Unit Cells and Symmetry Elements

The properties observed for large macroscopic crystals suggested to early observers that crystals were made up by the regular repetition in space of the same unit of structure. Each unit cell displays the full symmetry of the structure. 

Although cells A, B and C all show the full symmetry of this lattice, C is twice the size of the other cells. This introduces the principle of point counting and sharing to calculate how many lattice points a cell contains. In order to count points, the number of cells each point is shared between must be considered. Point i in cell A is on a comer, and if cells were drawn in both directions it could be seen that i is shared between four cells. This means that i is only worth one quarter of a whole point to the cell A. If we repeat this procedure for the whole of A, this generates one whole orange and one whole apple inside A. Thus for apples 4 x A = 1 apple for oranges =1x1 = 1 orange 

A Repeating this procedure for C produces two apples and two oranges in a unit cell. A unit cell which contains only one lattice point of a particular type is known as primitive. So here A is a primitive unit cell and C is non-primitive as it contains two lattice points (two apples and two oranges). 

How much of the sphere lies within each sugar cube in the three tasks  

Not all types of lattice are allowable within each crystal system, because the symmetrical relationships between cell parameters mean a smaller cell could be drawn in another crystal system. For example a C-centred cubic unit cell can be redrawn as a body-centred tetragonal cell. The fourteen allowable combinations for the lattices are given in Table 1.4. These lattices are called the Bravais lattices. 

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