Note
Go to the end to download the full example code.
(extra) Wulfric¶
Exercises
Create a crystal structure, in which conventional and primitive cells are the same. Compute k-points and k-path for it.
Create a crystal structure, in which original cell is neither conventional nor primitive. Compute k-points and k-path for it.
For the crystal structure from this page compute k-points and k-path for the (2, 3, 1) super-cell. Compare high-symmetry points and k-path with those of the original cell.
For the crystal structure from this page compute k-points and k-path for the (2, 3, 1) super-cell, modifying the
spglib_typesin such a way that same atoms of the original cell that are located in different "sub-cells" of the super-cell have differentspglib_types. Compare high-symmetry points and k-path with those of the original cell.
import numpy as np
import wulfric
Before moving on to the examples we define two plotting routines, that will be useful for us later
def plot_real_space_cells(cell, atoms, conv_cell, conv_atoms, prim_cell, prim_atoms):
pe = wulfric.PlotlyEngine(_sphinx_gallery_fix=True)
pe.plot_cell(cell=cell, legend_label="Original cell", color="#000000")
pe.plot_atoms(
cell=cell,
atoms=atoms,
legend_label="Original atoms",
colors=["#000000" for _ in atoms["names"]],
)
pe.plot_cell(cell=conv_cell, legend_label="Conventional cell", color="#0057B7")
pe.plot_atoms(
cell=conv_cell,
atoms=conv_atoms,
legend_label="Conventional atoms",
colors=["#0057B7" for _ in conv_atoms["names"]],
)
pe.plot_cell(cell=prim_cell, legend_label="Primitive cell", color="#FFDD00")
pe.plot_atoms(
cell=prim_cell,
atoms=prim_atoms,
legend_label="Primitive atoms",
colors=["#FFDD00" for _ in prim_atoms["names"]],
)
return pe
def plot_k_points(kp, cell, prim_cell):
kp_of_input_cell = wulfric.Kpoints.from_crystal(
cell=cell, atoms=dict(spglib_types=[1], positions=[[0, 0, 0]])
)
pe = wulfric.PlotlyEngine(_sphinx_gallery_fix=True)
pe.plot_kpoints(
kp, color="#C8102E", legend_label="High-symmetry points (crystal-based)"
)
pe.plot_kpath(
kp, color="#C8102E", legend_label="Recommended k-path (crystal-based)"
)
pe.plot_brillouin_zone(
cell=prim_cell,
color="#C8102E",
legend_label="Brillouin zone of the primitive cell",
)
pe.plot_kpoints(
kp_of_input_cell,
color="#003DA5",
legend_label="High-symmetry points (cell-based)",
)
pe.plot_kpath(
kp_of_input_cell,
color="#003DA5",
legend_label="Recommended k-path (cell-based)",
)
pe.plot_brillouin_zone(
cell=cell,
color="#003DA5",
legend_label="Brillouin zone of the input cell",
)
return pe
Exercise 1¶
Simplest example is a cubic cell with a single atom at the origin. We will show another one - a hexagonal cell with two atoms.
# Create a crystal
cell = np.array(
[
[1.0, 0.0, 0.0],
[-0.5, np.sqrt(3) / 2, 0.0],
[0.0, 0.0, 2.0],
]
)
atoms = dict(
names=["Cr", "Br", "Br"],
positions=[[0.0, 0.0, 0.5], [1 / 3, 2 / 3, 0.25], [2 / 3, 1 / 3, 0.75]],
spins=[3 / 2, None, None],
g_factors=[2.0, None, None],
)
Use wulfric to analyse the crystal symmetry and compute conventional and primitive cells.
spglib_data = wulfric.get_spglib_data(cell=cell, atoms=atoms)
print(f"Space group: {spglib_data.space_group_number}")
print(f"Bravais lattice: {spglib_data.crystal_family + spglib_data.centring_type}")
# Compute conventional and primitive cells
conv_cell, conv_atoms = wulfric.crystal.get_conventional(
cell=cell, atoms=atoms, spglib_data=spglib_data, convention="HPKOT"
)
prim_cell, prim_atoms = wulfric.crystal.get_primitive(
cell=cell, atoms=atoms, spglib_data=spglib_data, convention="HPKOT"
)
Space group: 164
Bravais lattice: hP
Then, we visualize and compare real-space cells
Finally, we compute high-symmetry k-points and k-path for the given crystal.
