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

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?

• Points in real space map to periodicities in reciprocal space and vice versa, so there is no sense in which a point in real space maps to a point in reciprocal space. Commented Jul 26, 2022 at 6:02

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 $$(h,k,l)$$.