Law of constancy of interfacial angles

The interfacial angle (red), is the angle between the normals (blue) to the two crystal faces.

The law of constancy of interfacial angles (German: Das Gesetz der Winkelkonstanz; French: Loi de constance des angles) is an empirical law in the fields of crystallography and mineralogy concerning the shape, or morphology, of crystals. The law states that the angles between adjacent corresponding faces of crystals of a particular substance are always constant despite the different shapes, sizes, and mode of growth of crystals. The law is also named the first law of crystallography or Steno's law.

Definition

The contact goniometer was the first instrument used to measure the interfacial angles of crystals

The International Union of Crystallography (IUCr) gives the following definition: "The law of the constancy of interfacial angles (or 'first law of crystallography') states that the angles between the crystal faces of a given species are constant, whatever the lateral extension of these faces and the origin of the crystal, and are characteristic of that species."[1] The law is valid at constant temperature and pressure.[2]

This law is important in identifying different mineral species as small changes in atomic structure can lead to large differences in the angles between crystal faces.

The sum of the interfacial angle (external angle) and the dihedral angle (internal angle) between two adjacent faces sharing a common edge is π radians (180°).

History

Dodecahedron built from smaller cubical units

The law of the constancy of interfacial angles was first observed by the Danish physician Nicolas Steno when studying quartz crystals[3][4] (De solido intra solidum naturaliter contento, Florence, 1669),[5][6] who noted that, although the crystals differed in appearance from one to another, the angles between corresponding faces were always the same.[7]

The law was also observed by Domenico Guglielmini (Riflessioni filosofiche dedotte dalle figure de Sali, Bologna, 1688),[8] but it was generalized and firmly established by Jean-Baptiste Romé de l'Isle (Cristallographie, Paris, 1783)[9] who accurately measured the interfacial angles of a great variety of crystals, using the goniometer designed by Arnould Carangeot and noted that the angles are characteristic of a substance.[10][11] Carangeot was a student of Romé de L’Isle at the time of his invention of the basic crystallographic measuring instrument.[12][13][14]

A French crystallographer, René Just Haüy, showed in 1774[15] that the known interfacial angles could be accounted for if the crystal were made up of minute building blocks (molécules intégrantes)[16] that correspond approximately to the present-day unit cells.

In the diagram, the green dodecahedron on the left is built from cubical units, with the faces having a Miller index of (210). Unlike the regular dodecahedron on the right, its faces are not regular pentagons, but they are close to regular in appearance. The piling of the cubical units forms the pentagonal dodecahedron of pyritohedral pyrite. The decrement of the layers is in the proportion of 2:1, which leads to a dihedral angle at the top edge pq of 126° 87′, closely corresponding to that of the empirical crystal, of 127° 56′. The diagram is based on an 1801 drawing by René Just Haüy.[17][18]

Crystal structure

Crystal faces vary in their underlying lattice density. Three faces are shown with their Miller indices

The phenomenon of the constancy of interfacial angles is important because it is an outward sign of the inherent symmetry and ordered arrangement of atoms, ions or molecules within a crystal structure. The faces of a crystal are parallel to the planes of the crystal lattice, and it is for this reason that the interfacial angles are the same in different crystal specimens.[19]

The angles between the various faces of a crystal remain unchanged throughout its growth. Crystals grow by addition of material to existing faces, this material being deposited parallel to the already existing surfaces. Consequently, if more material is added to one face than to another, the faces become unalike in size and shape, nevertheless the interfacial angles between them remain the same.[20]

Crystals generally exhibit anisotropy, that is their properties are dependent on their direction. In particular, crystals cleave in specific directions, namely those parallel to the planes of the lattice structure.[21] Cleavage preferentially occurs parallel to higher density planes[22] with low Miller indices.[23]

