Table 3095a. P3m1 (164) space group.
Name in the International Tables for Crystallography 
P3m1 

P 

164 

3m 
Crystal system 
Trigonal 

3m 
Asymm 
0≤ x ≤ 2/3 and 0 ≤ y ≤ 1/3 and 0 ≤ z ≤ 1 and x ≤ (1+y)/2 and y ≤ x/2 

12 
Symmetry (atomic coordinates) 
Y, X, Z; Y, X, Z; XY, Y, Z; X+Y, Y, Z; X, X+Y, Z; X, XY, Z.
X, Y, Z; Y, XY, Z; X+Y, X, Z; XY, X, Z; X, Y, Z; Y, YX, Z. 
Indices 
k, h, l; k, h, l; h, hk, l; h, h+k, l; hk, k, l; h+k, k, l. 
Table 3095b. Examples of materials with P3m1 (164) space group.

Parameters 
Angles 
α = 90.0°; β = 90.0°; γ = 120.0° 

Material 
Ba_{3}NiNb_{2}O_{9} [3] 
Lattice parameters 
a = b = 5.7550(5) Å; c = 7.0656(2) Å 
XRD pattern 
The (112) and (103) peaks in the inset indicates 1:2 ordering of B site ions (NiNbNb) 
HAADF Zcontrast 
The intensities of the Nb and Ni atom columns in [010] direction shows the NiNbNb ordering. Brightest contrast: Ba (Z=88); brighter contrast: Nb (Z = 73); weak contrast: Ni (Z=46). 

Material 
CaAl_{2}Si_{2 }[1] 
Structure with AlSi_{4} trigonal pyramids and CaSi_{6} trigonal
antiprisms (refer to MgAl_{2}Ge_{2} below) 


Material 
K_{3}Na(CrO_{4})_{2} 
Lattice parameters 
a = b = 5.858 Å; c = 7.523 Å 

Material 
Mg(OH)_{2} 
Lattice parameters 
a = b = 3.1420 Å; c = 4.7660 Å 

Material 
(K,Na)_{3}Na(SO_{4})_{2} (aphthitalite) 
Lattice parameters 
a = b = 5.68 Å; c = 7.31 Å 

Material 
MgAl_{2}Ge_{2} [1] 
Lattice parameters 
a = b = 4.11693(5) Å; c = 6.7873(1) Å 
Cell volume V 
99.627(3) Å^{3} 
Atomic coordinates 
Mg: Wyckoff position = 1a, x = 0, y = 0, z = 0; Al: Wyckoff position = 2d, x = ⅓, y = ⅔, z = 0.6353(3); Ge: Wyckoff position = 2d, x = ⅓, y = ⅔, z = 0.2419(2) 
Interatomic distances and coordination polyhedra 
Atoms 

δ (Å) 
Polyhedron 
Mg 
6 Ge 
2.8888(8) 

Al 
3 Ge
1 Ge 
2.5188(8)
2.6701(24) 

Ge 
3 Al
1 Al
3 Mg 
2.5188(8)
2.6701(24)
2.8888(8) 



Material 
αZr_{2}N_{2}S [2] 
Lattice parameters 
a = b = 3.605(2) Å; c = 6.412(3) Å 
HRTEM 
Inset: 0 0 1 zone axis electron diffraction pattern in hexagonal symmetry without superstructure reflections 
For hcp crystals, the (0 0 0 l) reflections, e.g. for the case with 164 (P3m1) space group, are forbidden when l is odd. However, those reflection positions often show diffraction intensity, which is probably caused by chemical order on the basal planes, or by double or multiple diffraction (scattering).
For [0 1 1 0] Sb_{3}Te_{2} in Figure 3095a, the ratio of AB/CD is the layer number (N) in a unit cell. We can know the Sb_{3}Te_{2} crystal belongs to the P3m1 space group as N is 9, a multiple of 3. Note that we can index diffraction patterns starting from comparing experimental dspacings and theoretical calculations obtained from known lattice parameters, for instance, use the excel file for crystals with P3m1 space group.



[0001] ηphase Cu_{3}Si [7]
(a = 4.06Å and c = 7.33Å) 





[0 1 1 0] Ge_{2}Bi_{2}Te_{5}. [4] 
[0 1 1 0] Sb_{3}Te_{2}. [4] 




[1 2 1 0] Ge_{2}Bi_{2}Te_{5}. [4] 
[2 1 1 0] Ge_{2}Bi_{2}Te_{5}
(space group p3m1) [4] 
[2 1 1 0] Ge_{2}Bi_{2}Te_{5}. [4] 



[1 1 2 0] Ge_{1}Sb_{2}Te_{4}. [5] 
[1 1 2 0] Ge_{1}Sb_{2}Te_{4}. [6]
(a' = 0.425 nm and c' = 4.10 nm) 
[1 1 2 0] Ge_{3}Sb_{2}Te_{6}. [6]
(a' = 0.425 nm and c' = 6.26 nm) 



[1 1 2 0] Ge_{2}Sb_{2}Te_{5}. [6]
(a' = 0.425 nm and c' = 1.827 nm) 





[3 3 0 1] ηphase Cu_{3}Si [7]
(a = 4.06Å and c = 7.33Å) 





[1 4 11 1] Ge_{2}Bi_{2}Te_{5}. [4] 
[1 0 10 1] Ge_{2}Bi_{2}Te_{5}. [4] 

Figure 3095a. Examples of indexed electron diffraction (or FFT) patterns of 164 (P3m1) HCP crystals.
[1] Svitlana Pukas, Liliya Pylypchak, Oksana Matselko, Pavlo Demchenko, Roman Gladyshevskii, MgAl2Ge2 – a new representative of the structure type CaAl2Si2, Chem. Met. Alloys 5 (2012) 5965.
[2] Chad Alan Stoltz, Synthesis of Layered Group IV Nitride Materials by Soft Chemical Anion Metathesis, thesis, 2005.
[3] Jungmin Hwang, Magnetoelectric and Multiferroic Properties in Layered 3d Transition Metal Oxides, thesis, 2012.
[4] Chang Woo Sun, A Transmission Electron Microscopy Study on the Microstructural Properties of Tebased Chalcogenide Thin Films, Doctoral Thesis, 2010.
[5] Enzo Rotunno, Advanced analytical transmission electron microscopy methodologies for the study of the chemical and physical properties of semiconducting nanostructures, Doctoral Thesis, 2014.
[6] B. J. Kooi and J. Th. M. De Hosson, Electron diffraction and highresolution transmission electron microscopy of the high temperature crystal structures of GexSb2Te3+x (x = 1, 2, 3
) phase change material, J. Appl. Phys.92 (7), 3584 (2002).
[7] ChengYen Wen & Frans Spaepen, Insitu electron microscopy of the phases of Cu3Si, Philosophical Magazine, 87(35), 55815599, 2007.
