Category Archives: crystal structures


The first densest lattice of chemical supertetrahedra?

In one of the last issues of the Angewandte, Wang and coworkers [1] presented an extremely interesting structure, in which metal-organic polyhedra (MOP) are assembled in different ways. These polyhedra consist of polyoxovanadate metal clusters and bridging dicarboxylate linkers. The authors call them VMOP. The overall shape of the MOP can be described as tetrahedron, or to be precise, as truncated tetrahedron see Fig. 1.

Fig. 1: VMOP, in atomistic representation (left) and simplified as a truncated tetrahedron (right).

In fact, the authors obtained two different phases, which differ in the kind of packing: in the VMOP-alpha isomer (a very low-density phase, which is thermodynamically less stable) each truncated tetrahedron makes perfect contact with four neighbors via the (small) four trigonal faces (the truncated faces), thus leading to a dia-like framework (corner-connected tetrahedra).

However, in VMOP-beta the MOPs are packed in a corner-to-face fashion, i.e. each
truncated tetrahedron has contact with eight neighboring tetrahedra (trigonal-to-hexagonal face-to-face-connection), see Fig. 2.

Fig. 2: The eight direct neighbors of a central truncated tetrahedron (in green) in the packing of VMOP-beta.

This packing mode of regular (non-truncated) tetrahedra is known as the densest lattice packing  (i.e. only translations are allowed) of tetrahedra, which was proven in 1969 by Hoylman [2]. The resulting packing density is 18/49 ≈ 36.73 %. Here, each tetrahedron is in contact with 14 others (4 corners + 4 faces + 6 edges). However, in VMOP-beta the 6 edge-edge connections are a bit further apart.

I think, the structure of VMOP-beta is very remarkable and I am not aware of any analogous chemical structure – do you?

PS: I would like to thank Ahmad Rafsanjani Abbasi (ETH Zürich) for bringing this structure to my attention.


[1] Y. Gong, Y. Zhang, C. Qin, C. Sun, X. Wang, Z. Su, Angew. Chem. Int. Ed. 201958, 780.

[2] D.J. Hoylman, Bull. Amer. Math. Soc. 1970, 76, 135.



Afwillite – a calcium nesosilicate


  • Named after the abbreviated discoverer Alpheus Fuller Williams (1874–1953), CEO of De Beers Consolidated Mines at that time
  • Formula: Ca3(SiO3OH)2 · 2 H2O
  • It belongs to the nesosilicates, i.e. there are only isolated SiO4 tetrahedra
  • Space group: Cc (No. 9)
  • Crystal system: monoclinic
  • Crystal class: m
  • Lattice parameters: a = 16.278 Å, b = 5.6321 Å, c = 13.236, α =  γ = 90°, β = 134.898°

Picture: Matteo Chinellato – | CC BY-SA 3.0

Crystal structure (click on the pictures to download the VESTA file):

(K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,” J. Appl. Crystallogr., 44, 1272-1276 (2011).)

  • Oxygen (red)
  • Hydrogen (white)
  • SiO4 tetrahedra (yellow)
  • CaO7 polyhedra (light blue)

For a 3D interactive version, see here:



  • Named after Wolfgang Sartorius von Waltershausen (1809 – 1876) Professor of Mineralogy, University of Göttingen, Germany. He was the first who described the mineral.
  • Formula: PbAs2S4
  • Space group: P21/(No. 14)
  • Crystal system: monoclinic
  • Crystal class: 2/m
  • Lattice parameters: a = 19.62 Å, b = 7.89 Å, c = 4.19 Å, α = γ = 90°,  β = 90° (!)

Picture: Rob Lavinsky, – CC BY-SA-3.0

Crystal structure (click on the picture to download the VESTA file):

(K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,”J. Appl. Crystallogr., 44, 1272-1276 (2011).)

