law of rational indices
{{short description|Law of crystallography}}
File:Law of rational indices.png
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."{{cite book |last1=Burke |first1=John G. |title=Origins of the science of crystals |publisher=University of California Press |location=Berkley and Los Angeles |year=1966 |pages=78–79 |url=https://archive.org/details/originsofscience0000burk/page/78/mode/2up |url-access=registration |access-date=8 January 2025}} The law is also named the law of rational intercepts{{cite book |last1=Phillips |first1=F. C. |title=An Introduction To Crystallography |date=1963 |publisher=John Wiley & Sons |location=New York |pages=40–43 |edition=3rd |url=https://archive.org/details/introductiontocr0000fcph/page/40/mode/2up |url-access=registration |access-date=13 January 2025}} or the second law of crystallography.
Definition
File:Miller indices of a plane and a direction.png of a plane (hkl) and a direction [hkl]. The intercepts on the axes are at a/{{math|h}}, b/{{math|k}} and c/{{math|l}}.]]
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, {{math|h}}, {{math|k}}, {{math|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."{{cite web |title=Law of rational indices |url=https://dictionary.iucr.org/Law_of_rational_indices |website=Online Dictionary of Crystallography |publisher=International Union of Crystallography |date=14 November 2017 |access-date=10 January 2025}}
History
File:Calcite scalenohedron constructed using Haüy's integrant molecules.png scalenohedron crystal constructed from small building blocks (molécules intégrantes) using the method of René Just Haüy (1801) in his Traité de Minéralogie.{{cite book |last1=Haüy |first1=René Just |title=Traité de minéralogie |date=1801 |publisher=Chez Louis |location=Paris |volume=Caractère Minéralogique |pages=I, fig. 6, III, fig. 17 |url=https://gallica.bnf.fr/ark:/12148/bpt6k9760845h/f11.item |access-date=8 January 2025 |language=French}}]]
The law of constancy of interfacial angles, first observed by Nicolas Steno,{{cite book |last1=Senechal |first1=Marjorie |editor1-last=Lima-de-Faria |editor1-first=J. |title=Historical atlas of crystallography |date=1990 |publisher=Published for International Union of Crystallography by Kluwer Academic Publishers |location=Dordrecht ; Boston |isbn=079230649X |chapter=Brief history of geometrical crystallography |chapter-url=https://archive.org/details/historicalatlaso0000unse_p0f2/page/44/mode/2up |chapter-url-access=registration |access-date=24 December 2024}}{{rp|page=44}}{{cite book |last1=Schuh |first1=Curtis P. |title=Mineralogy and Crystallography: An Annotated Biobibliography of Books Published 1469 to 1919 |section=Steno, Nicolaus |url=https://mineralogicalrecord.com/new_biobibliography/steno-nicolaus/ |url-status=live |access-date=8 January 2025 |archive-url=https://archive.org/details/BioBib_Mineralogy_2007_Vol_2/page/n611/mode/2up |archive-date=25 August 2007 |volume=2 |pages=1381–1382}} (De solido intra solidum naturaliter contento, Florence, 1669),{{cite book |last1=Steno |first1= Nicolas |year=1669 |url=https://archive.org/details/nicolaistenonisd00sten |language=Latin |access-date=8 January 2025 |title=De solido intra solidum naturaliter contento |publisher=Star |location=Florence}} and firmly established by Jean-Baptiste Romé de l'Isle (Cristallographie, Paris, 1783),{{cite book |last1=Romé de L'Isle |first1=Jean Baptiste Louis de |title=Cristallographie |date=1783 |publisher=De l'imprimerie de Monsieur |location=Paris |chapter=Préface |url=https://archive.org/details/cristallographi01unkngoog/page/n4/mode/2up |language=French |access-date=8 January 2025}} was a precursor to the law of rational indices.
