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Interpenetrating hcp lattices 

The hexagonal closest packed (hcp) structure has the same density as the ccp structure but it is less symmetrical, it is polarized. 

While the ccp is cuboctahedrally coordinated the hcp has the coordination of twinned cuboctahedron or twist cuboctahedron, which is obtained by reflecting one-half of a cuboctahedron at its hexagonal equator or twisting one-half by 60 degrees, (Fig. 10a). 

Note that we are using hcp lattices in a looser form of definition to show a similarity to ccp lattices. 

Due to hcp's polarized nature it exhibits more layering qualities and a more limited combinations of interpenetrating lattices. 
 
  
 
 

Fig. 10a. Twinned Cuboctahedron. 
 
  
 

Fig. 10b. Hexagonal Closest Packed Structure.
 

The four hcp lattices 

The four lattices are the hcp lattice itself and 3 other interstitial lattices derived from connecting the centers of the tetrahedra and octahedra voids in the hcp structure. Three of the hcp lattices are topologically identical, but C' lattice is a trigonal prismatic lattice. 

• A' is the lattice connecting the Tet+ interstitial centers (hcp)
• B' is the lattice connecting the Tet- interstitial centers (hcp)
• C' is the lattice connecting the Octahedra interstitial centers, this grid is not hcp, it is a trigonal prismatic lattice 
• D' is the lattice  from connecting all the adjacent sphere centers in the hcp structure.

Fig. 11a. The A' hcp lattice (red) with D' hcp lattice. 
 
  

Fig. 11b. The B' hcp lattice (red) with D' lattice. 
 

 

Fig. 11c. The C' trigonal prismatic lattice (blue) with D' lattice. 
 
  

 
 
Fig. 11d. The D' hcp lattice. It is a smaller section of (Fig. 10b). 
 
  

Structural combinations of the 4 hcp lattices 

These structural combinations can be of  the same or different atoms and of the same or different sizes. In the presentation to follow all atoms are at the vertices, and all vertices are occupied. This becomes an alternative system to identifying unit cells. 

1. The 4 hcp lattices: A', B', C', and D' 
2. Combinations of A'+D' or B'+D' is the hexagonal diamond structure or structure of wurtzite ZnS.  Half of tetrahedra are filled. 
3. Combinations of C'+D' is the structure of Ni As, where all octahedra are filled, pg. 753 (Ref. 3) 
4. Combinations of A'+B' where all tetrahedra are filled are not known. 
5. Combinations of A'+C' or B'+C' are not known. 
 
 

1. The Hexagonal Closest Packed Structure 
A', B', D' 
 
Elements: Mg, Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd, Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl (Ref. 4) 

Any elements in the C' lattice?  trigonal prismatic? 

Compounds: none?
 
See (Fig. 10b, 11c, 11d ) 

2. The Hexagonal Diamond or Wurtzite Structure 
A'+D' or B'+D' 
 
Elements: C (hexagonal Diamond) (Ref. 4) 

Compounds: ZnS (Wurtzite), ZnO, SiC, AlN, CiSe, BN 
 
 
 

 
 
Fig. 12a. Wurtzite structure, traditional view. 
 
  

 
 
Fig. 12b. A'+D' with the combined view. D' lattice with interconnecting hexagonal diamond net in blue. 
 

 

Fig. 12c. Combination A'+D' with A' lattice in red. 

 3. The Nickel Arsenide Structure 
C'+D' 

Elements: ? 

Compounds: NiAs, AuSn, CoTe, CrSe, CuSn, FeS, IrS, MnAs, NiSn, PdSb, PtB, RhSn, VP, ZrTe 
 
 

 
 Fig. 13a.  C'+D' combination. NiAs structure with C' lattice in red.  
 
  

 
Fig. 13b. View of NiAs from pg. 753 (Ref. 3). Ni = C' lattice and As = D' lattice in red. The interconnecting net (octahedral axes) is blue. 
 

 

 
Fig. 13c. Interconnecting net (octahedral axes) view. Ni atoms are green. 
 

 

 
Fig. 13d. Same as (Fig. 13c) combination C'+D' with D' lattice shown. Ni atoms are at the vertices of C' lattice.

This combination is not known to exist. The atomic distances between the two lattices are too close. 
 

 
Fig. 14a. A'+B' combination. No known structure. 
 
 

 
Fig. 14b. Same as (Fig. 14a) from a tetrahedral axial point of view. A' lattice is blue. 
 

5. Combination of A'+C' or B'+C' 

 No known structures in this configuration. 
 

Fig. 15. A'+C' combination. No known structures.
 

Other ccp structural combinations 
 
There are other combination patterns that have been noted. It is an area that needs further development. A few examples will be shown here. 

Compound ccp lattice (combinations within the same ccp lattice) 
cuboctahedral coordination 
Cu3Au  (cuboctahedrally coordinated Au) 
Is a compounded ccp lattice formed by cuboctahedrally coordinated Au. It is the D-lattice with all the centers of cuboctahedra occupied by Au atoms and the vertices of the cuboctahedra occupied by Cu atoms. 
 
  
 

 
 
 Fig. 20a. Cu3Au. Au atoms are green and Cu atoms are at the blue vertices. Au (green) atoms are cuboctahedrally coordinated. 
 
  
 

Fig. 20b. Cu3Au. Showing the compound ccp lattice.  Cu are at the blue vertices and Au are green.

Cuboctahedral and Octahedral Coordination 
Perovskites  CaTiO3 (Ref. 6), pg. 584 (a) (Ref. 3) 

CaTiO3 (the mineral Perovskite), SrTiO3, KNbO3, BaTiO3 and some of the high Tc superconductors (Ref. 6). 
CaTiO3 is of cuboctahedral and octahedral coordination. Ca (black) atoms are cuboctahedrally coordinated by O (blue) atoms, and Ti (green) atoms are octahedrally coordinated by O (blue) atoms, see figure below. 
 
