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






Lecture about topology

Topology concepts and the deconstruction of MOFs into their underlying nets

Two weeks ago I gave a talk at a nice little workshop about “Structure and Topology – at the heart of MOF chemistry”, which was organized by the Technical University of Munich and was held in the framework of the priority programme COORNETs of the German Research Foundation DFG, initiated by Prof. Roland A. Fischer.

I decided to make a screen-cast of it. It is now available on YouTube:




Wurtzite in solid-state chemistry textbooks

Captain, Captain Solid-State, we have a (didactic) problem…

Wurtzite (ZnS) is undoubtedly one of the most important structure types. Usually this structure type is described as based on a hexagonal-densest or hexagonal-closest packing (hdp or hcp) of anions, in which half of the tetrahedral voids are occupied by the cations, specifically all tetrahedral voids of one kind, namely those which are pointing upwards with their tips (sometimes also referred as to T+ sites).

At this level of description everything is correct. However, in many textbooks and numerous other sources at the internet the illustrations of Wurtzite are wrong or at least misleading, see, for example, this illustration at the English Wikipedia site:

Fig. 1: A typical but wrong representation of the unit cell of Wurtzite (yellow: sulfur, blue-gray: zinc). (Picture kindly released to the public domain by Wikipedia User Benjah-bmm27.)

What’s wrong with this picture?

Well, the relative positions of the zinc and sulfide ions are correct, but their positions with respect to the origin of the unit cell and with respect to the positions of the symmetry elements in the corresponding space group are wrong.

But this is at least consistent, because already many descriptions and representations of the hcp packing are wrong as soon as a unit cell is additionally shown, for example:

Fig. 2: A typical but at least misleading representation of the unit cell of the hcp packing.

The description to figures like the one shown in Fig. 2 usually reads as: we have a layer stacking sequence of ABAB… and the unit cell consists of two atoms, here located at (0,0,0) and (1/3, 2/3, 1/2). And a prototypical structure of this hcp arrangement is Magnesium (structure type A3).

Well, now, if the students look for real crystal structures and if they load a corresponding CIF file into a crystal structure visualization software (during my lectures I introduce to them VESTA) they will be completely confused, because they see something like that:

Fig. 3: The unit cell of the experimentally determined crystal structure of Magnesium visualized with VESTA. (BTW: a completely correct representation.)

Kind of shocking!

The first thing the students might notice is that the unit cell of Fig. 3 obviously has a center of inversion, while the unit cell of Fig. 2 lacks such inversion center. The choice of the origin of the cell in Fig. 2 ignores the convention that the unit cell should be chosen in such a way that it reflects the symmetry of the structure in an optimal manner. Why should we ignore this useful convention? There is no good reason to do so!

Looking at Fig. 2, the students notice that there are also two atoms per unit cell in the cell of their CIF and if they expand the structure they will also realize that this is indeed a hexagonal layered structure with a AB-stacking, but they are still confused, because the atoms are not located at the places where they were taught the atoms should be located. And if they study the CIF they realize that only one position is specified: (1/3, 2/3, 3/4) and they read something like: okey, the space group is P63/mmc (No. 194)… Hmmm, not sure what this should mean…

If they proceed to Wurtzite and again load a CIF into VESTA, they look at something like this:

Fig. 4: The unit cell of Wurtzite as determined experimentally by X-ray diffraction, visualized with VESTA. (yellow: sulfur, blue-gray: zinc)

Arrgh, again there is no atom at the origin located!

The specifications are as follows:

Space group P63mc (No. 186)

Zn at (1/3, 2/3, 0)
S   at (1/3, 2/3, 3/8)

Some of the students may begin to think that the lecturer is an idiot, probably other students will creep up on considerable self-doubts.

The more advanced students might check the diagram of the symmetry elements in the International Tables for Crystallography, Volume A. Slowly they may realize that something is wrong with the choice of the origin in Fig 1. and Fig. 2. Well that is indeed the case!

