Proof That The Reciprocal Lattice of The Reciprocal Lattice Is The Direct Lattice
From its definition we know that the vectors of the Bravais lattice must be closed under vector addition and subtraction. Thus it is sufficient to say that if we have
and
then the sum and difference satisfy the same.
Thus we have shown the reciprocal lattice is closed under vector addition and subtraction. Furthermore, we know that a vector K in the reciprocal lattice can be expressed as a linear combination of its primitive vectors.
From our earlier definition of, we can see that:
where is the Kronecker delta. We let R be a vector in the direct lattice, which we can express as a linear combination of its primitive vectors.
From this we can see that:
From our definition of the reciprocal lattice we have shown that must satisfy the following identity.
For this to hold we must have equal to times an integer. This is fulfilled because and . Therefore, the reciprocal lattice is also a Bravais lattice. Furthermore, if the vectors construct a reciprocal lattice, it is clear that any vector satisfying the equation:
...is a reciprocal lattice vector of the reciprocal lattice. Due to the definition of, when is the direct lattice vector, we have the same relationship.
And so we can conclude that the reciprocal lattice of the reciprocal lattice is the original direct lattice.
Read more about this topic: Reciprocal Lattice
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