Note that the crystal structure of ReO3 would be the octahedral coordination only. 
 

Interpenetrating allspace-filler nets 
NbO net 
 

The ccp lattice can be viewed as a tetrahedra and octahedra net, likewise there can be other 3D nets composed of different polyhedra that are allspace-fillers, such as nets composed of octahedra and cuboctahedra. The NbO net can be viewed as two interpenetrating nets of octahedra-cuboctahedra. 

Nb is the D ccp lattice with centers of cuboctahedra taken out and O is the C ccp lattice with centers of cuboctahedra taken out. 
 

Fig. 22a. The NbO net, traditional view. 
 
 

Fig. 22c. The NbO structure shown as 2 frequency or 8 unit cells. Nb and O are both octahedra-cuboctahedra nets. They are interpenetrating nets with Nb on the D ccp lattice and O on the C ccp lattice. 

References made to interpenetrating lattices/nets: 

1. Chemical Bonding in Solids, Jeremy K. Burdett, pg. 96. "The structure of diamond... Alternatively it may be regarded as being composed of two interpenetrating face-centered cubic lattices shifted by (1/4, 1/4, 1/4)." 

2. The geometrical Foundation of Natural Structure, Robert Williams, pg 120. "The bcc packing of spheres, then, can be considered as two interpenetrating tetrahedral sphere packings." 

3. Structural inorganic chemistry, A. F. Wells, fifth edition, pg 92. "Several examples of crystals built of two or more identical interpenetrating 3D nets will be mentioned in connection with the diamond structure." pg 127. "Structures based on systems of interpenetrating diamond nets." 

4. Crystals and Crystal Growing, Alan Holden and Phylis Morrison, MIT,1997. pg. 191. "It is sometimes convinient to think of the structure (bcc) as consisting of two of the simple cubic arrangements ... The two" interpenetrate" each other; each atom...", pg. 194. "The fact that you can consider the atomic arrangement in sodium choride as two interpenetrating face-centered cubic structures gives you an easy way to find a unit cell with only one sodium ion and one chloride ion...". pg. 195, "Another structure you can consider as two interpenetrating face-centered cubic structures is that of zinc sulfide in the form of the mineral sphalerite." pg. 197, "Figure 100 shows the fluorite structure. It is a face-centered cubic structure of calcium ions, interpenetrated by a simple cubic structure of fluoride ions." 
 
5. Crystal Structures, Web site by Professor Winston Chan, email: winston-chan@uiowa.edu. Many references to interpenetrating sublattices. 
 http://ostc.physics.uiowa.edu/~wkchan/SOLIDSTATE/CRYSTAL/index.html 

6. Connections: the geometric bridge between art and science, Jay Kappraff, pg. 369 - 371. A Unified Look at Nets Related to Cubic Lattices. pg. 370, "Now all basic point complexes, scp, fcc, bcc, and diamond, are defined by Fuller's system in a unified way." 
 
7. A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, Amy C. Edmondson, pg. 136, IVM and IVM'.
 
8. Synergetics, R. Buckminster Fuller, Fig. 931.10 Tetrahedral Characteristics of Chemical Bonding, D & E, "The quadri-bond and mid-edge-coordinate tetrahedron systems demonstrate the super strength of substances such as diamond and the metals." The illustration D refered to as the quadri-bond, showing a tetrahedra caged inside each other is the diamond structure shown by the interpenetrating A and D ccp lattices, and illustration E described as a mid-edge-coordinate tetrahedron system is the interpenetrating A and B ccp lattices.  sec. 931.50 Quadruple bond, quadrivalent. 

9. Synergetica, a journal of Synergetics, edited by Russell Chu. April 1985. Article on "Isotropic vector matrix (ivm) and chemical structures". 

1. R. Buckminster Fuller, Synergetics, Macmillan (1975) 
2. A. F. Wells, Models in Structural Inorganic Chemistry, Oxford (1970) 
3. A. F. Wells, Structural Inorganic Chemistry, Oxford, fifth edition (1984) 
4. R. Benjamin Young, and Michael Mehl, Crystal Lattice Structures, Web site 
5. James E. Huheey, Inorganic Chemistry, Harper &Row (1978) 
6. Winston Chan, Crystal Structures, Web site 
 
 

Links 
 

Crystal Structures: 
Crystal Lattice Structures: http://cst-www.nrl.navy.mil/lattice/index.html 
Prof. Winston Chan: http://ostc.physics.uiowa.edu/~wkchan/SOLIDSTATE/CRYSTAL/index.html 

Lectures on chemical structures: 
Four lectures by Dr Heyes: http://www.ncl.ox.ac.uk/icl/heyes/structure_of_solids/Lecture1/Lec1.html 
The structures of simple inorganic solids: http://neon.chem.ox.ac.uk/course/inorganicsolids/ 
Crystals and the periodic structure of solids: http://chem.ufl.edu/~itl/2041_u98/lectures/lec_h.html 

Synergetics: 

A Fuller Explanation, Amy Edmondson: http://www.angelfire.com/mt/marksomers/40.html
SpringSpace, Karl Erickson: http://www.critpath.org/idioverse/springspace/ 
Synergetics, R.B. Fuller: http://www.servtech.com/~rwgray/synergetics/synergetics.html 
Beautiful code, Gerald de Jong: http://www.beautifulcode.nl/ 
Synergetics on the web, Kirby Urner: http://www.grunch.net/synergetics/synhome.html