References

  1. ^ "Law of the constancy of interfacial angles". Online Dictionary of Crystallography. International Union of Crystallography. 23 March 2019. Retrieved 7 January 2025.
  2. ^ Borchardt-Ott, Walter (2012). Crystallography: An Introduction. Translated by Gould, Robert O. Berlin, Heidelberg: Springer. p. 35. doi:10.1007/978-3-642-16452-1. ISBN 9783642164514.
  3. ^ 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. p. 44. ISBN 079230649X. Retrieved 24 December 2024.
  4. ^ 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.
  5. ^ Steno, Nicolas (1669). De solido intra solidum naturaliter contento (in Latin). Florence: Star. Retrieved 8 January 2025.
  6. ^ Steno, Nicolaus (1916) [1669]. The prodromus of Nicolaus Steno's dissertation concerning a solid body enclosed by process of nature within a solid. Translated by Winter, John Garrett. New York, London: The Macmillan Company. p. 272. Retrieved 9 January 2025. Figures 5 and 6 belong to the class of those which I could present in countless numbers to prove that in the plane of the axis both the number and the length of the sides are changed in various ways without changing the angles.
  7. ^ Ladd, Marcus Frederick Charles (2014). Symmetry of crystals and molecules. Oxford: Oxford University Press. pp. 13–15. ISBN 9780199670888.
  8. ^ Whewell, William (2011) [1837]. "Prelude to the Epoch of De L'Isle and Haüy". History of the Inductive Sciences: From the Earliest to the Present Times (Online ed.). Cambridge: Cambridge University Press. pp. 191–197. doi:10.1017/cbo9780511734359.027. ISBN 978-1-108-01926-2. Nevertheless since there is here a principle of crystallization, the inclination of the planes and of the angles is always constant.
  9. ^ Romé de L'Isle, Jean Baptiste Louis de (1783). "Préface". Cristallographie (in French). Paris: De l'imprimerie de Monsieur. Retrieved 8 January 2025.
  10. ^ Metzger, Hélène (1969) [1918]. La Gènese de la Science de Cristaux (in French). Paris: Albert Blanchard. pp. 67–68. Archived from the original on 2 October 2021. Retrieved 8 January 2025.
  11. ^ Phillips, F. C. (1963). An Introduction To Crystallography (3rd ed.). New York: John Wiley & Sons. pp. 12–14. Retrieved 8 January 2025.
  12. ^ Burke, John G. (1966). Origins of the science of crystals. Berkley and Los Angeles: University of California Press. pp. 69–70. Retrieved 8 January 2025.
  13. ^ Scholz, Erhard (1989). Symmetrie, Gruppe, Dualität (in German). Basel Boston Berlin: Birkhäuser. pp. 19, 21. ISBN 3764319747.
  14. ^ Authier, André (2015). "11.11 J.-B. L. Romé de l'Isle and the law of the constancy of interfacial angles, 1783". Early days of X-ray crystallography. Oxford: Oxford University Press. pp. 313–317. doi:10.1093/acprof:oso/9780199659845.003.0011. ISBN 9780198754053. Retrieved 8 January 2025.
  15. ^ 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.
  16. ^ "Steno's law". Encyclopaedia Britannica. Retrieved 7 January 2025.
  17. ^ Haüy, René Just (1801). Traité de minéralogie (in French). Vol. Caractère Minéralogique. Paris: Chez Louis. p. II, fig. 16. Retrieved 8 January 2025.
  18. ^ Haüy, René Just (1822). Traité de cristallographie (in French). Vol. Atlas. Paris: Bachelier. p. 8, fig. 75. Retrieved 7 January 2025.
  19. ^ Bragg, W. L. (1933). "The Crystal Pattern". The Crystalline State. Vol. 1. A General Survey. London: G. Bell & Sons. p. 6. Retrieved 11 January 2025.
  20. ^ Sunagawa, Ichirō (2007). Crystals: growth, morphology, and perfection. Cambridge: Cambridge University Press. ISBN 9780521841894.
  21. ^ Hoffmann, Frank (2020). Introduction to crystallography (1st ed.). Cham, Switzerland: Springer. pp. 238–239. doi:10.1007/978-3-030-35110-6. ISBN 9783030351090.
  22. ^ Bravais, Auguste (1866). Études Cristallographiques (in French). Paris: Gauthier-Villars. p. 168. Retrieved 13 January 2025.
  23. ^ Vaknin, Uriel; Sherman, Dov; Gorfman, Semën (1 September 2021). "Geometrical prediction of cleavage planes in crystal structures" (PDF). IUCrJ. 8 (5): 793–804. Bibcode:2021IUCrJ...8..793V. doi:10.1107/S2052252521007272. PMC 8420770. PMID 34584740. Retrieved 8 January 2025.