If we disregard all the glide planes, then this diagram looks like this:

Fig. 5: The diagram of the symmetry elements of the space group P63mc, disregarding the glide planes.

The 63 screw axis is running along the c axis and through the origin! And an atom at the origin will therefore stay always on that axis! It is simply not possible to generate another, a second  position which is not located on that axis. This can be also seen by inspecting the Positions section within the Tables:

a     3m.      0, 0, z      0, 0, z+1/2

This means that we can’t put atoms at the origin (0,0,0), if we want to build a pattern, a motif that can be described by a six-fold helix-like arrangement as it is the case for both the zinc and sulfur ions!

So, and what about the tetrahedral coordination environment of the ions? How can we fit these tetrahedra with their three-fold axis into this cell? Ah! There is only one proper location: the tip of the tetrahedron and its center have to be located on a three-fold axis of the unit cell! This is the case if they are located at Wyckoff position 2b:

2   b   3m.      1/3, 2/3, z      2/3, 1/3, z+1/2

In the figure below the three-fold axes of rotation are shown in orange:

Fig. 6: The slightly extended unit cell of Wurtzite as determined experimentally by X-ray diffraction, visualized with Mercury (by CCDC) (yellow: sulfur, blue-gray: zink); the three-fold axes are shown as thin orange lines, running through the tips and centers of the ZnS4 tetrahedra.

This means, we are free to choose the fractional coordinate along the c axis (and if this is the case then it is common practice to choose z = 0 – compare the location of the zinc ion in Fig. 6) as it is, but with respect to the a and b axis we have no degree of freedom!

Issues of a hair-splitter?

Well, you could say, don’t be such a quibbler! The unit cell in these misleading pictures is only shifted by 1/3 and 2/3 in the a and b direction, respectively – come on!

Yes, but aren’t there thoughtful conventions how to choose the origin of a unit cell? A unit cell is not a thing that can be considered completely detached from the locations of symmetry elements within the unit cell. And why should we confuse the students in the way that I have sketched?

The mother of all problems…

…starts well before structure types are discussed. It usually begins with the nuisance that the primitive hexagonal unit cell is wrongly depicted, because it is frequently extended by a certain amount in order to show the hexagonal environment, the six-fold rotational symmetry of the underlying lattice – probably good intention, but bad consequences! Such a sketch of an extended unit cell is shown in Fig. 7:


Fig. 7: Common, but misleading sketch of the unit cell of a primitive hexagonal lattice (Picture: Wikipedia user Bor75, CC BY-SA 3.0)

If I present such a picture to my students, approximately 80 – 90 % of them think that these are 3 unit cells (in fact Fig. 7 shows 2 whole unit cells plus 2 times a half unit cell), although they actually know everything about the translation principle of unit cells and the fact that it is forbidden to rotate them around…  To be a little provocative, I could also ask, how many teachers think that Fig. 7 shows 3 unit cells…

Conclusions and Recommendations

1. A radical sanction would be to scrap all misleading representations. Well, this is probably not possible. It is interesting to note that in many areas the way in which knowledge is spread has a certain tradition that is hard to clean up, although almost all lecturers must have made the same experience with these contradictions and struggling students.

2. Crystallography and solid-state chemistry is hard to teach in a consistent way; in a way in which you can gradually build-up the knowledge without referring at some points to anything that cannot yet be understood. But I would recommend to teach solid-state chemistry only when the students have gained a significant level of knowledge in the field of symmetry of solids/crystals. In my opinion it is advisable to introduce first all about the conventions of choosing unit cells in a reasonable way, then to teach how to read the International Tables (what is a space group, how to read the diagram of the symmetry elements, the general position diagram, what is the systematic behind general and special positions, how to understand multiplicities and Wyckoff positions) and then to turn slowly to real structures but always in conjunction with the underlying symmetry specification of the respective space group.

