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Unit Cell

The smallest repeating structural unit of a crystalline solid is called unit cell. According to Bravais, “the study of three vectors and their angles shows morphologically that the unit cells can be classified into seven crystal systems based on the presence of certain rotation axes.” These seven crystal systems are classified according to geometrical and symmetrical considerations as
  1. cubic,
  2. orthorhombic,
  3. rhombohedral,
  4. monoclinic,
  5. triclinic,
  6. tetragonal, and
  7. hexagonal.
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There are only four possible ways in which constituent species of a unit cell can be arranged:
  1. Primitive or simple
  2. Body centered
  3. Face centered
  4. End centered
So, the total number of possible unit cells are 7 × 4 = 28. Bravais then went on to predict that out of these 28 possible unit cells, only 14 of them exist in nature.
 
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Primitive or simple cubic unit cell

In a primitive cubic (PC) unit cell, all the eight corners of the cube are occupied by atoms and they are not present anywhere else in the cube. If all the eight atoms are of the same type, the unit cell is called ideal primitive, and if they are not same, it is referred to as pseudo-primitive or primitive-like.
 
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The volume of a cubic unit cell is not completely occupied by atoms. The ratio of volume occupied by the effective atoms to the volume of the unit cell is called packing fraction (PF). The fraction of volume of a unit cell which is empty is called void fraction (VF).
 
Packing fraction Description: 25585.png
 
Void fraction = (1 – Packing fraction)
 
∴ Description: 25593.png

Body centered cubic unit cell

In body centered cubic (bcc) unit cell, the lattice points are corners and body centered. This implies that the atoms are present at all the corners and at the body centered position and are not present anywhere else in the cube. Since the body centered atom touches all eight corner atoms, so the body diagonal Description: 25613.png would be equal to 4r, where a represents the edge length and r is the radius of atom. The fraction of volume occupied by the atoms is given by
 
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Face centered cubic unit cell

In face centered cubic (fcc) unit cell, the lattice sites are corners and face centered. Thus, in fcc unit cell, the atoms are present at the corners and at the face centers and are not present anywhere else in the cube. When all the 14 atoms are of the same size, the unit cell is called ideal fcc, while if the atoms present at the corners and face centers are different, the unit cell is referred to as pseudo-fcc or fcc-like.
 
Description: Description: 27879.png
 
The fraction of volume occupied by the atoms is given by
 
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End centered unit cell

In an end centered unit cell, there are lattice points in the face centers of only one set of opposite faces, in addition to the lattice points at the corners of the unit cell.




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