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Conventional unit cell is defined in the following:

A definition of a conventional unit cell of a lattice is one that contains the same point group symmetries as the overall lattice and is the smallest such cell.

Suppose that we have a monatomic Bravais lattice. Does the parallelepiped having the primitive vectors as edges satisfy this condition?

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A definition of a conventional unit cell of a lattice is one that contains the same point group symmetries as the overall lattice and is the smallest such cell.

I am used to consider this the definition of the so-called Wigner-Seitz (WS) cell. Anyway, in the case of a monatomic Bravais lattice, the parallelepiped having the primitive vectors as edges does not satisfy this condition.

You can easily see it by considering the case of a 2D triangular lattice (which is a Bravais lattice in 2D). The 2D figure having two basis vectors as the edges of the cell is a rhombus with two opposite internal angles of 60º and the other two of 120º. Such rhombus does not have the full point-symmetry of the triangular lattice which is kept instead by the hexagonal shape of the WS cell.

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