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

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

Section 1.1.4. The Miller formulae

E. Kocha

aInstitut 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 (]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 ([link] 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 ([link] 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, ([link] may be rearranged to [p \cot\varphi_{12}-q\cot\varphi_{13}=(\, p-q)\cot\varphi_{14}.\eqno (]This equation allows the determination of one angle if two of the angles and the indices of all four faces are known.

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