kp = wulfric.Kpoints.from_crystal(cell=cell, atoms=atoms)
print(f"Recommended k-path: {kp.path_string}")
pe = plot_k_points(kp=kp, cell=cell, prim_cell=prim_cell)
pe.show(axes_visible=False, legend_position="left")
Recommended k-path: GAMMA-M-K-GAMMA-A-L-H-A|L-M|H-K
Exercise 2¶
Simplest example will be a supercell of any given cell. We will show another one, with one atom per unit cell.
# Create a crystal
cell = np.array(
[
[1.0, 0.0, 0.0],
[1, 1, 0.0],
[0.0, 0.0, 1.0],
]
)
atoms = dict(
names=["Fe"],
positions=[[0.0, 0.0, 0.0]],
spins=[5 / 2],
g_factors=[2.0],
)
Use wulfric to analyse the crystal symmetry and compute conventional and primitive cells.
spglib_data = wulfric.get_spglib_data(cell=cell, atoms=atoms)
print(f"Space group: {spglib_data.space_group_number}")
print(f"Bravais lattice: {spglib_data.crystal_family + spglib_data.centring_type}")
# Compute conventional and primitive cells
conv_cell, conv_atoms = wulfric.crystal.get_conventional(
cell=cell, atoms=atoms, spglib_data=spglib_data, convention="HPKOT"
)
prim_cell, prim_atoms = wulfric.crystal.get_primitive(
cell=cell, atoms=atoms, spglib_data=spglib_data, convention="HPKOT"
)
Space group: 221
Bravais lattice: cP
Then, we visualize and compare real-space cells
Finally, we compute high-symmetry k-points and k-path for the given crystal.
kp = wulfric.Kpoints.from_crystal(cell=cell, atoms=atoms)
print(f"Recommended k-path: {kp.path_string}")
pe = plot_k_points(kp=kp, cell=cell, prim_cell=prim_cell)
pe.show(axes_visible=False, legend_position="left")
Recommended k-path: GAMMA-X-M-GAMMA-R-X|R-M
Note
Even though the input cell and primitive cell are different, they both contain one lattice point per cell, therefore they span the same real-space and reciprocal space lattices. As a result, they have the same Brillouin zones.
Exercise 3¶
# Get a cubic cell with a = 4
cell = 4 * np.eye(3)
# Eight atoms: four will be magnetic, and four will not
atoms = dict(
names=["Fe1", "Fe2", "Fe3", "Fe4", "O1", "O2", "O3", "O4"],
positions=[
[0.0, 0.0, 0.0],
[0.5, 0.5, 0.0],
[0.0, 0.5, 0.5],
[0.5, 0.0, 0.5],
[0.5, 0.5, 0.5],
[0.5, 0.0, 0.0],
[0.0, 0.5, 0.0],
[0.0, 0.0, 0.5],
],
spins=[2.5, 2.5, 2.5, 2.5, None, None, None, None],
g_factors=[2, 2, 2, 2, None, None, None, None],
)
# Get spglib data for the original cell
spglib_data = wulfric.get_spglib_data(cell=cell, atoms=atoms)
# Make a supercell
super_cell = cell @ np.diag([2, 3, 1])
# Generate atoms of the super-cell
super_atoms = {}
for key in atoms:
super_atoms[key] = []
for i in range(2):
for j in range(3):
for name, pos, spin, g_factor in zip(
atoms["names"], atoms["positions"], atoms["spins"], atoms["g_factors"]
):
super_atoms["names"].append(name)
super_pos = [
(pos[0] + i) / 2,
(pos[1] + j) / 3,
pos[2] / 1,
]
super_atoms["positions"].append(super_pos)
super_atoms["spins"].append(spin)
super_atoms["g_factors"].append(g_factor)
# Get spglib data for the super-cell
super_spglib_data = wulfric.get_spglib_data(cell=super_cell, atoms=super_atoms)
Visualize original cell and supercell in real space
pe = wulfric.PlotlyEngine(_sphinx_gallery_fix=True)
pe.plot_cell(cell=cell, legend_label="Original cell", color="#000000")
pe.plot_atoms(
cell=cell,
atoms=atoms,
legend_label="Original atoms",
colors=["#000000" for _ in atoms["names"]],
)
pe.plot_cell(cell=super_cell, legend_label="Super-cell", color="#0057B7")
pe.plot_atoms(
cell=super_cell,
atoms=super_atoms,
legend_label="Atoms of the super-cell",
colors=["#0057B7" for _ in super_atoms["names"]],
)
pe.show(axes_visible=False, legend_position="left")
There are only two types of atoms in the super-cell, therefore original cell and super-cell generate the same crystal.
