In this example, [disorder variable](random link) which was introduced to measure the disorder of a system is explored. We start by importing the necessary modules. We will use :mod:~pyscal.crystal_structures
to create the necessary crystal structures.
import pyscal.core as pc
import pyscal.crystal_structures as pcs
import matplotlib.pyplot as plt
import numpy as np
First an fcc structure with a lattice constant of 4.00 is created.
fcc_atoms, fcc_box = pcs.make_crystal('fcc', lattice_constant=4, repetitions=[4,4,4])
The created atoms and box are assigned to a :class:~pyscal.core.System
object.
fcc = pc.System()
fcc.atoms = fcc_atoms
fcc.box = fcc_box
The next step is find the neighbors, and the calculate the Steinhardt parameter based on which we could calculate the disorder variable.
fcc.find_neighbors(method='cutoff', cutoff='adaptive')
Once the neighbors are found, we can calculate the Steinhardt parameter value. In this example $q=6$ will be used.
fcc.calculate_q(6)
Finally, disorder parameter can be calculated.
fcc.calculate_disorder()
The calculated disorder value can be accessed for each atom using the :attr:~pyscal.catom.disorder
variable.
atoms = fcc.atoms
disorder = [atom.disorder for atom in atoms]
np.mean(disorder)
-1.041556887034408e-16
As expected, for a perfect fcc structure, we can see that the disorder is zero. The variation of disorder variable on a distorted lattice can be explored now. We will once again use the noise
keyword along with :func:~pyscal.crystal_structures.make_crystal
to create a distorted lattice.
fcc_atoms_d1, fcc_box_d1 = pcs.make_crystal('fcc', lattice_constant=4, repetitions=[4,4,4], noise=0.01)
fcc_d1 = pc.System()
fcc_d1.atoms = fcc_atoms_d1
fcc_d1.box = fcc_box_d1
Once again, find neighbors and then calculate disorder
fcc_d1.find_neighbors(method='cutoff', cutoff='adaptive')
fcc_d1.calculate_q(6)
fcc_d1.calculate_disorder()
Check the value of disorder
atoms_d1 = fcc_d1.atoms
disorder = [atom.disorder for atom in atoms_d1]
np.mean(disorder)
0.013889967380485688
The value of average disorder for the system has increased with noise. Finally trying with a high amount of noise.
fcc_atoms_d2, fcc_box_d2 = pcs.make_crystal('fcc', lattice_constant=4, repetitions=[4,4,4], noise=0.1)
fcc_d2 = pc.System()
fcc_d2.atoms = fcc_atoms_d2
fcc_d2.box = fcc_box_d2
fcc_d2.find_neighbors(method='cutoff', cutoff='adaptive')
fcc_d2.calculate_q(6)
fcc_d2.calculate_disorder()
atoms_d2 = fcc_d2.atoms
disorder = [atom.disorder for atom in atoms_d2]
np.mean(disorder)
1.8469165876016702
The value of disorder parameter shows an increase with the amount of lattice distortion. An averaged version of disorder parameter, averaged over the neighbors for each atom can also be calculated as shown below.
fcc_d2.calculate_disorder(averaged=True)
atoms_d2 = fcc_d2.atoms
disorder = [atom.avg_disorder for atom in atoms_d2]
np.mean(disorder)
1.850630088115515
The disorder parameter can also be calculated for values of Steinhardt parameter other than 6. For example,
fcc_d2.find_neighbors(method='cutoff', cutoff='adaptive')
fcc_d2.calculate_q([4, 6])
fcc_d2.calculate_disorder(q=4, averaged=True)
atoms_d2 = fcc_d2.atoms
disorder = [atom.disorder for atom in atoms_d2]
np.mean(disorder)
1.880741277448693
$q=4$, for example, can be useful when measuring disorder in bcc crystals