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Law of rational indices
Law of crystallography

The law of rational indices is an empirical law in the field of crystallography concerning crystal structure. The law states that "when referred to three intersecting axes all faces occurring on a crystal can be described by numerical indices which are integers, and that these integers are usually small numbers." The law is also named the law of rational intercepts or the second law of crystallography.

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Definition

The International Union of Crystallography (IUCr) gives the following definition: "The law of rational indices states that the intercepts, OP, OQ, OR, of the natural faces of a crystal form with the unit-cell axes a, b, c are inversely proportional to prime integers, h, k, l. They are called the Miller indices of the face. They are usually small because the corresponding lattice planes are among the densest and have therefore a high interplanar spacing and low indices."3

History

The law of constancy of interfacial angles, first observed by Nicolas Steno,4: 44 5 (De solido intra solidum naturaliter contento, Florence, 1669),6 and firmly established by Jean-Baptiste Romé de l'Isle (Cristallographie, Paris, 1783),7 was a precursor to the law of rational indices.

René Just Haüy showed in 17848 that the known interfacial angles could be accounted for if a crystal were made up of minute building blocks (molécules intégrantes), such as cubes, parallelepipeds, or rhombohedra. The 'rise-to-run' ratio of the stepped faces of the crystal was a simple rational number p/q, where p and q are small multiples of units of length (generally different and not more than 6).9: 46 10 Haüy's method is named the law of decrements, law of simple rational truncations, or Haüy's law.11: 322  The law of rational indices was not stated in its modern form by Haüy, but it is directly implied by his law of decrements.12: 333 

In 1830, Johann Hessel13 proved that, as a consequence of the law of rational indices, morphological forms can combine to give exactly 32 kinds of crystal symmetry in Euclidean space, since only two-, three-, four-, and six-fold rotation axes can occur.1415: 796  However, Hessel's work remained practically unknown for over 60 years and, in 1867, Axel Gadolin independently rediscovered his results.16

Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller,17 although a similar system (Weiss parameters) had already been used by the German mineralogist Christian Samuel Weiss since 1817.18

In 1866, Auguste Bravais19 showed that crystals preferentially cleaved parallel to lattice planes of high density.20 This is sometimes referred to as Bravais's law or the law of reticular density and is an equivalent statement to the law of rational indices.21: 333 22: 48 

Crystal structure

The law of rational indices is implied by the three-dimensional lattice structure of crystals. A crystal structure is periodic, and invariant under translations in three linearly independent directions.23

Quasicrystals do not have translational symmetry, and therefore do not obey the law of rational indices.

See also

References

  1. Burke, John G. (1966). Origins of the science of crystals. Berkley and Los Angeles: University of California Press. pp. 78–79. Retrieved 8 January 2025. https://archive.org/details/originsofscience0000burk/page/78/mode/2up ↩

  2. Phillips, F. C. (1963). An Introduction To Crystallography (3rd ed.). New York: John Wiley & Sons. pp. 40–43. Retrieved 13 January 2025. https://archive.org/details/introductiontocr0000fcph/page/40/mode/2up ↩

  3. "Law of rational indices". Online Dictionary of Crystallography. International Union of Crystallography. 14 November 2017. Retrieved 10 January 2025. https://dictionary.iucr.org/Law_of_rational_indices ↩

  4. Senechal, Marjorie (1990). "Brief history of geometrical crystallography". In Lima-de-Faria, J. (ed.). Historical atlas of crystallography. Dordrecht ; Boston: Published for International Union of Crystallography by Kluwer Academic Publishers. ISBN 079230649X. Retrieved 24 December 2024. 079230649X ↩

  5. Schuh, Curtis P. "Steno, Nicolaus". Mineralogy and Crystallography: An Annotated Biobibliography of Books Published 1469 to 1919. Vol. 2. pp. 1381–1382. Archived from the original on 25 August 2007. Retrieved 8 January 2025. https://mineralogicalrecord.com/new_biobibliography/steno-nicolaus/ ↩

  6. Steno, Nicolas (1669). De solido intra solidum naturaliter contento (in Latin). Florence: Star. Retrieved 8 January 2025. https://archive.org/details/nicolaistenonisd00sten ↩

  7. Romé de L'Isle, Jean Baptiste Louis de (1783). "Préface". Cristallographie (in French). Paris: De l'imprimerie de Monsieur. Retrieved 8 January 2025. https://archive.org/details/cristallographi01unkngoog/page/n4/mode/2up ↩

  8. Haüy, René-Just (1784). Essai d'une théorie sur la structure des crystaux, appliquée à plusieurs genres de substances crystallisées (in French). Paris: Gogué et Née de La Rochelle. Archived from the original on 26 September 2016. Retrieved 8 January 2025. https://gallica.bnf.fr/ark:/12148/bpt6k1060890.r=.langFR ↩

