K points

For the full reference see Kpoints

Kpoints is a relatively small class, which represent the set of k-points of a particular path of a given lattice.

It is used to generate full kpoints path both for calculation and for plotting.

Import

>>> # Exact import
>>> from radtools.crystal.kpoints import Kpoints
>>> # Explicit import
>>> from radtools.crystal import Kpoints
>>> # Recommended import
>>> from radtools import Kpoints

For the examples in this page we need additional import and some predefined variables:

>>> from radtools import lattice_example

Creation

Usually it is created from some :py:class`.Lattice` (or :py:class`.Crystal`):

>>> lattice = lattice_example("CUB")
>>> kp = lattice.kpoints
>>> kp.hs_names
['G', 'M', 'R', 'X']

However, it could be created explicitly:

>>> b1, b2, b3 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>>> names = ["G", "X"]
>>> coordinates = [[0, 0, 0], [0.5, 0, 0]]
>>> labels = [R"$\Gamma$", "X"]
>>> kp = Kpoints(b1, b2, b3, names=names, coordinates=coordinates, labels=labels)
>>> kp.hs_names
['G', 'X']

For the full list of constructor parameters see Kpoints documentation.

High-symmetry points

Information about high symmetry points is accessible through the following properties:

• Kpoints.hs_names

List of names of high symmetry points.

>>> kp.hs_names
['G', 'X']

• Kpoints.hs_coordinates

Dictionary of coordinates of high symmetry points.

>>> kp.hs_coordinates
{'G': array([0, 0, 0]), 'X': array([0.5, 0. , 0. ])}

• Kpoints.hs_labels

Dictionary of labels of high symmetry points. Usually used for plotting.

>>> kp.hs_labels
{'G': '$\\Gamma$', 'X': 'X'}

Adding a point

>>> kp.add_hs_point(name="M", coordinates=[0.5, 0.5, 0], label="M")
>>> kp.hs_names
['G', 'X', 'M']
>>> kp.hs_coordinates
{'G': array([0, 0, 0]), 'X': array([0.5, 0. , 0. ]), 'M': array([0.5, 0.5, 0. ])}
>>> kp.hs_labels
{'G': '$\\Gamma$', 'X': 'X', 'M': 'M'}

Path

The path is the route in the reciprocal space, defined by the high symmetry points.

We use a specific format in the package: "G-K-X|R-S". "-" separates high symmetry points in each subpath, "|" separates sections of the path. In the example n points are generated between "G" and "K", between "K" ans "X", between "R" and "S", but not between "X" and "R". By default path is constructed from the list of high symmetry points.

>>> b1, b2, b3 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>>> names = ["G", "K", "X", "R"]
>>> coordinates = [[0, 0, 0], [0.5, 0.5, 0], [0.5, 0, 0], [0.5, 0.5, 0.5]]
>>> labels = ["$\Gamma$", "K", "X", "R"]
>>> kp = Kpoints(b1, b2, b3, names=names, coordinates=coordinates, labels=labels)
>>> kp.path
[['G', 'K', 'X', 'R']]
>>> # It cause an Error, because high symmetry point "S" is not defined
>>> kp.path = "G-K-X|R-S"
Traceback (most recent call last):
...
ValueError: Point 'S' is not defined. Defined points are:
  G : [0 0 0]
  K : [0.5 0.5 0. ]
  X : [0.5 0.  0. ]
  R : [0.5 0.5 0.5]
>>> kp.path = "G-K-X|R-G"
>>> kp.path
[['G', 'K', 'X'], ['R', 'G']]
>>> kp.add_hs_point(name="S", coordinates=[0.5, 0.5, 0.5], label="S")
>>> kp.path = "G-K-X|R-S"
>>> kp.path
[['G', 'K', 'X'], ['R', 'S']]
>>> kp.path_string
'G-K-X|R-S'

Note

Internally RAD-tools stores the path as a list of subpaths, where each subpath is a list of high symmetry points. This format is also correct for assigning the path attribute.

Configuration

The amount of kpoints to be generated between each pair of high symmetry points in the path is controlled by the Kpoints.n property.

>>> # Default value is 100
>>> kp.n
100
>>> kp.n = 10
>>> kp.n
10

Usage

Once the configuration of the Kpoints are done, it can be used for calculation or plotting.

Calculation

There is one property suitable for calculation: Kpoints.points. which is an array of all generated kpoints. For each pair of high symmetry points it generates Kpoints.n between them. The first and the last points are always the high symmetry points of this section of the path.

>>> b1, b2, b3 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
>>> names = ["G", "K", "X"]
>>> coordinates = [[0, 0, 0], [0.5, 0.5, 0], [0.5, 0, 0]]
>>> labels = ["$\Gamma$", "K", "X"]
>>> kp = Kpoints(b1, b2, b3, names=names, coordinates=coordinates, labels=labels, n=4)
>>> kp.points()
array([[0. , 0. , 0. ],
       [0.1, 0.1, 0. ],
       [0.2, 0.2, 0. ],
       [0.3, 0.3, 0. ],
       [0.4, 0.4, 0. ],
       [0.5, 0.5, 0. ],
       [0.5, 0.5, 0. ],
       [0.5, 0.4, 0. ],
       [0.5, 0.3, 0. ],
       [0.5, 0.2, 0. ],
       [0.5, 0.1, 0. ],
       [0.5, 0. , 0. ]])

Note

For each section the last point is repeated twice, because it is the first point of the next section of the path.

array([[0. , 0. , 0. ], # <--- Gamma
       [0.1, 0.1, 0. ],
       [0.2, 0.2, 0. ],
       [0.3, 0.3, 0. ],
       [0.4, 0.4, 0. ],
       [0.5, 0.5, 0. ], # <--- K
       [0.5, 0.5, 0. ], # <--- K
       [0.5, 0.4, 0. ],
       [0.5, 0.3, 0. ],
       [0.5, 0.2, 0. ],
       [0.5, 0.1, 0. ],
       [0.5, 0. , 0. ]]) # <--- X

Plotting

For plotting there are three properties. Two of them are for the high symmetry points and describe the labels and position of ticks on the x-axis:

>>> kp.labels
['$\\Gamma$', 'K', 'X']
>>> import numpy as np
>>> np.around(kp.coordinates(), decimals=4)
array([0.    , 0.7071, 1.2071])

The third property gives the coordinates of the Kpoints.points for the plot:

>>> for point in kp.flatten_points():
...     print(round(point, 4))
...
0.0
0.1414
0.2828
0.4243
0.5657
0.7071
0.7071
0.8071
0.9071
1.0071
1.1071
1.2071

Note

Those coordinates are directly corresponds to the k-points from the previous subsection.

0.0    # <--- Gamma
0.1414
0.2828
0.4243
0.5657
0.7071 # <--- K
0.7071 # <--- K
0.8071
0.9071
1.0071
1.1071
1.2071 # <--- X

Hint

Repeated Kpoints.points or Kpoints.flatten_points can be used to restore the position of high symmetry points in the path.