print(spglib_data.original_types)
print(super_spglib_data.original_types)
[1, 1, 1, 1, 2, 2, 2, 2]
[1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2]
Original cell and super-cell are of the same space group
print(f"Original cell space group: {spglib_data.space_group_number}")
print(
f"Original cell Bravais lattice: {spglib_data.crystal_family + spglib_data.centring_type}"
)
print(f"Super-cell space group: {super_spglib_data.space_group_number}")
print(
f"Super-cell Bravais lattice: {super_spglib_data.crystal_family + super_spglib_data.centring_type}"
)
Original cell space group: 225
Original cell Bravais lattice: cF
Super-cell space group: 225
Super-cell Bravais lattice: cF
They have the same primitive cell
prim_cell, _ = wulfric.crystal.get_primitive(
cell=cell, atoms=atoms, convention="HPKOT", spglib_data=spglib_data
)
super_prim_cell, _ = wulfric.crystal.get_primitive(
cell=super_cell,
atoms=super_atoms,
convention="HPKOT",
spglib_data=super_spglib_data,
)
print(f"Primitive cells are the same? {np.allclose(prim_cell, super_prim_cell)}")
Primitive cells are the same? True
They have the same high-symmetry k-points and k-path
kp = wulfric.Kpoints.from_crystal(cell=cell, atoms=atoms, spglib_data=spglib_data)
super_kp = wulfric.Kpoints.from_crystal(
cell=super_cell, atoms=super_atoms, spglib_data=super_spglib_data
)
print(f"Recommended k-path for the original cell: {kp.path_string}")
print(f"Recommended k-path for the super-cell: {super_kp.path_string}")
pe = wulfric.PlotlyEngine(_sphinx_gallery_fix=True)
pe.plot_kpoints(
kp, color="#C8102E", legend_label="High-symmetry points (original cell)"
)
pe.plot_kpath(kp, color="#C8102E", legend_label="Recommended k-path (original cell)")
pe.plot_brillouin_zone(
cell=cell,
color="#C8102E",
legend_label="Brillouin zone of the original cell",
)
pe.plot_kpoints(
super_kp, color="#003DA5", legend_label="High-symmetry points (super-cell)"
)
pe.plot_kpath(super_kp, color="#003DA5", legend_label="Recommended k-path (super-cell)")
pe.plot_brillouin_zone(
cell=super_cell,
color="#003DA5",
legend_label="Brillouin zone of the super-cell",
)
pe.plot_brillouin_zone(
cell=prim_cell,
color="#00A86B",
legend_label="Brillouin zone of the primitive cell",
)
pe.show(axes_visible=False, legend_position="left")
Recommended k-path for the original cell: GAMMA-X-U|K-GAMMA-L-W-X
Recommended k-path for the super-cell: GAMMA-X-U|K-GAMMA-L-W-X
Exercise 4¶
We re-use the super-cell from the previous exercise
# Change spglib_types of all atoms to reflect sub-cells of the super-cell
new_types = [1, 1, 1, 1, 2, 2, 2, 2] # original types for the 8 atoms
for i in range(1, 6): # 6 sub-cells in the super-cell
new_types += [
1 + 2 * i,
1 + 2 * i,
1 + 2 * i,
1 + 2 * i,
2 + 2 * i,
2 + 2 * i,
2 + 2 * i,
2 + 2 * i,
]
super_atoms["spglib_types"] = new_types
print("spglib_types from exercise 3 -> spglib_types for exercise 4")
for ex_3_type, ex_4_type in zip(
