P42/mnm (D4h14, 136) Space Group

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P4₂/mnm (D_4h¹⁴, 136) Space Group

Table 3018a. P4₂/mnm (D_4h¹⁴, 136) space group.

Name in the International Tables for Crystallography

P4₂/mnm (simplified from P4₂/m2_l/n2/m: the symbol 4₂ means that an axis of the fourth order is a screw axis, with translation on half a period along this axis on rotation by angle π/2 or 360/4 degrees, and the symbol n means that two of the four vertical planes are planes of sliding reflection.)

Lattice type

Patterson space group

123

Point group

4/mmm

Crystal system

Tetragonal

Laue class

4/mmm

Asymm

0 ≤ x ≤ 1/2 and 0 ≤ y ≤ 1/2 and 0 ≤ z ≤ 1/2 and x ≤ y

Number of symmetry operators

Symmetry (atomic coordinates)

X, Y, Z; -X, -Y, Z; 1/2-Y, X+1/2, Z+1/2; Y+1/2, 1/2-X, Z+1/2; 1/2-X, Y+1/2, 1/2-Z; X+1/2, 1/2-Y, 1/2-Z; Y, X, -Z; -Y, -X, -Z; -X, -Y, -Z; X, Y, -Z; Y+1/2, 1/2-X, 1/2-Z; 1/2-Y, X+1/2, 1/2-Z; X+1/2, 1/2-Y, Z+1/2; 1/2-X, Y+1/2, Z+1/2; -Y, -X, Z; Y, X, Z

Wyckoff positions

Wyckoff positions	Oriented site symmetry groups (E_j)	Coordinates of the Wyckoff positions in the unit cell
2a	m.mm	(0, 0, 0), (1/2, 1/2, 1/2)
2b	m.mm	(0, 0, 1/2), (1/2, 1/2, 0)
4c	2/m..	(0, 1/2, 0), (0, 1/2, 1/2), (1/2, 0, 1/2), (1/2, 0, 0)
4d	-4..	(0, 1/2, 1/4), (0, 1/2, 3/4), (1/2, 0, 1/4), (1/2, 0, 3/4)
4e	2.mm	(0, 0, z), (1/2, 1/2, z+1/2), (1/2, 1/2, 1/2-z), (0, 0, -z)
4f	m.2m	(x, x, 0), (-x, -x, 0), (1/2-x, 1/2+x, 1/2), (1/2+x, 1/2-x, 1/2)
4g	m.2m	(x, -x, 0), (-x, x, 0), (1/2+x, 1/2+x, 1/2), (1/2-x, 1/2-x, 1/2)
8h	2..	(0, 1/2, z), (0, 1/2, z + 1/2), (1/2, 0,−z + 1/2), (1/2, 0,−z),
		(0, 1/2,−z), (0, 1/2,−z + 1/2), (1/2, 0, z + 1/2), (1/2, 0, z).
8i	m..	(x, y, 0), (−x,−y, 0), (−y + 1/2, x + 1/2, 1/2), (y + 1/2,−x + 1/2, 1/2),
		(−x + 1/2, y + 1/2, 1/2), (x + 1/2,−y + 1/2, 1/2), (y, x, 0), (−y,−x, 0).
8j	..m	(x, x, z), (−x,−x, z), (−x + 1/2, x + 1/2, z + 1/2), (x + 1/2,−x + 1/2, z + 1/2),
		(−x + 1/2, x + 1/2,−z + 1/2), (x + 1/2,−x + 1/2,−z + 1/2), (x, x,−z), (−x,−x,−z)
16k	1	(x, y, z), (-x, -y, z), (1/2-y, 1/2+x, 1/2+z), (1/2+y, 1/2-x, 1/2+z),
		(1/2-x, 1/2+y, 1/2-z), (1/2+x, 1/2-y, 1/2-z), (y, x, -z), (-y, -x, -z),
		(-x, -y, -z), (x, y, -z), (1/2+y, 1/2-x, 1/2-z), (1/2-y, 1/2+x, 1/2-z),
		(1/2+x, 1/2-y, 1/2+z), (1/2-x, 1/2+y, 1/2+z), (-y, -x, z), (y, x, z).

Material examples

MnF₂: a = 0.4874 nm and c = 0.3310 nm.
NiF₂: a = 0.4650 nm and c = 0.3084 nm.
SnO₂: a = 0.4738 nm and c = 0.3187 nm.
TiO₂: a = 0.4593 nm and c = 0.2959 nm: Ti at 0, 0, 0 and O at 0.3053, 0.3053, 0.
CoF₂: a = 0.4695 nm and c = 0.3182 nm.
MgF₂
α-MgH₂

Table 3018b. Example of compounds with similar structures to BCC.

	BCC	TiO₂ (Rutile)	CsCl	Cu₂O (Cuprite)
Space group	Im-3m	P42/mnm	Pm-3m	Pn-3m
Compounds	α-Fe, W, Nb, Ta, Na, K, Mo, V	GeO₂, SnO₂, PbO₂	CsI, CsBr, CsCl	Pb₂O, Ag₂O


[-1 1 1] tetragonal rutile TiO₂ [1]

Figure 3018. Examples of indexed electron diffraction (or FFT) patterns of P4₂/mnm (136) crystals.

[1] Chaolun Liang, Wenxia Zhao, Xianfeng Yang, Mingmei Wu and Yexiang Tong, Applications of High Resolution Electron Microscopy in Structural Analysis of Nanoarrays, Current Microscopy Contributions to Advances in Science and Technology, 1271, 2012.

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P4₂/mnm (D_4h¹⁴, 136) Space Group