  9. Senechal, Marjorie (1990). "Brief history of geometrical crystallography". In Lima-de-Faria, J. (ed.). Historical atlas of crystallography. Dordrecht ; Boston: Published for International Union of Crystallography by Kluwer Academic Publishers. ISBN 079230649X. Retrieved 24 December 2024. 079230649X ↩

  10. Rogers, Austin F. (1912). "The validity of the law of rational indices, and the analogy between the fundamental laws of chemistry and crystallography". Proceedings of the American Philosophical Society. 51 (204): 103–117. JSTOR 984098. Retrieved 10 January 2025. https://www.jstor.org/stable/984098 ↩

  11. Authier, André (2015). "The Birth and Rise of the Space-Lattice Concept". Early days of X-ray crystallography. Oxford: Oxford University Press. pp. 318–400. doi:10.1093/acprof:oso/9780199659845.003.0011. ISBN 9780198754053. Retrieved 13 January 2025. 9780198754053 ↩

  12. Authier, André (2015). "The Birth and Rise of the Space-Lattice Concept". Early days of X-ray crystallography. Oxford: Oxford University Press. pp. 318–400. doi:10.1093/acprof:oso/9780199659845.003.0011. ISBN 9780198754053. Retrieved 13 January 2025. 9780198754053 ↩

  13. Hessel, Johann Friedrich Christian (1897) [1830]. Krystallometrie, oder, Krystallonomie und Krystallographie (in German). Leipzig: Wilhelm Engelmann. Retrieved 14 January 2025. https://archive.org/details/krystallometrie01hessgoog/page/n3/mode/2up ↩

  14. Whitlock, Herbert P. (1934). "A century of progress in crystallography" (PDF). American Mineralogist. 19 (3): 93–100. Retrieved 14 January 2025. https://msaweb.org/AmMin/AM19/AM19_93.pdf ↩

  15. Wigner, E. P. (September 1968). "Symmetry Principles in Old and New Physics" (PDF). Bulletin of the American Mathematical Society. 74 (5): 793–815. doi:10.1090/S0002-9904-1968-12047-6. Retrieved 14 January 2025. https://www.ams.org/bull/1968-74-05/S0002-9904-1968-12047-6/S0002-9904-1968-12047-6.pdf ↩

  16. Barlow, W.; Miers, H. A. (1901). "The Structure of Crystals". Report of The Seventy-First Meeting of the British Association for the Advancement of Science. London: John Murray. pp. 303, 309–310. Retrieved 14 January 2025. https://archive.org/details/reportoftheseven030432mbp/page/302/mode/2up ↩

  17. Miller, William Hallowes (1839). A treatise on crystallography. Cambridge: J. & J. J. Deighton. p. 1. Retrieved 13 January 2025. https://archive.org/details/treatiseoncrysta00millrich/page/n11/mode/2up ↩

  18. Weiss, Christian Samuel (1817). "Ueber eine verbesserte Methode für die Bezeichnung der verschiedenen Flächen eines Krystallisationssystems, nebst Bemerkungen über den Zustand der Polarisierung der Seiten in den Linien der krystallinischen Structur". Abhandlungen der physikalischen Klasse der Königlich-Preussischen Akademie der Wissenschaften (in German): 286–336. Retrieved 13 January 2025. https://archive.org/stream/abhandlungenderp16akad#page/286/mode/2up ↩

  19. Bravais, Auguste (1866). Études Cristallographiques (in French). Paris: Gauthier-Villars. p. 168. Retrieved 13 January 2025. https://books.google.com/books?id=zGWuwwEACAAJ&pg=PA168 ↩

  20. Ladd, Marcus Frederick Charles (2014). Symmetry of crystals and molecules. Oxford: Oxford University Press. pp. 14–15 133–135. ISBN 9780199670888. 9780199670888 ↩

  21. Authier, André (2015). "The Birth and Rise of the Space-Lattice Concept". Early days of X-ray crystallography. Oxford: Oxford University Press. pp. 318–400. doi:10.1093/acprof:oso/9780199659845.003.0011. ISBN 9780198754053. Retrieved 13 January 2025. 9780198754053 ↩

  22. Senechal, Marjorie (1990). "Brief history of geometrical crystallography". In Lima-de-Faria, J. (ed.). Historical atlas of crystallography. Dordrecht ; Boston: Published for International Union of Crystallography by Kluwer Academic Publishers. ISBN 079230649X. Retrieved 24 December 2024. 079230649X ↩

  23. Souvignier, B. (2016). "A general introduction to space groups". In Aroyo, Mois I. (ed.). International tables for crystallography. Vol. A. Space Group Symmetry (6th ed.). Dordrecht, Holland; Boston, U.S.A.; Hingham, MA: D. Reidel Pub. Co.; Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers Group. p. 22. ISBN 978-0-470-97423-0. 978-0-470-97423-0 ↩