  • PbS9 polyhedra (gray)
  • AsS3 trigonal pyramids (green)
  • Sulfur (yellow)

For a 3D interactive version on sketchfab, see here:

Ice III and IX

Ice III and IX (Ice three and Ice nine)

  • Ice III can be formed from liquid water at 300 MPa (3000 bar) by lowering the temperature to approx. 250 K.
  • The relative permittivity is very high (~ 117)
  • Density: 1.16 g/cm3

Structural features

  • Ice III is also called Keatite Ice, because the oxygen atoms are located at analogous positions of the silicon atoms in the SiO2 phase Keatite.
  • Ice IX is the proton-ordered form of Ice III.
  • The tetrahedral environment of the water molecules are considerably distorted.
  • Interestingly, there are no 6-membered rings present anymore, but 5, 7, and 8 –membered rings instead. The 5-membered rings can be best seen if you look along the a or b direction:


    • Two-third of the water molecules are forming 41 helices/screws running along the c axis, which means that Ice III is chiral! The other water molecules connect these helices and are forming a 21 helix.


  • Space group: P41212 (No. 92)
  • Crystal system: tetragonal
  • Lattice parameters:
    • a = b = 6.73(1) Å, c = 6.83(1) Å
    • α = β = γ = 90°


 Here, you can download the CIF.

[Atomic structure figures created with

K. Momma and F. Izumi, “VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data,” J. Appl. Crystallogr., 44, 1272-1276 (2011).



Selected Beauties: Silver Oxide Nitrate

Revealing the Beautifulness of Silver Oxide Nitrate

Delving among some books about crystal structures I came upon the remarkable and beautiful crystal structure of silver oxide nitrate: Ag(Ag6O8)NO3.

Space Group: Fm-3m (No. 225)
Crystal system: cubic
Crystal class: m-3m

Lattice parameters: a = b = c = 9.8893 Å, α = β = γ = 90°


Let’s have a first look how its crystal structure looks like:

Fig. 1: Crystal structure of silver oxide nitrate (Blue: nitrogen, red: oxygen, grey: silver, purple: silver).

Uhh, interesting, but partly a mess! But we can turn this into something more beautiful.

Step 1: There are these heavily disordered nitrate anions. Let’s represent them simply with one nitrogen atom in blue.

Fig. 2: The disordered nitrate anions are represented by the blue spheres.

Step 2: One sort of the silver cations are surrounded by 4 oxygen anions in a square-planar fashion. Let’s highlight this feature and slightly change the viewing direction:

Fig. 3: Highlighting the square-planar AgO4 coordination polygons with blue squares.

Ah, doesn’t it then look like an arrangement of rhombicuboctahedra, one of the Archemedian solids?

Step 3: There are additional squares built by 4 oxide anions, but in which no silver cations are located in the center. And then there are these triangular faces built by 3 oxygen atoms, which will complete this rhombicuboctahedron (a rhombicuboctahedron has 24 vertices (here realized by 24 oxygen anions), 26 faces and 48 edges). Let’s highlight these squares in orange and the triangular faces in grey :

Fig. 4: Completing the rhombicuboctahedra…

Now, we see in which way the face-centered cubic arrangement of the rhombicuboctahedra are connected to each other – by orange cubes!

This means, we have two cavities, both forming a fcc-like arrangement. In the center of the rhombicuboctahedra ( = at the corners and face centers of the unit cell) the disordered nitrate anions are located and in the center of the cubes ( = at the middle of the edges of the unit cell) there are silver ions. In this sense, silver oxide nitrate is only a kind of a ‘decorated’ rock salt structure 🙂

Fig. 5: Visualization of the silver and nitrate ions, being located in the center of the rhombicuboctahedra and cubes, respectively.

Isn’t this beautiful?

Further fun facts

1. While mixed-valence compounds are very common, here we have a mixed-valence compound in which the silver ions that are covalently bonded to oxide anions have two different oxidation states, i.e. Ag(I) and Ag(III). So the actual formula can be written as


2. Silver oxide nitrate has a relatively high electrical conductivity of 2.1 x 102 S/cm.


[1] C.H. Wong, T.-H. Lu, C.N. Chen, T.-J. Lee, Journal of Inorganic and Nuclear Chemistry (1972), 34, 3253-3257
DOI: 10.1016/0022-1902(72)80125-8

[2] I. Náray-Szabó, G. Argay, P. Szabó, Acta Crystallogr. (1965), 19, 180-184 DOI: 10.1107/S0365110X65003043

[3] W. Levason, M.D. Spicer, Coord. Chem. Rev. (1987), 76, 45-120
DOI: 10.1016/0010-8545(87)85002-6