Cowley short range order parameter¶

The Cowley short range order parameter can be used to find if an alloy is ordered or not. The order parameter is given by,

$\alpha_i = 1 - \frac{n_i}{m_A c_i}$

where $$n_i$$ is the number of atoms of the non reference type among the $$c_i$$ atoms in the $$i$$th shell. $$m_A$$ is the concentration of the non reference atom.

We can start by importing the necessary modules

[1]:

import pyscal as pc
import pyscal.crystal_structures as pcs
import matplotlib.pyplot as plt


We need a binary alloy structure to calculate the order parameter. We will use the crystal structures modules to do this. Here, we will create a L12 structure.

[2]:

atoms, box = pcs.make_crystal('l12', lattice_constant=4.00, repetitions=[2,2,2])


In order to use the order parameter, we need to have two shells of neighbors around the atom. In order to get two shells of neighbors, we will first estimate a cutoff using the radial distribution function.

[4]:

sys = pc.System()
sys.box = box
sys.atoms = atoms

[5]:

val, dist = sys.calculate_rdf()


We can plot the rdf,

[7]:

plt.plot(dist, val)
plt.xlabel(r"distance $\AA$")
plt.ylabel(r"$g(r)$")
plt.xlim(0, 5)

[7]:

(0.0, 5.0)


In this case, a cutoff of about 4.5 will make sure that two shells are included. Now the neighbors are calculated using this cutoff.

[8]:

sys.find_neighbors(method='cutoff', cutoff=4.5)


Finally we can calculate the short range order. We will use the reference type as 1 and also specify the average keyword as True. This will allow us to get an average value for the whole simulation box.

[9]:

sys.calculate_sro(reference_type=1, average=True)

[9]:

array([-0.33333333,  1.        ])


Value for individual atoms can be accessed by,

[10]:

atoms = sys.atoms

[11]:

atoms[4].sro

[11]:

[-0.33333333333333326, 1.0]


Only atoms of the non reference type will have this value!