This can be best done with the software Mercury, which is the only software that is able to visually overlay the structure with the symmetry elements – superb! See also Fig. 6!

3. There a few positive exceptions, in which you can find exclusively correct descriptions of structure types, for instance in the book Inorganic Structural Chemistry by Prof. Dr. Ulrich Müller, John Wiley & Sons, 2009, 2nd Edition,

4. My mentor Prof. Dr. Ulrich Behrens occasionally quoted a former colleague with the statement that the hexagonal and trigonal crystal system was invented by the devil – probably he is right.


PS: I would like to thank @ZeoliteMiFi for helpful discussions


Creating images of augmented nets

Making beautiful pictures of beautiful nets in augmented version

There is one requisite in order to be able to create and make great looking images of augmented nets: A software package that has the option to make and show customizable coordination polyhedra. Two of the most frequently used (commercial) packages for this purpose are Diamond and CrystalMaker. Although the gorgeous freeware VESTA has also some capabilities in that respect it is, unfortunately, not suitable.

Okey, let’s begin. In this tutorial we want to make an image of the net reo-a.


1. In general, you need the structural information of the net-a net, i.e. the space group of the net and the fractional coordinates of the nodes. There are two possible sources:

a) RCSR (please checkmark the “including augmented version” option) – for reo the search is unsuccessfull

b) ToposPro is delivered with a database called “idealnets.cmp” (search with your File Explorer to find this file). Open this database, which contains approx. 3000 nets, within ToposPro and look-up, if you can find the desired net. Again, for reo-a there is no entry. But, if the net from which you want to make a nice-looking figure is present, proceed with step # 15.

2. Even if the coordinates of the augmented version of a net is not in your database “idealnets.cmp”, it is very likely that the coordinates of the basic net are available in the ToposPro database “idealnets.cmp”.

3. Create a new database in ToposPro. Name it “net-a.cmp”.

4. Copy the entry of choice (here reo) from the “idealnets.cmp” into the newly created database. Now we can create the augmented variant within ToposPro.

5. Go to “Compound – Auto Determine – Bond Midpoints”. Repeat this procedure, choose “Compound – Auto Determine – Bond Midpoints” again.

6. Run ADS in the standard simplification mode (Save Simplified Net/Standard), but with the option “Contract Atom” under the tab “Topology” checked.

– select all atoms except carbons
– select all nitrogen atoms
– select all carbon atoms

7. You will get a new database “net-a_c” containing one entry. This entry has to be modified a little bit.

8. Remove all 2-c boron atoms: Choose “Compound – Auto Determine – Simplify Adjacency Matrix” – remove all 0-, 1- and 2-coordinated vertices.

In the case of the reo net the result will look like this:

9. It looks quite okey, but two things are not correct. There are some additional bonds and it is probably not the most symmetrical representation. Remove the additional bonds by modifying the corresponding adjacency matrix. Turn the additional bonds into “No bond”. This looks better:


10. Now, try to maximize the symmetry with the help of Systre (which is part of the package Gavrog). Export the file from ToposPro into the Systre format (*.cgd).

11. Run Systre and open the file net-a.cgd. Systre tries to relax the nodes so that a) the average bond length is 1.000 Å and b) that the symmetry of the net is maximized.

12. After a few seconds Systre has finished and you can save the net with the highest possible symmetry by choosing “Save as…” and by specifying the format as “”Embedded nets type (*.cgd)”. Name it for instance “net-a_systre_out.cgd” to distinguish between the input and output file.

13. Import the Systre file into a new database in ToposPro. Run IsoCryst to check your results:

That looks good.

