The Miller formulae

Koch, E.

doi:10.1107/97809553602060000572

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 1.1, p. 5

Section 1.1.4. The Miller formulae

E. Koch^a

^a Institut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

1.1.4. The Miller formulae

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Consider four faces of a crystal that belong to the same zone in consecutive order: [(h_1k_1l_1)] , [(h_2k_2l_2)] , [(h_3k_3l_3)] , and [(h_4k_4l_4)] . The angles between the ith and the jth face normals are designated $[\varphi_{ij}]$ . Then the Miller formulae relate the indices of these faces to the angles $[\varphi_{ij}]$ : $[{\sin\varphi_{12}\sin\varphi_{43}\over\sin\varphi_{13}\sin\varphi_{42}}={u_{12}u_{43}\over u_{13}u_{42}}={v_{12}v_{43}\over v_{13}v_{42}}={w_{12}w_{43}\over w_{13}w_{42}}\eqno (1.1.4.1)]$ with $[u_{ij}=\left|k_il_i\atop k_jl_j\right|,\quad v_{ij}=\left|l_ih_i\atop l_jh_j\right|,\quad w_{ij}=\left|h_ik_i\atop h_jk_j\right|.]$ If all angles between the face normals and also the indices for three of the faces are known, the indices of the fourth face may be calculated. Equation (1.1.4.1) cannot be used if two of the faces are parallel.

From the definition of $[u_{ij}]$ , $[v_{ij}]$ , and $[w_{ij}]$ , it follows that all fractions in (1.1.4.1) are rational: $[{\sin\varphi_{12}\sin\varphi_{43}\over \sin\varphi_{13}\sin\varphi_{42}}={p\over q}\quad{\rm with\ }p,q{\ \rm integers}.]$ Therefore, (1.1.4.1) may be rearranged to $[p \cot\varphi_{12}-q\cot\varphi_{13}=(\, p-q)\cot\varphi_{14}.\eqno (1.1.4.2)]$ This equation allows the determination of one angle if two of the angles and the indices of all four faces are known.

References

International Tables for Crystallography (2006). Vol. C. ch. 1.1, p. 5