Author Archives: doktorholz

A hitherto unrecognized 2-periodic net derived from a circle packing

Recently, I have been immersed in the vast world of intermetallic phases, in particular those with an ordered superstructure of either cubic-closest packed (ccp) or body-centred cubic (bcc) parent structure. They caught my interest, because some of the researchers in that field are paying particular attention to how the densest-packed layers/planes look like and how they are organized, i.e., in which direction they are stacked and if they build an cubic (ABC…) or hexagonal (AB…) stacking sequence. They also partly give network-like descriptions of the atomic configurations of the densest layers.

Since densest layers are considered, one enters the field of circle packings. And the densest circle packing of equally sized circles is, of course, the hexagonal packing. If we draw lines from each circle midpoint to every midpoint of the next-neighboring circles, we get the well known hxl net, a uninodal 6-c net. The vertex symbol is 36.

In those densest layers of ordered alloys the atoms within one densest plane are of different kind. If we ignore this – as it is usually the case, when we analyze the topology of an atomic arrangement – then we obtain again the hxl net. But what happens, if we focus only on a subnet, for instance, the net that is built by the major component?

Let’s look at an example.

In the ordered binary intermetallic phase Cu3Au (space group Pm-3m), the Cu atoms in the densest layers, (111) planes, form a continous net as shown in Fig. 1, in which the Au atoms are a kind of decoration, located in the centres of the hexagons. The Cu subnet is the well-known, again a uninodal net, the kagome net kgm. The vertex symbol is 3.6.3.6.

Fig. 1: Kagome net of Cu atoms in the densest layers of Cu3Au, decorated by Au atoms.

Two different binodal nets based on circle packings are realized in the subnets of densest layers in TiAl3 and MoNi4, respectively, shown in Fig. 2 and 3. In TiAl3 the Al atoms form a bew net, and the Ni atoms in MoNi4 form a krh net. The vertex symbols are given in the figures.

Fig. 2: The subnet formed by Al atoms in the binary intermetallic phase TiAl3.

Fig. 3: The subnet formed by Ni atoms in the binary intermetallic phase MoNi4.

Probably, it is hard to find a new uninodal or binodal net that is based on circle packings as it is believed that the RCSR contains all uninodal and binodal (stable) circle packings. In stable circle packings all tiles are strictly convex (angles between adjacent edges < 180°).

But on the other hand, this also means that one or the other higher-nodal net based on circle packings can still be discovered. Indeed, it turned out that the trinodal net that is formed by the Al atoms (see Fig. 4) in the ordered intermetallic phase ZrAl3 is a hitherto unrecognized net, which, of course, is only due to the fact that no one has cared about it yet.

Fig. 4: The subnet of Al atoms in the binary intermetallic phase ZrAl3.

Who knows if there are more undiscovered networks when looking at densest layers of further intermetallic phases. Maybe there is even a previously undiscovered binodal one 🙂

PS: I would like thank Davide Proserpio for confirming that this a new net by carrying out an appropriate analysis with ToposPro, and I would like to thank Michael Fischer for the inspiring exchange we had concerning this topic.

Visualisation of the Period Table

Michael Fischer, known as @ZeoliteMiFi on twitter, created a mock .cif file that allows visualisation of the periodic table in a structure visualisation software like VESTA. With that, one can see at one glance how different elements are shown.

He kindly provided the CIF and you can download it here. May be reused for all purposes, but no warranty is given. Comments and corrections should be send to: michael.fischer@uni-bremen.de.

Three examples in VESTA:

1. Default settings, vdW radii + labels

2. Default settings, atomic radii, no labels

3. Default settings, ionic radii, no labels

This should work equally well with other software packages. The main issue might be an artificial generation of “bonds” between neighbouring elements that one should suppress in the settings (which is straightforward in VESTA).

This is how it looks in Mercury applying the “Ball-and-Stick” style (bond radius set to zero):

Pentagonal Bipyramid vs. Capped Octahedron (Updated)

In a very interesting article by Giese and Seppelt from 1994 [1] the question of the preferred coordination geometry for the coordination number 7 is explored. Until then, it has been shown that the pentagonal bipyramid is preferred for main group elements (for instance in the compound IF7), although this geometry results in an overall slightly higher ligand repulsion than the alternative arrangements according to a capped octahedron or capped tigonal prism, respectively.

In this article they raised the question if this is also valid for transition elements, and the ideal candidate to answer this question would be the homoleptic compound ReF7. Unfortunately, for various reasons it is very difficult to determine the crystal structure of ReF7.

Alternatively, they synthesized several ionic compounds comprising of anions with transition elements with the coordination number 7 and varied the nature of the cation to exclude lattice energy effects that might influence the arrangement of the ligands.

