# Adaptive cutoff methods¶

A fixed cutoff radius can introduce limitations to explore the local environment of the particle in some cases:

- At finite temperatures, when thermal fluctuations take place, the selection of a fixed cutoff may result in an inaccurate description of the local environment.
- If there is more than one structure present in the system, for example, bcc and fcc, the selection of cutoff such that it includes the first shell of both structures can be difficult.

In order to achieve a more accurate description of the local environment, various adaptive approaches have been proposed. Two of the methods implemented in the module are discussed below.

## Solid angle based nearest neighbor algorithm (SANN)¶

SANN algorithm [1] determines the cutoff radius by counting the solid angles around an atom and equating it to \(4\pi\). The algorithm solves the following equation iteratively.

where \(i\) is the host atom, \(j\) are its neighbors with \(r_{ij}\) is the distance between atoms \(i\) and \(j\). \(R_i\) is the cutoff radius for each particle \(i\) which is found by increasing the neighbor of neighbors \(m\) iteratively. For a description of the algorithm and more details, please check the reference [2]. SANN algorithm can be used to find the neighbors by,

```
import pyscal.core as pc
sys = pc.System()
sys.read_inputfile('conf.dump')
sys.find_neighbors(method='cutoff', cutoff='sann')
```

Since SANN algorithm involves sorting, a sufficiently large cutoff is used in the beginning to reduce the number entries to be sorted. This parameter is calculated by,

\[r_{initial} = \mathrm{threshold} \times \bigg(\frac{\mathrm{Simulation~box~volume}}{\mathrm{Number~of~particles}}\bigg)^{\frac{1}{3}}\]

a tunable `threshold`

parameter can be set through function arguments.

## Adaptive cutoff method¶

An adaptive cutoff specific for each atom can also be found using an algorithm similar to adaptive common neighbor analysis [2]. This adaptive cutoff is calculated by first making a list of all neighbor distances for each atom similar to SANN method. Once this list is available, then the cutoff is calculated from,

\[r_{cut}(i) = \mathrm{padding}\times \bigg(\frac{1}{\mathrm{nlimit}} \sum_{j=1}^{\mathrm{nlimit}} r_{ij} \bigg)\]

This method can be chosen by,

```
import pyscal.core as pc
sys = pc.System()
sys.read_inputfile('conf.dump')
sys.find_neighbors(method='cutoff', cutoff='adaptive')
```

The `padding`

and `nlimit`

parameters in the above equation can be
tuned using the respective keywords.

Either of the adaptive method can be used to find neighbors, which can then be used to calculate Steinhardt’s parameters or their averaged version.

[1] | van Meel, JA, Filion, L, Valeriani, C, Frenkel, D, J Chem Phys 234107, 2012. |

[2] | (1, 2) Stukowski, A, Model Simul Mater SC 20, 2012. |