Crystal Lattice#

A crystal lattice describes the periodic arrangement of atoms in a crystal. Its geometry is characterised by six parameters: three unit-cell edge lengths \(a, b, c\) (in ångströms) and three inter-edge angles \(\alpha, \beta, \gamma\) (in degrees).

The lattice matters to diffractometer operation for two reasons:

  1. UB matrix — the \(B\) matrix is computed directly from the lattice parameters and transforms Miller indices \((h, k, l)\) into Cartesian reciprocal-space coordinates. Without accurate lattice parameters the \(UB\) matrix, and therefore every forward() and inverse() calculation, will be wrong.

  2. Crystal symmetry — the crystal system (cubic, hexagonal, …) constrains which lattice parameters are independent. hklpy2 determines the crystal system from the supplied parameters automatically and reports it; if the parameters are inconsistent with any known system an exception is raised.

In hklpy2, lattice parameters are stored in an instance of Lattice, which is part of the sample.

See also

Sample — the sample that owns the lattice.

How to Compute and Set the UB Matrix — computing the UB matrix from the lattice and orientation reflections.

Glossary — definitions of lattice, UB matrix, and related terms.

The Seven 3-D Crystal Systems (highest to lowest symmetry)

system

command

a

b

c

alpha

beta

gamma

cubic

Lattice(5.)

5

5

5

90

90

90

hexagonal

Lattice(4., c=3., gamma=120)

4

4

3

90

90

120

rhombohedral

Lattice(4., alpha=80.2)

4

4

4

80.2

80.2

80.2

tetragonal

Lattice(4, c=3)

4

4

3

90

90

90

orthorhombic

Lattice(4, 5, 3)

4

5

3

90

90

90

monoclinic

Lattice(4, 5, 3, beta=75)

4

5

3

90

75

90

triclinic

Lattice(4, 5, 3, 75., 85., 95.)

4

5

3

75

85

95

Default values are italicized. It is not necessary to supply the default parameters.

Examples

 1>>> from hklpy2 import Lattice
 2>>> Lattice(5.)
 3Lattice(a=5.0, b=5.0, c=5.0, alpha=90.0, beta=90.0, gamma=90.0)
 4
 5>>> Lattice(4., c=3., gamma=120)
 6Lattice(a=4.0, b=4.0, c=3.0, alpha=90.0, beta=90.0, gamma=120)
 7
 8>>> Lattice(4., alpha=80.2)
 9Lattice(a=4.0, b=4.0, c=4.0, alpha=80.2, beta=80.2, gamma=80.2)
10
11>>> Lattice(4, c=3)
12Lattice(a=4, b=4, c=3, alpha=90.0, beta=90.0, gamma=90.0)
13
14>>> Lattice(4, 5, 3)
15Lattice(a=4, b=5, c=3, alpha=90.0, beta=90.0, gamma=90.0)
16
17>>> Lattice(4, 5, 3, beta=75)
18Lattice(a=4, b=5, c=3, alpha=90.0, beta=75, gamma=90.0)
19
20>>> Lattice(4, 5, 3, 75., 85., 95.)
21Lattice(a=4, b=5, c=3, alpha=75.0, beta=85.0, gamma=95.0)

See also

The Seven 3-D Crystal Systems (highest to lowest symmetry) is also cross-referenced from Diffractometers.

IUCr: Crystal system

Wikipedia: Crystal system