Can somebody explain how to establish the connection between Miller indices $(h,k,l)$ of a crystal plane and the triplet of integers $(m_1,m_2,m_3)$ that appear in the linear combination $$\vec{G}=m_1\vec{b}_1+m_2\vec{b}_2+m_3\vec{b}_3.$$ Here, ${\vec G}$ is the reciprocal lattice vector.
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asked Jun 15, 2020 at 3:40
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A crystal plane with Miller indices $(hkl)$ is orthogonal to the reciprocal lattice vector $\vec G = h \vec b_1 + k \vec b_2 + l \vec b_3$.
answered Jun 15, 2020 at 14:10
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$\begingroup$ Can you explain what it means for a vector in the reciprocal lattice space to be orthogonal to a plane in the direct lattice space? $\endgroup$ Commented Jun 16, 2020 at 13:49
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$\begingroup$ @mithusengupta123 In crystallography, the direct and reciprocal lattices and their corresponding vectors are identified as subsets of $\mathbb R^2$ or $\mathbb R^3$, and they inherit the obvious inner product structure from that. $\endgroup$ Commented Jun 17, 2020 at 3:38