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A book I am reading states that one possible definition of a Bravais Lattice is that the surroundings will look the same from whichever lattice point you observe from.

Consider for example the simple cubic lattice - this definition clearly holds here for any lattice point.

However when considering a BCC lattice it would seem to me that the central lattice points are no longer equivalent to those on the outside - the connections and relative positions to the other lattice points are completely different.

Likewise for an FCC lattice it would seem to me that the face centred lattice points are no longer equivalent to those on the outside in a similar way.

However these structures are called lattices, so there appears to be a contradiction - am I completely wrong or is there something I am missing about the concept of a Bravais lattice?

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  • $\begingroup$ You just need to look more carefully. Everyone of the "face-centered" points is also an "outside" point of another lattice cell (obtained by shifting by half a diagonal). $\endgroup$
    – ACuriousMind ♦
    Commented Mar 1, 2016 at 22:26
  • $\begingroup$ Am I supposed to ignore the lines that are usually drawn to connect the points? If not, it wouldn't seem to work. $\endgroup$
    – Watw
    Commented Mar 1, 2016 at 22:28
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    $\begingroup$ The lines are just visual aids. The lattice only consists of the points. $\endgroup$
    – ACuriousMind ♦
    Commented Mar 1, 2016 at 22:30

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A crystal structure has both a Bravais lattice, and a unit cell, which is a set of atoms (the 'basis') positioned around each lattice point. The key concept here is that the atoms are arrayed NEAR a lattice point, not that every atomic position is always a lattice point. In an FCC crystal (let's say an iron crystal) there is a basis which contains four atoms, and the origin of the lattice displacement vectors for those four atoms is the 'lattice point'. It would be entirely consistent to take the lattice point to be the unoccupied location at the cube center. The 'lattice' defines the positions of the cubes (one cube per four atoms) not the positions of the atoms within.

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