Mixed Product and Cell Multiplicity

The mixed product between three vectors is a scalar quantity represented as (V, V, V ) and is defined to be equal to  

The vector product can also be written as a matrix product  

The multiplicity of the cell, m, is a dimensionless physical quantity equcd to the number of entities (e.g., electrons, ions, atoms, molecules) contained in the crystal lattice structure. 

Therefore the multiplicity, m, of the cell can be easily calculated from the number, N, of entities in each particular location (i.e., corners, edges, faces, interior)  

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