# Introduction to Pyboo¶

A Python package to compute bond orientational order parameters as defined by Steinhardt Physical Review B (1983) doi:10.1103/PhysRevB.28.784.

Steinhardt’s bond orientational order parameter is a popular method (>20k citations of the original paper) to identify local symmetries in an assembly of particles in 3D. It can be used in particle-based simulations (typically molecular dynamics, brownian dynamics, monte-carlo, etc.) or in particle-tracking experiments (colloids, granular materials) where the coordinates of all particles are known.

## Licence, citations, contact¶

This code is under GPL 3.0 licence. See LICENCE file.

Please cite Pyboo and it’s author(s) in any scientific publication using this software.

@misc{
pyboo,
title={Pyboo: A Python package to compute bond orientational order parameters},
author={Mathieu Leocmach},
year={2017},
doi={10.5281/zenodo.1066568},
url={https://github.com/MathieuLeocmach/pyboo}
}

Contact
Mathieu LEOCMACH, Institut Lumière Matière, UMR-CNRS 5306, Lyon, France mathieu.leocmach AT univ-lyon1.fr

## Installation¶

Dependencies are numpy, scipy and numba. Tested with python 2.7 and python 3.5.

You can install with pip:

pip install pyboo


## Input¶

The present library takes as input a (N, 3) array of float coordinates named pos and a (M, 2) array of integers named bonds that defines the bond network. If bonds contains the couple (10, 55) it means that there is a bond between the particles which coordinates can be found at pos[10] and pos[55]. The bonds are supposed unique and bidirectional, therefore if the bond (10, 55) is in bonds, the bond (55, 10) exists implicitely and should not be part of bonds.

The library is agnostic on how the bonds were determined. Possible choices are (among others):
• two particles closer than a maximum distance,
• Delaunay triangulation.

Other libraries have very efficient implementations of theses methods. See for example scipy.spatial.KDTree for fast spatial query or scipy.spatial.Delaunay.

Alternatively Pyboo can take as input arrays of k nearest neighbours (that can also be obtained from scipy.spatial.KDTree). For example in a dense assembly of hard spheres, one expects the important structures (FCC, HCP, icosahedron) to be made of a central particle and its 12 neighbours. The k=12 first neighbours are stored in ngbs, a (N, 12) array of indicies. If for a reason or an other some particles have less than 12 neighbours, the remaining slots can be filled by negative values, e.g. -1.

## Spherical harmonics¶

The 3D orientation of each bond can be projected on a base of spherical harmonics $$Y_{\ell m}(\theta,\phi)$$, where $$\ell \geq 0$$ indicates the order of symmetry, and $$m$$, with $$-\ell \geq m \geq \ell$$, indicates the orientation with respect to the referent set of orthonormal axes. Since our bonds are bidirectional $$\ell$$ is even. Averaging these coefficients over a set of bonds yields a measure of the symmetry of this set.

For example the whole system is characterised by the coefficients

$\bar{q}_{\ell m} = \frac{1}{M} \sum_{i,j} Y_{\ell m}(\theta_{ij},\phi_{ij})$

where the average is taken over all the bonds. More locally, each particle $$i$$ is characterised by the coefficients

$q_{\ell m}(i) = \frac{1}{N_i}\sum_{0}^{N_i} Y_{\ell m}(\theta_{ij},\phi_{ij})$

where there are $$N_i$$ bonds starting from $$i$$. We call the $$q_{\ell m}$$ coefficients the local tensorial bond orientational order parameter.

The function bonds2qlm() computes the $$q_{\ell m}$$ coefficients for the $$\ell$$-fold symetry. The extra parameter periods allows to do so in periodic boundary conditions.

The function ngbs2qlm() does the same from a k neighbours array. For all particles $$N_i=k$$, even for particles with less than k neighbours.

