Polyhedral Models

Introduction to Polyhedral Models



An alternative to represent ionic compounds is to use polyhedral models. When we can see the way in which three=dimensional shapes represent the individual units, it helps visualize long range order. The crystal structure model becomes unwieldy when you represent long range order in the form of multiples of unit cells.




The Two Basic Polyhedrons




polyhedral shapes
Figure 1: Two primary polyhedral shapes: octahedron and tetrahedron


The two most common shapes are octahedrons and tetrahedrons, Figure 1. Cations occupy the center of a polyhedron and anions take vertices. 



Other shapes exist, but these two suffice for now. Then you can see the character of long range order often results from how polyhedral shapes connect to one another.




Polyhedrons Have More Than One View



multiple representation polyhedral models
Figure 2: multiple views of octahedral and tetrahedral polyhedrons.



Either one of these polyhedral models can have more than one way to depict them, Figure 2.


The stereo view shows a three-dimensional form turned in a three quarter angle. The shading indicates the direction of the faces.


Octahedral forms also have an end on view. This shows an octahedron looking down a top axis. The octahedron rests on one of its triangular faces.


The inverted view of a tetrahedron represents a tetrahedron turned up onto an edge with an opposing edge perpendicular to the first edge.






Sodium Chloride (Halite) Example of Polyhedral Models



Sodium chloride acts as a representative example of how a crystal structure can also be shown as a combination of polyhedral models. The crystal structure of NaCl has chloride ions at the vertices of a cube with occupied by chloride ions. Smaller sodium cations sit in each octahedral interstitial site.




There is no specific reason to choose sodium chloride other than the fact it is a simple crystal structure which is reasonably easy to understand.




Relationship of Polyhedrons to Crystal Structure




polyhedral model superimposed on NaCl lattice
Figure 3: Polyhedral model of sodium chloride superimposed onto an NaCl lattice shows how one model relates to another model.

Superimposing the shape formed by an octahedron onto a sodium chloride lattice results in an image which shows the relationship of lattice point models to polyhedral models, Figure 3.




In the lattice, green spheres represent chloride anions and red spheres denote sodium cations. Each sodium atom has six chlorine atoms as its nearest neighbor. 



The polyhed formed with six atoms distributed around a single central sodium cation forms an octahedron.



The red spheres represent a sodium cation while green atoms show chloride atoms.




How Octahedra Fit Together



Octahedra can share part of their atoms with neighboring octahedron in one of three ways:




Figure 4: The thee ways octahedra can share atoms: point sharing, edge sharing, and face sharing


  • point sharing


each octahedra shares a vertices with an adjacent nearest neighbor join octahedra at each one of its lattice points


  • edge sharing


octahedra attach to its neighbors through two of its atoms with an edge shared between each adjoining octahedra 

  • face sharing

three atoms are shared between each octahedra joined by its neighbors by its triangular face



Polyhedral Models Compared to Crystal Structure of NaCl




NaCl crystal structure compared to octahedral structure
Figure 5: The crystal structure of NaCl shown next to the polyhedral model of polyhedral structures

The side by side views of the crystal lattice model and the polyhedral model show how the two of models together provide a more complete concept of the structure of NaCl. Figure 5 shows the crystal structure of halite compared to the a polyhedral model of NaCl.






Built Up Polyhedral Models of NaCl




two ways to represent NaCl
Figure 6: Polyhedral models of NaCl can be shown as a stereo view and end on view

Building up a polyhedral model has two possibilities: the polyhedral models viewed from the side or viewed from the end looking down, Figure 6.



The depiction on the left shows a stereo view of sodium chloride where the octahedra cluster has been built from octahedra with a three-quarter aspect.



On the right a top down view shows how polyhedral models reveal an otherwise difficult to visualize a feature present in the long range order of NaCl. Between the octahedral polyhedrons a tetrahedral hole results.


Example Tetrahedral Polyhedral Models



Sphalerite (Zinc Sulfide)



ZnS crystal structure and polyhedral models
Figure 7: Crystal structure of ZnS and its structure represented by polyhedral models

Zinc sulfide adopts one of two common structures: sphalerite and Wurtzite. Figure 7 shows sphalerite adjacent to a space filling model built from tetrahedral polyhedral models. 


The crystal structure is composed of chloride ions which occupy the lattice points of a face center cubic close packed structure. Zn+2 ions sit in every other tetrahedral hole.




Polyhedral models form from a Zn+2 ion at the center of a tetrahedron while S-2 ions reside at each vertex. The tetrahedral shape connect together by sharing S-2 ions in a corner sharing scheme.


Many Polyhedral Models



Other ionic compounds adopt a variety of polyhedral models. These simple examples serve to illustrate how to understand these kind of representations. Between crystal structures and polyhedral models, a more complete concept of ionic structures provide enhanced insight.




Many more structures exist, which will be elaborated when needed. This should provide a start the next time polyhedral models are encountered.