Can a point $\vec R$ in direct lattice be uniquely mapped to a point in the reciprocal space?

Question

Given a point in the direct lattice $\vec R=\vec a_1+\vec a_2+\vec a_3$ (say), what is the reciprocal lattice vector $\vec G$ corresponding to $\vec R=\vec a_1+\vec a_2+\vec a_3$?

The reciprocal lattice vector is defined by the relation $\vec G\cdot\vec R=2\pi\times\text{integer}$. Using $\vec G=m_1\vec b_1+m_2\vec b_2+m_3\vec b_3$ and $\vec b_i\cdot\vec a_j=2\pi\delta_{ij}$, we get, $$m_1+m_2+m_3=\text{integer}$$ Since the "integer" on the right is arbitrary, $m_i$'s (therefore, $\vec G$) cannot be uniquely determined. Even if we choose one particular value of the "integer" on the right, it is possible to choose $m_1, m_2,m_3$ (therefore, $\vec G$) in more than one way.

Does it mean that a given point $\vec R$ cannot be uniquely mapped to a point in the reciprocal space?

Answer 1

Any direct lattice point (or direction vector) corresponds to infinitely many planes in the reciprocal lattice whose indices are $(m_1,m_2,m_3)$. This is exactly equivalent to say that any reciprocal lattice vector (or direction vector) $\vec G = h \vec b^1 + k \vec b^2 + \ell \vec b^3$ corresponds to the infinitely many planes with indices $(h,k,l)$.