Last Update 27/ 03/ 2001
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The figure shows a set of objects represented by blue points regularly distributed on the plane. It is possible to select a subset in parallelogram shape out of it, which can build the whole set by simple shift, like a net where each parallelogram contains exactly the same objects in the same relative positions. This can be observed after clicking on button a on the figure. The origin of the net can be defined arbitrarily at any place, for example the vertex with coordinates x=104, y=102, readable on the bottom of the page if the cursor is pointing that vertex. The dimension of the parallelogram shaped cell can be calculated by vertex coordinate difference.
If the objects are organized in three dimensions the plane cells are substituted by parallelepipeds with edge length a, b, c and angle a, b and g.
In crystallography the space limited by the parallelepiped is named unit cell. Its study may clarify the crystal structure. This study can be shortened if any symmetry element is noticed, for example the reflection plane visible as a white line across the unit cell after clicking on button b. Only half of the cell needs to be solved in this case, the other half may be known by means of plane reflection operation.
Curiosity suggests to seek for another net, as that visible after a click on button c. Now the origin can be displaced (arbitrarily) to the point with coordinates x=94, y=115. The actual cell size can be compared with the former, just using the described coordinate system. Two reflection planes one vertical and other horizontal across the cell and through the centre of the rectangle can be seen as white lines after a click on button d. With this selected net the investigation is reduced to only 1/4 of the cell. The rest of the cell can be solved by reflection operations due to the mentioned perpendicular reflection planes. Three objects are inside this cell as in the first cell (after a click in a).
If a lattice point (black point) is used to represent three objects (blue points) a related crystal lattice is obtained as can be seen after a click on button e. It is common to represent a complete molecule by a single lattice point, for example the 12 atoms in the benzene molecule in a benzene crystal by one lattice point. In this case the distance between the lattice points will be equal to the distance between the benzene molecules in its crystal.
A click on button f shows the original sequence.
Bibliography
1. Keer, H.V., Principles of the Solid State, John Wiley & Sons, N.Y., 1993.
2. Kittel, C., Introduction to Solid State Physics, John Wiley & Sons, New York, 1996.
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