14. Export this net as a CIF. Now, CrystalMaker comes into play…

15. Run CrystalMaker and Import the file net-a.cif. It will look like this:

16. Expand the view so that all polyhedra are clearly visible, remove additional bonds by adjusting the bond length. Then select all atoms of one kind of polyhedron and choose “Selection – Make Polyhedron…”. A new Centroid is created, here coloured in blue:

17. Now, only some slight setting modifications have to be done – choose “Model – Model Inspector…”. In the Atom tab all new centroids (Zz and so on), i.e. the polyhedra have to be rendered as “Rendered solid”, all atoms as “Stick bonds”.

18. In the Bond tab, all bonds in which centroids are involved should be blanked out, and for the bonds between atoms choose an eye-catching colour:

19. A last step is necessary – choose “Model – Polyhedral” (and adjust the stick percentage…).

  1. Done!

Cycles, Rings, Strong rings

In chemical graph theory or chemical-related topology a distinction is made between a cycle, a ring, and a strong ring. It is relatively easy to define a cycle:

A cycle is a continuous path (along edges) in which the start and end vertex are the same.

So this means a cycle (also sometimes called a circuit) can be of any size, regardless if there are shorter (or also longer) paths to get back to the ‘home vertex’. We can also say that a cycle is a closed path.

What might be a little more difficult to understand is the formal definition of a ring:

A ring is a cycle that is not the sum of any two smaller cycles.

It is important to emphasize both attributes: two and smaller. The first conclusion is: Okey, in a graph or net there might be rings that are the sum of three, four, five (smaller) cycles, but a cycle is not a ring if it is the sum of exactly two (smaller) cycles.

In the following figure the graph of a cube is shown and some cycles (highlighted in red) within this graph:

Which of these cycles are rings? I think it is clear, that the two cycles in the top row are rings. Furthermore, the cycle in the middle in the bottom row is a ring, while the left and right cycle are not rings.  Why? Well, the 6-cycle on the left is the sum of two smaller 4-cycles. The 6-cycle in the middle is the sum of three(!) smaller cycles but not the sum of two smaller cycles. And the 8-cycle on the right can be considered as the sum of a 4-cycle and a 6-cycle (the latter can be formed as the sum of two smaller 4-cycles, the top-most and the central one):

Well, but shouldn’t it be possible to build the 6-cycle in the middle as the sum of a 4-cycle (top-most) and a 6-cycle (which itself be the sum of the central and right 4-cycles)? Yes, that is correct, but the definition states that the cycles of a sum of cycles have to be smaller! This is not the case here!

Interestingly, there is an alternative definition of a ring which is, at least for me, much more intuitive and comprehensible:

A ring is a cycle that contains along its path no potential short-cut (of the length of exactly one edge) to the ‘home vertex’.

It is easy to see that the cycle on the left has one short-cut and the cycle on the right even two (dashed edges):

So, what are now strong rings? They can be defined as follows:

Strong rings are rings that are not the sum of any number of smaller cycles.

From that definition It follows that only the two cycles in the top row are strong rings.

I would like to thank Michael Fischer (@ZeoliteMiFi) for helpful discussions.





  • Named after Johann August Streng (1830-1897), German mineralogist, University of Giessen, Germany
  • Formula: FePO4 · 2 H2O
  • Space group: Pbca (No. 61)
  • Crystal system: orthorhombic
  • Crystal class: mmm
  • Lattice parameters: a = 8.722 Å, = 9.878 Å, c = 10.8117 Å, αβ = γ = 90°

Picture: Christian Rewitzer  | 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).)

  • FeO6 octahedra (dark-red)
  • POtetrahedra (green-blue)
  • Oxygen (red)
  • Hydrogen (white)

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



  • Named after its type locality Spessart (Bavaria, Germany)
  • Formula: Mn3Al2[SiO4]3
  • Space group: Ia-3(No. 230)
  • Crystal system: cubic
  • Crystal class: m-3m
  • Lattice parameters: a = = c = 11.621 Å, αβ = γ = 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).)

  • MnO8 polyhedra (purple)
  • AlO6 octahedra (blue)
  • SiOtetrahedra (orange)

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