René Just Haüy showed in 1784{{cite book |last1=Haüy |first1=René-Just |title=Essai d'une théorie sur la structure des crystaux, appliquée à plusieurs genres de substances crystallisées |date=1784 |publisher=Gogué et Née de La Rochelle |location=Paris |url=https://gallica.bnf.fr/ark:/12148/bpt6k1060890.r=.langFR |url-status=live |access-date=8 January 2025 |archive-url=https://archive.org/details/b28762411_0001/page/n5/mode/2up |archive-date=26 September 2016 |language=French}} 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).{{rp|page=46}}{{cite journal |last1=Rogers |first1=Austin F. |title=The validity of the law of rational indices, and the analogy between the fundamental laws of chemistry and crystallography |journal=Proceedings of the American Philosophical Society |date=1912 |volume=51 |issue=204 |pages=103–117 |url=https://www.jstor.org/stable/984098 |jstor=984098 |jstor-access=free |access-date=10 January 2025}} Haüy's method is named the law of decrements, law of simple rational truncations, or Haüy's law.{{rp|page=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.{{cite book |last1=Authier |first1=André |title=Early days of X-ray crystallography |date=2015 |publisher=Oxford University Press |location=Oxford |isbn=9780198754053 |doi=10.1093/acprof:oso/9780199659845.003.0011 |chapter=The Birth and Rise of the Space-Lattice Concept |pages=318–400 |url=https://academic.oup.com/book/8011/chapter/153381347 |url-access=registration |access-date=13 January 2025}}{{rp|page=333}}
In 1830, Johann Hessel{{cite book |last1=Hessel |first1=Johann Friedrich Christian |title=Krystallometrie, oder, Krystallonomie und Krystallographie |date=1897 |orig-date=1830 |publisher=Wilhelm Engelmann |location=Leipzig |url=https://archive.org/details/krystallometrie01hessgoog/page/n3/mode/2up |access-date=14 January 2025 |language=German}} 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.{{cite journal |last1=Whitlock |first1=Herbert P. |title=A century of progress in crystallography |journal=American Mineralogist |date=1934 |volume=19 |issue=3 |pages=93–100 |url=https://msaweb.org/AmMin/AM19/AM19_93.pdf |access-date=14 January 2025}}{{cite journal |last1=Wigner |first1=E. P. |title=Symmetry Principles in Old and New Physics |journal=Bulletin of the American Mathematical Society |date=September 1968 |volume=74 |issue=5 |pages=793–815 |doi=10.1090/S0002-9904-1968-12047-6 |url=https://www.ams.org/bull/1968-74-05/S0002-9904-1968-12047-6/S0002-9904-1968-12047-6.pdf |access-date=14 January 2025}}{{rp|page=796}} However, Hessel's work remained practically unknown for over 60 years and, in 1867, Axel Gadolin independently rediscovered his results.{{cite book |last1=Barlow |first1=W. |last2=Miers |first2=H. A. |title=Report of The Seventy-First Meeting of the British Association for the Advancement of Science |date=1901 |publisher=John Murray |location=London |pages=303,309–310 |chapter-url=https://archive.org/details/reportoftheseven030432mbp/page/302/mode/2up |access-date=14 January 2025 |chapter=The Structure of Crystals}}
Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller,{{cite book |last1=Miller |first1=William Hallowes |title=A treatise on crystallography |date=1839 |publisher=J. & J. J. Deighton |location=Cambridge |page=1 |url=https://archive.org/details/treatiseoncrysta00millrich/page/n11/mode/2up |access-date=13 January 2025}} although a similar system (Weiss parameters) had already been used by the German mineralogist Christian Samuel Weiss since 1817.{{cite journal |last1=Weiss |first1=Christian Samuel |title=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 |journal=Abhandlungen der physikalischen Klasse der Königlich-Preussischen Akademie der Wissenschaften |date=1817 |pages=286–336 |language=German |access-date=13 January 2025 |url=https://archive.org/stream/abhandlungenderp16akad#page/286/mode/2up}}
In 1866, Auguste Bravais{{cite book |last1=Bravais |first1=Auguste |title=Études Cristallographiques |date=1866 |publisher=Gauthier-Villars |location=Paris |page=168 |url=https://books.google.com/books?id=zGWuwwEACAAJ&pg=PA168 |access-date=13 January 2025 |language=French}} showed that crystals preferentially cleaved parallel to lattice planes of high density.{{cite book |last1=Ladd |first1=Marcus Frederick Charles |title=Symmetry of crystals and molecules |date=2014 |publisher=Oxford University Press |location=Oxford |isbn=9780199670888 |pages=14–15 133–135}} 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.{{rp|page=333}}{{rp|page=48}}
Crystal structure
File:Rhombic dodecahedron assembled from cubic blocks.png assembled from progressively smaller cubic building blocks. Garnet has this crystal habit with {110} crystal faces.{{cite journal |last1=Geiger |first1=Charles A. |title=A tale of two garnets: The role of solid solution in the development toward a modern mineralogy |journal=American Mineralogist |date=August 2016 |volume=101 |issue=8 |pages=1735–1749 |doi=10.2138/am-2016-5522|doi-access=free }}]]
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.{{cite book |last1=Souvignier |first1=B. |editor1-last=Aroyo |editor1-first=Mois I. |title=International tables for crystallography |date=2016 |publisher=D. Reidel Pub. Co.; Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers Group |location=Dordrecht, Holland; Boston, U.S.A.; Hingham, MA |isbn=978-0-470-97423-0 |page=22 |edition=6th |volume=A. Space Group Symmetry |chapter=A general introduction to space groups}}
Quasicrystals do not have translational symmetry, and therefore do not obey the law of rational indices.
See also
References
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{{Crystallography}}