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Does a reciprocal lattice have a common point with its direct lattice? (possibly at the origin $\vec{K}=\vec{R}=\vec{0}$)

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You are comparing apples with oranges. $\vec k$ has the unit $1/\mathrm{Length}$, and $\vec r$ has the unit $\mathrm{Length}$.

It is true that every lattice and reciprocal lattice contain the origin. When the lattice is cubic with lattice constant $1$, the real and the reciprocal lattice are indeed completely identical, but this is of no importance whatsoever.

Imagine that a car is resting at the origin. Then, $\vec v=(0,0,0)$ and $\vec r=(0,0,0)$. They are the same, but the units as well as the meaning are different. It's the same with direct and reciprocal lattice vectors.

The only relation between reciprocal lattice vectors and real lattice vectors is

$$\vec k_i\cdot\vec r_j=n\delta_{ij},$$

where $n$ is an integer.

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That is correct. It does not make sense to compare points in position space with points in momentum space. – Frederic Brünner Feb 24 '13 at 15:33
:) the scales are NOT comparable but the directions ARE. For example a (normal) vector $\vec{k}$ represents parallel crystal planes. – richard Feb 24 '13 at 16:23
If you are talking about the Miller indices: You can also use the real lattice vectors for that. – Rafael Reiter Feb 24 '13 at 20:06
+1 Pretty much the same answer I would have given. – Wouter Feb 24 '13 at 22:58

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