Last Update 08/ 8/ 2014
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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