OCTAHEDRAL AND TETRAHEDRAL HOLE

Last Update 08/ 8/ 2014

in Portuguese
 

The application on this page presents a scratch of 86 panels on the screen to show with a dynamical sequence how to build an inexpensive model with 22 little polystyrene balls 25 mm in diameter and indicates one octahedral hole and one tetrahedral hole in the hexagonal close-packed structure. The Java ® applet  will show up in a computer with Internet Explorer 10, Konqueror 4.10.5, Firefox 24.0 and jdk6, open-jdk6 or Safari 5.1.10. The balls and polystyrene adhesive to build this model can be purchased in student stores.
Panel 1 presents the set of 7 styrofoam spheres joined with the proper glue and labeled one by one 0 to 6 respectively with an overhead projection marker. Two more sets of 7 spheres will be needed, joined with glue and labeled  7 to 13 and 14 to 20, respectively. The 21st sphere does not require a label, it will be named here as the extra sphere.
The three sets with the seven balls assembling needs to follow exactly the order and sequence as presented in the plates 1 to 3 of the application.
After the first rotation, as shown on plate 26 of the application which can be followed with the model on hand, the student can observe the tetrahedron hole exhibited among the spheres 9, 20, 13 and 17. For the more advanced student and familiar with the symmetry elements it will be evident the symmetry axis of greater order of this tetrahedron in plate 26 is practically perpendicular to this screen.
As plate 49 is presented it means the model was rotated 90 o from the initial position shown on plate 3.
Now, on plate 86 the model was rotated  60 o about the axis defined by the center of ball 3 and ball 17.
Note with good attention the square defined by spheres 4, 6, 11 and 13. These four tangent spheres and tangent to sphere 3 define an open octahedral hole. The extra sphere, represented in plate 86 as a white circle, can be exactly fitted over this open octahedral hole and among balls 4, 6, 11, 13. In this stage the octahedral hole will now be unveiled, just among balls 4, 6, 11, 13, 3 and the extra sphere. Notice the extra sphere could be painted yellow, according to the colored pattern of the whole model.
This model also represents a crystal of the alkaline earth metal magnesium.
Solid magnesium crystal belongs to space group number 194 of the reference given below.
The application on this page is a modification of the resource available on page Rotation of Objects About an Arbitrary Axis and this resource can be used to obtain a precision figure to complete the presentation of exercise 2) suggested below.

Exercise

1) Define the orientation of the axis used to rotate the model until plate 49.

2) Find the parameters of the unit cel of magnesium, draw a projection of the unit cell of magnesium to scale and compare the calculated density with calculated or experimental density with values on other references.

3) Find the respective numbers on spheres according with this model that define a second octahedron which has the center of the extra sphere in one of its vertices

4) Describe the position and orientation of the octahedron found in 3) related to the octahedron defined by the spheres 4, 6, 11, 13, 3 and the extra sphere.

5) What is, define the axis of greater order of a tetrahedron.

Bibliography

International Tables for Crystallography (2005). Vol. A, edited by T. Hahn, Dordrecht: Springer

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Table of subjects.
Presentation
Chemistry Analytical Chromatography
Elemental organic analysis
Volumetric analysis, simulation
Crystallography 3 fold screw axis
4 fold inversion axis on tetrahedron
5 fold rotation axis absent in crystallography
Binary axis and reflection plane in stereographic projection
Bravais lattices
Conic sections under symmetry operators
Converting from spherical coordinates to stereographic projection
Crystal lattice and unit cell
Determination of unit cell
Elements of symmetry in action - animation
Elements of symmetry in action - cube game
Elements of symmetry in action - dodecahedron game
Elements of symmetry in action - icosahedron game
Elements of symmetry in action - octahedron game
Elements of symmetry in action - tetrahedron game
Ewald sphere and crystal measurements
Extinctions
Five classes in the cubic system
Five classes in the rhombohedral system
From tetrahedron to prism
Gnomonic projection
Improper symmetry axis
Miller indices
Miller indices - animation
Miller indices - cube game
Miller indices - octahedron game
Miller indices - rhombic dodecahedron game
Miller indices - tetrahedron game
Mirror plane
Mirror planes and Miller indices game - tetrahedron
Octahedral and tetrahedral hole
Orientations of the cube
p2mm
Plane symmetry groups
Question on point group
Rotation axis in octahedron and Werner compounds
Rotation axis on tetrahedron and organic molecules
Rotation of objects about an arbitrary axis
Rotation of the parallel and stereographic projections of the cube
Rotation of the stereographic and parallel projection of the cube III
Seven faces in stereographic projection
Seven classes in the hexagonal system
Seven classes in the tetragonal system
Six elements of symmetry in seven orientations
Spherical projection of the octahedron
Stereographic projection
Stereographic projection of six polyhedra in different orientations
Straight line equations and symmetry elements
Symmetry, 2 fold axis
Symmetry, 2, 3 and 6 fold axis in benzene
Symmetry, 3 fold axis in the cube
Symmetry, 4 fold axis in the cube
Symmetry, 4 fold axis in the unit cell of gold
Symmetry elements and Miller indices game
Symmetry elements and Miller indices game - octahedron
Symmetry in art and in crystallography
Three classes in the monoclinic system
Three classes in the orthorhombic system
Twin crystals
Two classes in the triclinic system
Unit cell in hexagonal net
General Butane conformations
Density
Electrochemical cell
Ethane conformations
Oxidation and reduction
Resources of chemical-ICT: water, health and symmetry
Solid and liquid gold