## Invarients¶

The tensorial $$q_{\ell m}$$ coefficients are dependent of the orientation of the reference axis. That is why we have to compute quantities that are rotationally invarients:
• the second order invarient indicates the strength of the $$\ell$$-fold symetry.
$q_\ell = \sqrt{\frac{4\pi}{2l+1} \sum_{m=-\ell}^{\ell} |q_{\ell m}|^2 }$
• the third order invarient allows to discriminate different types of $$\ell$$-fold symetric structures.
$\begin{split}w_\ell = \sum_{m_1+m_2+m_3=0} \left( \begin{array}{ccc} \ell & \ell & \ell \\ m_1 & m_2 & m_3 \end{array} \right) q_{\ell m_1} q_{\ell m_2} q_{\ell m_3}\end{split}$

where the term in brackets is the Wigner 3-j symbol. For example $$w_6$$ allows to disctiminate icosahedral structures, see Leocmach & Tanaka, Nature Com. (2012) doi: 10.1038/ncomms1974.

Invarients can be computed respectively by ql() and wl().

## Coarse-graining¶

It is possible to coarse-grain the tensorial bond orientational order to get more information about the second shell of neighbours and thus discriminate better crystal structures, see Lechner & Delago J. Chem. Phys. (2008) doi:10.1063/1.2977970:

$Q_{\ell m}(i) = \frac{1}{N_i+1}\left( q_{\ell m}(i) + \sum_{j=0}^{N_i} q_{\ell m}(j)\right)$

here the central particle is included in the sum.

Coarse-graining can be done with coarsegrain_qlm() or coarsegrain_qlm_ngbs(). The parameter inside is a (N) array of booleans indicating particles where the original $$q_{\ell m}$$ coefficients were truthfully determined. Counter examples (where inside takes the value False) are particles that were too close to one edge of the experimental window, so that some of their neighbours were not dectected, causing a incomplete bond set. Coarse-grained invariants $$Q_\ell$$ and $$W_\ell$$ can be computed in the same way by ql() and wl() respectively.

## Cross product¶

The similarity between the symetry and the orientation of two neighbourhoods can be estimated by the normalized scalar product

$s_\ell(i,j) = \frac{4\pi}{2\ell + 1} \frac{\sum_{m=-\ell}^{\ell} q_{\ell m}(i) q_{\ell m}^{*}(j)}{q_\ell(i) q_\ell(j)}$

This quantity is the result of product() divided by ql(qlm1) * ql(qlm2). The similarity between all neighbouring particles can be obtained from bond_normed_product().

Typical use: when $$s_6(i,j)$$ is larger than a threshold value (typically 0.7) the bond can be considered crystalline. A particle is considered crystalline when it has at least 7 crystalline bonds. See Auer & Frenkel, J.Chem.Phys. (2004) doi: 10.1063/1.1638740. This procedure is implemented in x_particles().

In a more continuous manner, the crystallinity parameter is defined as the average of the cross products over the neighbours, see Russo & Tanaka, Sci Rep. (2012) doi:10.1038/srep00505.

$C_\ell(i) = \frac{1}{N_i} \sum_{j=0}{N_i} s_\ell (i,j)$

Crystallinity parameter is computed by crystallinity().

## Spatial correlation¶

To know how spatially extended is the local symmetry and orientation, one can look at the average cross product at a certain distance.

$g_\ell(r) = \frac{\sum_{i,j} s_\ell(i,j)\delta(r_{ij}-r)}{\sum_{i,j} \delta(r_{ij}-r)}$

where $$\delta$$ is a binning function equal to one between 0 and $$dr$$ and zero elsewhere.

The function gG_l() returns separately the numerator and denominator of the above expression to ease further averaging. maxdist is the maximum range to consider and Nbins the number of bins between 0 and maxdist. qlms is a list of bond orientational order coefficients that can have different values of $$\ell$$, some coarse-grained or not. is_center is a (N) array of boolean marking the particles that are further than maxdist from any edge of the experimental window in order to avoid edge effects.