transverse preferential alignment,
- and how scalar order parameters relate to full angular statistics.
The Hermans Orientation Parameter is defined as:
[ HOP = \frac{1}{2}\left(3\langle \cos^2\phi \rangle - 1\right) ]
Its theoretical range is:
- −0.5 → perfect perpendicular alignment
- 0 → isotropic (random orientation)
- 1 → perfect alignment with the reference direction
Interpretation:
- HOP > 0: preferential alignment with the reference axis
- HOP = 0: no preferred direction
- HOP < 0: preferential alignment perpendicular to the reference axis
- Dependencies are pinned via Pixi (
pixi.tomlandpixi.lock) - Random seeds are fixed for deterministic outputs
- Figures are generated algorithmically from synthetic orientation distributions
pixi run hop-odf
Arvin (Fazel) Mirzaei
Postdoctoral Researcher ; fazel.mirzaei@psi.ch
Paul Scherrer Institute (PSI), Switzerland.
Paul Scherrer Institute (PSI) is Switzerland’s largest research institute for natural and engineering sciences, focusing on cutting-edge research in energy, materials, and health.
This project was developed as part of the research paper Multi-directional neutron dark-field tomography of nanostructural orientation in cellulose foams published in Materials Today Advances DOI: https://doi.org/10.1016/j.mtadv.2026.100824
Corresponding author: Prof. Dr. Markus Strobl markus.strobl@psi.ch