# What is the connection between reciprocal lattice vectors $\vec G$ and the Miller indices?

We know that a family of crystal planes with Miller indices $$(hk\ell)$$ is orthogonal to the reciprocal lattice vector $$\vec G = h \vec b_1 + k \vec b_2 + \ell\vec b_3$$. My question is the converse of this.

For any given set of three integers $$m_1, m_2, m_3$$, is $$\vec G=m_1\vec b_1+m_2\vec b_2+m_3\vec b_3$$ always associated with a family of crystal planes in the direct lattice whose Miller indices are $$m_1=h, m_2=k$$ and $$m_3=\ell$$? In other words, for every $$\vec G$$ (or equivalently, every set $$m_1,m_2,m_3$$) that we can think of, is there a set of parallel planes in the direct lattice to which $$\vec G$$ is orthogonal?

You can represent a generic position vector of the direct space unit-cell in the basis of the direct lattice, i.e, $$\vec r = r_1 ~\vec a_1 + r_2 ~\vec a_2 + r_3 ~\vec a_3$$, where $$r_i \in \mathbb{R}$$ and the Bravais lattice $$V=\text{span}\lbrace\vec a_i\rbrace$$ whose dual is the reciprocal lattice $$V^* = \text{span}\lbrace\vec{b}^j\rbrace$$ satisfying $$\vec a_i \cdot \vec b^j = 2\pi ~\delta_{ij}$$. Any $$\vec G\in V^*$$ has the representation $$\vec G = m_j~ \vec b^j \equiv \vec G_m$$, since $$V^*$$ is a lattice. Now, consider this equality $$\vec G_m \cdot \vec r = m_i r_i =0$$. It is the equation of a plane passing through the origin, and intercepts the lattice vectors $$\vec a_i$$ at $$\frac{1}{m_i}$$ away from the origin. Therefore, by definition, $$m_i$$ coincides with Miller indices $$h_i$$. (Notice that any common factor among $$m_i$$'s is irrelevant to the intercepts, and you can get rid of them at this point.) It is easy to see that $$\vec G_m$$ is normal that plane. And, since any two parallel planes have the same normal vector, the same $$\vec G_m$$ applies for them, hence the same Miller indecis $$h_i$$. You can prove this explicitly by considering a plane shifted from the origin along some vector $$\vec d$$ and write its equation of loci.