super_spglib_data.original_types, super_atoms["spglib_types"]
):
print(f"{ex_3_type:3} -> {ex_4_type}")
# Re-compute spglib data for the super-cell
super_spglib_data = wulfric.get_spglib_data(cell=super_cell, atoms=super_atoms)
spglib_types from exercise 3 -> spglib_types for exercise 4
1 -> 1
1 -> 1
1 -> 1
1 -> 1
2 -> 2
2 -> 2
2 -> 2
2 -> 2
1 -> 3
1 -> 3
1 -> 3
1 -> 3
2 -> 4
2 -> 4
2 -> 4
2 -> 4
1 -> 5
1 -> 5
1 -> 5
1 -> 5
2 -> 6
2 -> 6
2 -> 6
2 -> 6
1 -> 7
1 -> 7
1 -> 7
1 -> 7
2 -> 8
2 -> 8
2 -> 8
2 -> 8
1 -> 9
1 -> 9
1 -> 9
1 -> 9
2 -> 10
2 -> 10
2 -> 10
2 -> 10
1 -> 11
1 -> 11
1 -> 11
1 -> 11
2 -> 12
2 -> 12
2 -> 12
2 -> 12
Now original cell and supercell essentially describe different crystals. They have different space groups.
print(f"Original cell space group: {spglib_data.space_group_number}")
print(
f"Original cell Bravais lattice: {spglib_data.crystal_family + spglib_data.centring_type}"
)
print(f"Super-cell space group: {super_spglib_data.space_group_number}")
print(
f"Super-cell Bravais lattice: {super_spglib_data.crystal_family + super_spglib_data.centring_type}"
)
Original cell space group: 225
Original cell Bravais lattice: cF
Super-cell space group: 28
Super-cell Bravais lattice: oP
They have different primitive cells
prim_cell, _ = wulfric.crystal.get_primitive(
cell=cell, atoms=atoms, convention="HPKOT", spglib_data=spglib_data
)
super_prim_cell, _ = wulfric.crystal.get_primitive(
cell=super_cell,
atoms=super_atoms,
convention="HPKOT",
spglib_data=super_spglib_data,
)
print(f"Primitive cells are the same? {np.allclose(prim_cell, super_prim_cell)}")
Primitive cells are the same? False
They have different high-symmetry k-points and k-path
kp = wulfric.Kpoints.from_crystal(cell=cell, atoms=atoms, spglib_data=spglib_data)
super_kp = wulfric.Kpoints.from_crystal(
cell=super_cell, atoms=super_atoms, spglib_data=super_spglib_data
)
print(f"Recommended k-path for the original cell: {kp.path_string}")
print(f"Recommended k-path for the super-cell: {super_kp.path_string}")
pe = wulfric.PlotlyEngine(_sphinx_gallery_fix=True)
pe.plot_kpoints(
kp, color="#C8102E", legend_label="High-symmetry points (original cell)"
)
pe.plot_kpath(kp, color="#C8102E", legend_label="Recommended k-path (original cell)")
pe.plot_brillouin_zone(
cell=cell,
color="#C8102E",
legend_label="Brillouin zone of the original cell",
)
pe.plot_kpoints(
super_kp, color="#003DA5", legend_label="High-symmetry points (super-cell)"
)
pe.plot_kpath(super_kp, color="#003DA5", legend_label="Recommended k-path (super-cell)")
pe.plot_brillouin_zone(
cell=super_cell,
color="#003DA5",
legend_label="Brillouin zone of the super-cell",
)
pe.show(axes_visible=False, legend_position="left")
Recommended k-path for the original cell: GAMMA-X-U|K-GAMMA-L-W-X
Recommended k-path for the super-cell: GAMMA-X-S-Y-GAMMA-Z-U-R-T-Z|X-U|Y-T|S-R
Total running time of the script: (0 minutes 3.872 seconds)