Steno’s law: constancy of interfacial angles

Nicolas Steno (1669) found that quartz crystals, however unevenly grown, always meet at the same angles between corresponding faces. The angle between two faces depends only on their Miller indices – not on how big the faces are – which interplanar_angle_cubic() computes for a cubic crystal from the face normals.

import matplotlib.pyplot as plt
import numpy as np

from chemistrykit.crystal.systems.crystal_systems import interplanar_angle_cubic

pairs = [
    ((1, 0, 0), (0, 1, 0), "cube / cube"),
    ((1, 0, 0), (1, 1, 1), "cube / octahedron"),
    ((1, 1, 1), (1, 1, -1), "octahedron / octahedron"),
    ((1, 0, 0), (1, 1, 0), "cube / dodecahedron"),
    ((1, 1, 0), (1, 1, 1), "dodecahedron / octahedron"),
]
for hkl_1, hkl_2, label in pairs:
    print(f"{str(hkl_1):12s} {str(hkl_2):12s} {label:28s} {interplanar_angle_cubic(hkl_1, hkl_2):7.2f} deg")

(1, 0, 0)    (0, 1, 0)    cube / cube                    90.00 deg
(1, 0, 0)    (1, 1, 1)    cube / octahedron              54.74 deg
(1, 1, 1)    (1, 1, -1)   octahedron / octahedron        70.53 deg
(1, 0, 0)    (1, 1, 0)    cube / dodecahedron            45.00 deg
(1, 1, 0)    (1, 1, 1)    dodecahedron / octahedron      35.26 deg

Steno’s point: growing one face bigger than another (a distorted crystal) changes the face sizes, never the angles. Scaling the face normals – the same faces pushed outward by different amounts – leaves every interfacial angle unchanged:

rng = np.random.default_rng(0)
for _ in range(3):
    s1, s2 = rng.uniform(0.5, 3.0, size=2)
    angle = np.degrees(np.arccos(np.dot(s1 * np.array([1, 0, 0]), s2 * np.array([1, 1, 1])) / (s1 * s2 * np.sqrt(3.0))))
    print(f"face distances {s1:.2f}, {s2:.2f}: cube/octahedron angle = {angle:.4f} deg")
    assert np.isclose(angle, interplanar_angle_cubic((1, 0, 0), (1, 1, 1)))

face distances 2.09, 1.17: cube/octahedron angle = 54.7356 deg
face distances 0.60, 0.54: cube/octahedron angle = 54.7356 deg
face distances 2.53, 2.78: cube/octahedron angle = 54.7356 deg

A regular and a distorted cuboctahedron-like cross section (the {100} and {110} faces cut in the (001) plane): same angles, different shapes.

fig, axes = plt.subplots(1, 2, figsize=(9, 4.5))
normals = [np.array([np.cos(t), np.sin(t)]) for t in np.radians(np.arange(0, 360, 45))]
for ax, distances, title in (
    (axes[0], np.ones(8), "regular crystal"),
    (axes[1], np.array([1.0, 0.8, 1.6, 1.2, 1.3, 0.9, 1.1, 1.4]), "distorted crystal"),
):
    corners = []
    for i in range(8):
        n1, n2 = normals[i], normals[(i + 1) % 8]
        corners.append(np.linalg.solve(np.array([n1, n2]), np.array([distances[i], distances[(i + 1) % 8]])))
    corners = np.array(corners + [corners[0]])
    ax.plot(corners[:, 0], corners[:, 1], "k-")
    ax.fill(corners[:, 0], corners[:, 1], alpha=0.2)
    ax.set_aspect("equal")
    ax.set_title(f"{title}: every interior angle = 135 deg")
    ax.axis("off")
plt.tight_layout()
plt.show()