Define the Sample Lattice#

This guide shows how to specify unit-cell parameters for any crystal system and inspect the resulting B matrix and reciprocal lattice.

Create a Lattice#

Pass the minimum parameters needed for your crystal system:

import ad_hoc_diffractometer as ahd

# Cubic: a only
silicon = ahd.Lattice(a=5.431)

# Tetragonal: a and c
tio2 = ahd.Lattice(a=4.594, c=2.959)

# Hexagonal: a, c, and gamma=120 (must be explicit)
sapphire = ahd.Lattice(a=4.785, c=12.991, gamma=120)

# Trigonal: a and alpha ≠ 90
calcite = ahd.Lattice(a=6.375, alpha=46.1)

# Orthorhombic: a, b, c
aragonite = ahd.Lattice(a=4.962, b=7.967, c=5.741)

# Monoclinic: a, b, c, beta
gypsum = ahd.Lattice(a=5.68, b=15.18, c=6.29, beta=118.4)

# Triclinic: a, b, c, alpha, beta, gamma
albite = ahd.Lattice(a=8.144, b=12.787, c=7.160,
                     alpha=94.33, beta=116.57, gamma=87.65)

The crystal system is deduced automatically from the supplied parameters:

print(silicon.crystal_system)  # 'cubic'
print(sapphire.crystal_system) # 'hexagonal'

Crystal system summary#

System	Minimum parameters required
Cubic	`a`
Tetragonal	`a`, `c`
Hexagonal	`a`, `c`, `gamma=120` (must be explicit)
Trigonal	`a`, `alpha` (alpha ≠ 90)
Orthorhombic	`a`, `b`, `c`
Monoclinic	`a`, `b`, `c`, `beta`
Triclinic	`a`, `b`, `c`, `alpha`, `beta`, `gamma`

Assign the lattice to a sample#

g = ahd.presets.fourcv()
g.sample.lattice = silicon

Inspect the B matrix#

import numpy as np

B = g.sample.lattice.B
print(B)

The B matrix satisfies \(|\mathbf{B}\,\mathbf{h}| = 2\pi / d_{hkl}\):

h = [0, 0, 1]
Q_mag = np.linalg.norm(B @ h)
d_hkl = 2 * np.pi / Q_mag
print(f"d(001) = {d_hkl:.4f} Å")

Inspect the reciprocal lattice#

b1, b2, b3 = g.sample.lattice.reciprocal_lattice_vectors
print(f"b1 = {b1}")

Compute d-spacing and Q for a reflection#

d   = ahd.hkl_to_d(g, 1, 1, 0)
Q   = ahd.hkl_to_Q(g, 1, 1, 0)
tth = ahd.conversions.hkl_to_two_theta(g, 1, 1, 0)
print(f"d(110) = {d:.4f} Å, 2θ = {tth:.3f}°")

See also#