The results were as follows:

The anions in Cs+MoF7 (see Fig. 1), Cs+WF7, NO2+MoF7 ∙ CH3CN, and C11H24N+MoF7 (C11H24N+ = 1,1,3,3,5,5-hexamethylpiperidinium) (all belong to the cubic crystal system) as well as (H3C)4N+MoF7 (tetragonal, space group P4/nmm, ) form a capped octahedron coordination polyhedron. The latter result is of special interest as the 7 fluorine atoms around Tellurium in (H3C)4N+TeF7 (again space group P4/nmm) form a pentagonal bipyramid instead.

Fig. 1: The crystal structure of Cs+MoF7 (space group Pa-3, ICSD deposition number = 78390); Cs = purple, Mo = gray, F = green. (Image made with VESTA [2]).

According to the authors, it can therefore be concluded that neither the lattice type nor the crystal packing have an influence on the different structures of the anions.

Now, what will happen, if one of the 7 fluorine ligands is exchanged with a larger atom? As expected, a pentagonal bipyramid is then realised, as in Cs+ReOF6 (space group P21/a), where the larger ligand occupies one of the two axial positions (see Fig. 2).

Fig. 2: The crystal structure of Cs+ReOF6 (space group P21/a, ICSD deposition number = 78392); Cs = purple, Re = gray, O = red, F = green. (Image made with VESTA [2]).

Update:

I have to admit that I was too lazy to check if the structure of ReF7 is known by now. In fact – thanks to Robert McMeeking (from The Chemical Database Service/CrystalWorks, @cds_daresbury) for the hint – the crystal structure of ReF7 was published in Science only a single(!) day after the publication of the Angewandte paper by Giese & Seppelt: ReF7 builds a (slightly distorted) pentagonal bipyramid [3], see Fig. 3!

Fig. 3: The crystal structure of ReF7 (space group C-1, ICSD deposition number = 78311); Re = gray, F = green. (Image made with VESTA [2]).

References:

[1] S. Giese, K. Seppelt, Angew. Chem. Int. Ed Engl. 1994, 33, 461. http://dx.doi.org/10.1002/anie.199404611

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

[3] T. Vogt, A. N. Fitch, J. K. Cockcroft, Science 1994, 263, 1265.
https://doi.org/10.1126/science.263.5151.1265

Despujolsite

Despujolsite

  • Named in honor of Pierre Despujols (1888–1981), the founder of the Moroccan Geologic Survey (“Service de la carte géologique du Maroc”)
  • The mineral was first observed in 1962 in manganese ore samples from Tachgagalt (Anti-Atlas, Morocco).
  • Formula: Ca3Mn(SO4)2(OH)6  · 3 H2O
  • Space group: P-62c
  • Crystal system: hexagonal
  • Crystal class: -6m2
  • Lattice parameters: a = b =  8.5405(5) Å, c = 10.8094(9) Å, α = β = 90°, γ  120°
Despujolsite_0438792001298611550

Picture: Rob Lavinsky, iRocks.com – 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).)

Despujolsite
  • CaO8 polyhedra (blue)
  • Mn(OH)6 octahedra (purple)
  • SO4 tetrahedra (yellow)
  • Oxygen (red)
  • Hydrogen (white)

For a 3D interactive version, see here:

https://skfb.ly/onvGo

Refs:

[1] M.C. Barkley, H. Yang, S.H. Evans, R.T. Downs, M.J. Origlieri, Acta Cryst E 2011, 67, i47-i48.
DOI: 10.1107/S1600536811030911

Space Group Diagrams (not only) for Lecturers

I’ve started another ‘230 project’.

This time it is concerned with space group diagrams.

Of course, the No. 1 source for such diagrams, i.e. symmetry element and general position diagrams, is Volume A of the International Tables for Crystallography (ITA). As valuable as they are for the daily life of a crystallographer, they are unsuitable when it comes to teaching.

Hitherto, there is another extremely valuable online resource for these diagrams:

The Hypertext Book of Crystallographic Space Group Diagrams and Tables

However, one of the features I don’t like about these diagrams is that they decided to present both diagrams (symmetry elements and general positions) in a superimposed fashion.

For this reason I decided to draw all diagrams again and to make them publicly available (CC license) in various formats (as a PNG picture, a PDF, and a PPT file) – ready for use for teaching purposes.

However, there will be limitations. Different origin choices will be taken into account, but further different settings will be disregarded. And I do not know, if I will be able to manage the drawings of all diagrams for the cubic space groups. Lets see 🙂

Up to now, the first two space groups of the triclinic crystal system are ready. Until the end of the month the diagrams for all monoclinic space groups should be available. Then further diagrams will be added from time to time.

Have fun!

Kapellasite

Kapellasite

  • Named after Christo Kapellas (1938-2004), collector and mineral dealer of Kamariza, Lavrion, Greece
  • Kapellasite is isostructural with Haydeeite [Cu3Mg(OH)6Cl2]
  • Kapellasite is a metastable polymorph of Herbertsmithite
  • Formula: Cu3Zn(OH)6Cl2
  • Space group: P-3m1 (No. 164)
  • Crystal system: trigonal
  • Crystal class: -3m
  • Lattice parameters: a = b = 6.300(1) Å, c =  5.733(1) Å, α = β = 90°, γ = 120°

Crystal structure[1] (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).)

