Space group

The complete group of spatial symmetry operations of a periodic crystal, including translations and combinations such as screw axes and glide planes.

A crystallographic orbit is the set of points generated from one point

r

by all operations of a specified space group

G

G r = {g r : g \in G} .

(1)

Because the space group includes lattice translations, the full orbit is infinite. The familiar finite list of equivalent fractional positions lists representatives within one conventional unit cell.^[ref]

For an original example in

P \overset{ˉ}{1}

, take

r = (0.2, 0.3, 0.4)

. Inversion gives

(- 0.2, - 0.3, - 0.4)

, represented in the same cell as

(0.8, 0.7, 0.6)

. Lattice translations repeat both sites throughout space. At the origin, inversion produces no additional site, so there is only one representative per primitive cell. The smaller crystallographic site multiplicity comes from the larger site-symmetry group.

s = g r

, their site-symmetry groups obey

G_{s} = g G_{r} g^{- 1} .

(2)

Indeed, an operation fixing

r

becomes an operation fixing

g r

after conjugation. This connects spatial equivalence to conjugate subgroups.

Several distinct crystallographic orbits can belong to the same Wyckoff position. For example, replacing the generic coordinates above by

(0.1, 0.2, 0.3)

usually produces a different pair, while both pairs have trivial site-symmetry groups. A Wyckoff position classifies the conjugacy type; an orbit supplies one particular symmetry-generated set of positions.^[ref]

Site-symmetry group ( $G_{r}$ )

 0  0

The site-symmetry group of a point

r

in a crystal structure is the subgroup of its space group

G

that fixes that point:

G_{r} = {g \in G : g r = r} .

(1)

It is the stabilizer subgroup for the action on points. Although the space group contains infinitely many lattice translations, the site-symmetry group is finite and is isomorphic to a subgroup of the corresponding crystallographic point group.^[ref]

For an original example, take a lattice with inversion symmetry and no other point operations, the space-group type conventionally denoted

P \overset{ˉ}{1}

. At the origin, both the identity and inversion

r \mapsto - r

fix the site, giving a group of order2. A generic site has only the identity in its site-symmetry group; inversion instead produces a distinct site at

- r

The equality in the definition fixes the actual point. At

r = a /2

, inversion alone moves it to

- a /2

. Inversion followed by translation by

a

fixes it. Reducing fractional coordinates modulo integers can conceal that translation, but the full space group operation retains it.

The crystallographic site multiplicity consequently satisfies

m = \frac{m _{general}}{∣ G _{r} ∣} .

(2)

In this primitive inversion example the general multiplicity is2, so the origin has multiplicity1 while a generic site has multiplicity2. This is why high site symmetry reduces the number of distinct sites generated in a cell.^[ref]

Wyckoff position

 0  0

A Wyckoff position of a space group consists of all points whose site-symmetry groups are conjugate subgroups of that space group. It classifies sites by their symmetry relationship within the whole group. A Wyckoff letter labels each such class in the conventional tables; the label is accompanied by its crystallographic site multiplicity.^[ref]

A Wyckoff position need not specify one particular atom or one fixed coordinate triplet. A general position usually has free fractional parameters. Choosing those parameters produces a crystallographic orbit, the collection of equivalent sites generated from that choice by the space group. Different parameter choices can give distinct orbits belonging to the same Wyckoff position.

An original comparison uses

P \overset{ˉ}{1}

, with operations

r \mapsto \pm r + n

, where

n

is a lattice vector. Generic positions form pairs

{r, - r}

modulo lattice translations and have trivial site-symmetry groups. Many different pairs therefore belong to the same general Wyckoff position.

The origin and the half-cell point

(1/2, 0, 0)

instead each have an inversion-containing site-symmetry group of order2. Their groups are isomorphic but not conjugate within this space group: a group operation sends the origin to an integer lattice point, never to that half-cell point. They occupy different special Wyckoff positions. Group order or multiplicity alone therefore does not uniquely identify a Wyckoff position.

This classification helps constrain independent atomic coordinates in a crystal structure; it describes allowed symmetry, not which chemical species occupies a site or its fractional occupation.

Wyckoff letter

 0  0

A Wyckoff letter is the conventional alphabetic label identifying a Wyckoff position within the tables for a space group. It is normally written beside the crystallographic site multiplicity, as in

1 a

. Letters begin with

a

for a position with maximal site-symmetry group order and proceed toward the general position. The letter is a label, not a coordinate or a chemical-species symbol.^[ref]

For example, the space group

P 1

has only translations. Every point has a trivial site-symmetry group, and the general position is labeled

1 a

: one equivalent position per primitive unit cell, with free coordinates

(x, y, z)

.^[ref]

An original model makes the distinction from atom counting clear. Place one species at

(0.1, 0.2, 0.3)

and another at

(0.4, 0.1, 0.7)

in a generic

P 1

crystal structure. Both sites are described by

1 a

, but they define different crystallographic orbits. The notation does not say that the entire structure contains only one atom per cell.

The space group, its setting and coordinate convention must accompany the letter. The same letter in another group need not denote the same symmetry or coordinates. Nor does matching multiplicity by itself identify the Wyckoff position, since distinct positions can have equal multiplicity.

Nonsymmorphic space group

 0  0

A space group whose point-group operations cannot all be represented without essential fractional translations for any origin. screw axes and glide planes are characteristic examples. Its point group alone omits these translational data.

Symmorphic space group

 0  0

A space group for which an origin can be chosen so that representatives of all point-group operations have zero translation part. It is a semidirect product of its lattice translations and point group. Presence of centring translations in a conventional cell does not alone make a group nonsymmorphic.

Seitz notation ( ${R ∣ t}$ )

 0  0

Write a Euclidean isometry as

{R ∣ t}

, acting by

x \mapsto R x + t

. Composition is

{R ∣ t} {R^{'} ∣ t^{'}} = {R R^{'} ∣ R t^{'} + t}

. The translation subgroup is normal; the quotient is the crystallographic point group.

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A space group is a mathematical classification used in crystallography that describes the symmetries of a crystal structure. Specifically, it combines the symmetries of both the lattice (the periodic arrangement of points in space) and the motif (the group of atoms associated with each lattice point). In other words, a space group encapsulates how a crystal can be transformed into itself through operations like translations, rotations, reflections, and inversions.

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