CRYSTAL LATTICE AND UNIT CELL

Last Update 27/ 03/ 2001

in English/ in Esperanto/ in French/ in Portuguese

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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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
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