  • CuO4 square-planar coordination polygons (blue)
  • ZnO6 distorted octahedra (gray)
  • Oxygen (red)
  • Chlorine (green)

For a 3D interactive version, see here:

https://skfb.ly/6ZNHM

References:

[1] W. Krause, H.-J. Bernhardt, R.S.W. Braithwaite, U. Kolitsch, R. Pritchard
Kapellasite, Cu3Zn(OH)6Cl2, a new mineral from Lavrion, Greece, and its crystal structure
Mineralogical Magazine, 2006, 70, 329-340
DOI: 10.1180/0026461067030336

Bloedite

Blödite (also Bloedite)

  • named after the German mineralogist and chemist Karl August Blöde (1773 – 1820)
  • fun fact: the German adjective “blöd(e)” means “stupid”
  • Formula: Na2Mg(SO4)2  · 4 H2O
  • Space group: P21/a
  • Crystal system: monoclinic
  • Crystal class: 2/m
  • Lattice parameters: a = 11.126(2), b =  8.242(1) Å, c = 5.539(1) Å, α = 90°, β = 100.84(1)°, γ  90°

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).)

  • SO4 tetrahedra (yellow)
  • (distorted) NaO6 octahedra (purple)
  • MgO6 octahedra (green)
  • Oxygen (red)
  • Hydrogen (white)

For a 3D interactive version, see here:

https://skfb.ly/6ZEPS

Refs:

[1] F. C. Hawthorne, The Canadian Mineralogist 1985, 23, 669-674.
(PDF)

Ice VI

Phase diagram of water
  • Can be formed from liquid water at 11 kbar by lowering the temperature to approx. -3 °C
  • Density: 1.31 g/cm3

Structural features

Crystal structure of Ice VI
  • Ice VI is a proton-disordered phase
  • it is composed of two independent interpenetrating networks of H-bonded water molecules (shown above in blue and red, respectively)
  • the main structural motif is a tricyclic, cage-like water hexamer, similar as in liquid water
A tricyclic water hexamer composed of four-membered rings
  • This motif is also found for the silicon atoms in the zeolite edingtonite, see here for comparison.
  • The respective topology of the underlying net is edi, a binodal (4,4)-c net with transitivity pqrs = 2343
  • Space group: P42/nmc (No. 137)
  • Crystal system: Tetragonal
  • Lattice parameters:
    • a = b = 6.116(1) Å, c = 5.689(1) Å
    • α = β = γ = 90°

Literature:

[1] W. F. Kuhs, J. L. Finney, C. Vettier and D. V. Bliss, Structure and hydrogen ordering in ices VI, VII and VIII by neutron powder diffraction. J. Chem. Phys. 1984, 81, 3612-3623.
DOI: 10.1063/1.448109

Here, you can download the CIF.

Atomistic structure figures were created with:

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

Pentlandite – the most important nickel ore

Pentlandite

  • Named after the Irish natural historian, J. B. Pentland (1797-1873)
  • Pentlandite is the most important nickel ore
  • Formula: (Fe,Ni)9S8
  • Space group: Fm-3m (No. 225)
  • Crystal system: cubic
  • Crystal class: m-3m
  • Lattice parameters: a = b = c = 10.1075(1) Å, α = β = γ = 90°

Picture by: John Sobolewski (JSS) – http://www.mindat.org/photo-192760.html, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=9578696


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).)

  • For both metal cation positions there is a complete disorder between Ni and Fe.
  • There are two distinct coordination environments; octahedrally coordinated metals at the center and all edge centers and tetrahedrally coordinated metals for the others.
  • Eight tetrahedra each form edge-connected Fe/Ni8(µ-S)6S8 motifs, that means cubes of metal ions with six face-capping and eight terminal S atom. If we take now these cubes and octahedra as building blocks they form a NaCl-like structure.
  • Fe/NiS4 tetrahedra (blue)
  • Fe/NiS6 octahedra (orange)
  • Fe (brown)
  • Ni (green)

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

https://skfb.ly/6UPTq

References:

[1] Tenailleau, C., Etschmann, B., Ibberson, R. M. & Pring, A.
A neutron powder diffraction study of Fe and Ni distributions in synthetic pentlandite and violarite using 60Ni isotope.
Am. Mineral. 91, 1442–1447 (2006)

[2] Stacey, T. E., Borg, C. K. H., Zavalij, P. J. & Rodriguez, E. E.
Magnetically stabilized Fe8(µ-S)6S8 clusters in Ba6Fe25S27.
Dalton Trans. 43, 14612–14624 (2014)

Textbook available

I’m very happy that the English edition of my book on #crystallography is finally available!

I know it is a bit expensive, but at least you get the eBook when you order the hardcover version.

You can order it here:

https://www.springer.com/us/book/9783030351090

On this page you can find some electronic supplementary materials, for instance CIF files of the crystal structures that are mentioned in the book, sheets for paper models of typical crystal shapes and some animations.