Weiss’s crystal systems: classifying a unit cell by its axes

Christian Samuel Weiss (1815) sorted crystals into systems by the lengths of, and angles between, their crystallographic axes – the idea that survives as today’s 7 crystal systems. classify_crystal_system() classifies a unit cell purely from the equalities/inequalities among its six lattice parameters \((a,b,c,\alpha,\beta,\gamma)\), and unit_cell_volume() gives the general-cell volume formula that reduces to familiar shortcuts (\(abc\), \(\frac{\sqrt3}{2}a^2c\)) in special cases.

import numpy as np

from chemistrykit.crystal.systems.crystal_systems import classify_crystal_system, unit_cell_volume

cells = {
    "NaCl (rock salt)": (5.640, 5.640, 5.640, 90.0, 90.0, 90.0),
    "TiO2 (rutile)": (4.593, 4.593, 2.959, 90.0, 90.0, 90.0),
    "alpha-quartz": (4.913, 4.913, 5.405, 90.0, 90.0, 120.0),
    "calcite (rhombohedral setting)": (6.375, 6.375, 6.375, 46.08, 46.08, 46.08),
    "gypsum": (5.679, 15.202, 6.522, 90.0, 113.83, 90.0),
    "K2Cr2O7 (potassium dichromate)": (7.418, 7.318, 13.379, 98.6, 90.4, 95.5),
}

for name, (a, b, c, alpha, beta, gamma) in cells.items():
    system = classify_crystal_system(a, b, c, alpha, beta, gamma)
    V = unit_cell_volume(a, b, c, alpha, beta, gamma)
    print(f"{name:35s} -> {system:14s} V = {V:8.2f} A^3")

NaCl (rock salt)                    -> cubic          V =   179.41 A^3
TiO2 (rutile)                       -> tetragonal     V =    62.42 A^3
alpha-quartz                        -> hexagonal      V =   112.98 A^3
calcite (rhombohedral setting)      -> trigonal       V =   122.63 A^3
gypsum                              -> monoclinic     V =   515.06 A^3
K2Cr2O7 (potassium dichromate)      -> triclinic      V =   714.64 A^3

A right-angle (orthorhombic/tetragonal/cubic) cell’s volume is exactly a*b*c – the general formula’s simplest special case:

V_general = unit_cell_volume(4.593, 4.593, 2.959, 90.0, 90.0, 90.0)
V_shortcut = 4.593 * 4.593 * 2.959
print(f"\nRutile: general formula = {V_general:.4f}, a*b*c shortcut = {V_shortcut:.4f}")
assert np.isclose(V_general, V_shortcut)

Rutile: general formula = 62.4220, a*b*c shortcut = 62.4220

The hexagonal shortcut (sqrt(3)/2 * a^2 * c) matches the general formula too, for quartz’s hexagonal cell:

a, c = 4.913, 5.405
V_general = unit_cell_volume(a, a, c, 90.0, 90.0, 120.0)
V_shortcut = (np.sqrt(3.0) / 2.0) * a**2 * c
print(f"Quartz:  general formula = {V_general:.4f}, hexagonal shortcut = {V_shortcut:.4f}")
assert np.isclose(V_general, V_shortcut)

Quartz:  general formula = 112.9848, hexagonal shortcut = 112.9848