Lattice

For the full reference see Lattice

This guide describe attributes and methods of the general lattice class. Each Bravais lattice class is a child of Lattice, thus all it`s method and attribute are available. For the description of each Bravais lattice class see:

• Bravais lattices
  • Creation
  • Cubic lattice system
  • Tetragonal lattice system
  • Orthorhombic lattice system
  • Hexagonal lattice system
  • Rhombohedral lattice system
  • Monoclinic lattice system
  • Triclinic lattice system
• References

Creation

Lattice can be created in three different ways:

• From cell matrix:

>>> from radtools import Lattice
>>> cell = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>>> lattice = Lattice(cell)
>>> lattice.cell
array([[1, 0, 0],
       [0, 1, 0],
       [0, 0, 1]])

• From three lattice vectors \(\vec{a}_1\), \(\vec{a}_2\), \(\vec{a}_3\):

>>> from radtools import Lattice
>>> a1 = [1, 0, 0]
>>> a2 = [0, 1, 0]
>>> a3 = [0, 0, 1]
>>> lattice = Lattice(a1, a2, a3)
>>> lattice.cell
array([[1, 0, 0],
       [0, 1, 0],
       [0, 0, 1]])

• From lattice parameters \(a\), \(b\), \(c\), \(\alpha\), \(\beta\), \(\gamma\):

>>> from radtools import Lattice
>>> lattice = Lattice(1, 1, 1, 90, 90, 90)
>>> import numpy as np
>>> np.round(lattice.cell, decimals=1)
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])

Identification of the lattice

Bravais lattice type can be calculated via Lattice.identify() method:

>>> from radtools import Lattice
>>> lattice = Lattice(1, 1, 1, 90, 90, 90)
>>> lattice.identify()
'CUB'

Identification procedure is realised in the lepage() function. For the algorithm description and reference see LePage algorithm.

Lattice.identify method returns a string with the lattice type, but does not change the lattice itself. Therefore, if you want to get an instance of CUB lattice, you should create it manually:

>>> from radtools import Lattice
>>> lattice = Lattice(1, 1, 1, 90, 90, 90)
>>> lattice.identify()
'CUB'
>>> type(lattice)
<class 'radtools.crystal.lattice.Lattice'>
>>> from radtools import bravais_lattice_from_cell
>>> lattice = bravais_lattice_from_cell(lattice.cell)
>>> type(lattice)
<class 'radtools.crystal.bravais_lattice.CUB'>

Note

Lattice identification is not a trivial task. The algorithm is based on the assumption that the lattice`s unit cell is primitive.

Reference attributes

If lattice is an instance of one of the Bravais lattice classes, then you can use the following attributes for the information about the lattice:

>>> from radtools import bravais_lattice_from_cell
>>> lattice = bravais_lattice_from_cell([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
>>> lattice.pearson_symbol
'cP'
>>> lattice.crystal_family
'c'
>>> lattice.centring_type
'P'

Lattice parameters

All lattice parameters can be accessed as attributes:

• Real space

>>> from radtools import bravais_lattice_from_cell
>>> lattice = bravais_lattice_from_cell([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
>>> lattice.a1
array([1, 0, 0])
>>> lattice.a2
array([0, 1, 0])
>>> lattice.a3
array([0, 0, 1])
>>> lattice.cell
array([[1, 0, 0],
       [0, 1, 0],
       [0, 0, 1]])
>>> lattice.a
1.0
>>> lattice.b
1.0
>>> lattice.c
1.0
>>> lattice.alpha
90.0
>>> lattice.beta
90.0
>>> lattice.gamma
90.0
>>> lattice.unit_cell_volume
1
>>> lattice.parameters
(1.0, 1.0, 1.0, 90.0, 90.0, 90.0)

• Reciprocal space

>>> from radtools import bravais_lattice_from_cell
>>> from math import pi
>>> lattice = bravais_lattice_from_cell([[2*pi, 0, 0], [0, 2*pi, 0], [0, 0, 2*pi]])
>>> lattice.b1
array([1., 0., 0.])
>>> lattice.b2
array([0., 1., 0.])
>>> lattice.b3
array([0., 0., 1.])
>>> lattice.reciprocal_cell
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])
>>> lattice.k_a
1.0
>>> lattice.k_b
1.0
>>> lattice.k_c
1.0
>>> lattice.k_alpha
90.0
>>> lattice.k_beta
90.0
>>> lattice.k_gamma
90.0

Variation of the lattice

Some of the Bravais lattice classes have several variations.

To check the variation of the lattice use Lattice.variation attribute:

>>> from radtools import lattice_example, Lattice
>>> lattice = lattice_example("BCT")
>>> lattice.variation
'BCT1'
>>> lattice = Lattice(1, 1, 1, 90, 90, 90)
>>> lattice.variation
'Lattice'

Plotting of the lattice

Lattice primitive, conventional unit cells as well as the Wigner-Seitz cell and Brillouin zone can be plotted using Lattice.plot() method:

>>> from radtools import lattice_example
>>> lattice = lattice_example("BCT")
>>> lattice.plot("brillouin")
>>> lattice.plot("kpath")

To show the plot use Lattice.show() method:

>>> lattice.show()

More examples of the plots with the code snippets can be found in the Bravais lattices guide.

Full list of the available plotting methods can be found in the Lattice reference.

K points

Path in reciprocal space and k points for plotting and calculation are partially implemented in a separate class Kpoints.

High symmetry points, path are the part of the Lattice class, but once they are set an instance of Kpoints class is expected to be created in order to access labels of the plot, flatten coordinates, whole list of k points, etc.

>>> from radtools import Lattice
>>> lattice = Lattice(1, 1, 1, 90, 90, 90)
>>> lattice.add_kpoint("Gamma", [0, 0, 0])
>>> lattice.add_kpoint("X", [0.5, 0, 0])
>>> lattice.add_kpoint("M", [0.5, 0.5, 0])
>>> lattice.add_kpoint("CP", [0.5, 0.5, 0.5], plot_name="Custom plot name")
>>> lattice.path = "Gamma-X|M-CP-X"
>>> # n = 100 by default, it could be changed on the kp instance.
>>> kp = lattice.get_kpoints(n=100)

Note

For each Bravais lattice class there is a predefined path and set of kpoints in reciprocal space. See Bravais lattices for more details.

For the full guide on how to use Kpoints class see K points.