Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T19:25:28.479Z Has data issue: false hasContentIssue false
  • English
  • Français

History of the reciprocal lattice

Published online by Cambridge University Press:  03 July 2019

Mohammad Bagher Fathi Open the ORCID record for Mohammad Bagher Fathi [Opens in a new window]
Mohammad Bagher Fathi*
Affiliation:
Department of Condensed Matter, Faculty of Physics, Kharazmi University, Tehran, Iran
*
a)Author to whom correspondence should be addressed. Electronic mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

History of the development of the reciprocal lattice is reviewed. The reciprocal lattice as an essential tool for the study of diffraction experiments by ordered structures and characterization of their structural properties is widely taught in any text of solid state or chemistry, but usually without discussion of its history. This article aims to give a coherent historical perspective on the reciprocal lattice. First, a basic introduction to the reciprocal lattice concept, its mathematical foundation and physical origin, and its relationship with the direct lattice is provided. Then a detailed chronicle of ideas leading to the concept of the reciprocal lattice is presented, including a review of the contributions of Gibbs, Ewald, and others. The polar lattice concept, the great ancestor of the reciprocal lattice, is presented.

Keywords

reciprocal latticeGibbsEwaldpolar latticediffraction experiment
Type
Crystallography Education Article
Information
Powder Diffraction , Volume 34 , Issue 3 , September 2019 , pp. 260 - 266
Copyright
Copyright © International Centre for Diffraction Data 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aroyo, M. I. (Ed.) (2016). International Tables for Crystallography Vol. A: Space-Group Symmetry (Wiley, New York), ch. 1.3, pp. 22–28.Google Scholar
Ashcroft, N. W., and Mermin, N. D. (1976). Solid State Physics (Holt, Rinehart and Winston, New York), p. 86.Google Scholar
Authier, A. (1981). The Reciprocal Lattice. Teaching pamphlet: https://www.iucr.org/education/pamphlets, the official site of International Union of Crystallography.Google Scholar
Authier, A. (2013). Early Days of X-ray Crystallography, International Union of Crystallography (Oxford University Press, Oxford).Google Scholar
Berberian, S. K. (Managing Eds. Halmos, P. R. and Moore, C. C.) (1973). Lectures in Functional Analysis and Operator Theory (Springer-Verlag).Google Scholar
Bragg, W. H. (1915). “IX. Bakerian lecture- X-rays and crystal structure,” Philos. Trans. R. Soc., A 215, 253–274.Google Scholar
Bragg, W. H., and Bragg, W. L. (1913). “The reflexion of X-rays by crystals,” Proc. R. Soc. London, Ser. A 88 (605), 428–438.Google Scholar
Bragg, W. L., and West, J. (1929). “X. A technique for the X-ray examination of crystal structures with many parameters,” Z. Kristallogr. – Cryst. Mater. 69(1–6), 118–148.Google Scholar
Bravais, M. A. (1848). “Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans l'espace, Memoir on the systems formed by points regularly distributed on a plane or in space,” J. Ec. Polytech. 19, 1–128. Its English translation can be found in: Bravais, A. (1949) “On the systems formed by points regularly distributed on a plane or in space,” Crystallographic Society of America, translated by Amos J. Shaler.Google Scholar
Brillouin, L. N. (1931). Die Quantenstitistik (Springer Verlag, Berlin).Google Scholar
Brillouin, L. N. (1962). Science and Information Theory (Dover Phoenix Editions), ch. 8, sec. 11, p. 105.Google Scholar
Brown, J. W., and Churchill, R. V. (2009). Complex Variables and Applications (McGraw-Hill), 8th ed.Google Scholar
Buerger, M. J. (1956). Elementary Crystallography: An introduction to the Fundamental Geometrical Features of Crystals (John Wiley Sons, Inc.).Google Scholar
Burns, G. (1985). Solid State Physics (Academic Press), International ed., p. 78.Google Scholar
Champeney, D. C. (1973). Fourier Transforms and Their Physical Applications (Academic Press, London, LTD.).Google Scholar
Darwin, C. G. (1914a). “XXXIV. The theory of X-ray reflection,” Philos. Mag. 27(158), 315–333.Google Scholar
Darwin, C. G. (1914b). “LXXVIII. The theory of X-ray reflection. Part II,” Philos. Mag. 27(160), 675–690.Google Scholar
Evarestov, R. A., and Smirnov, V. P. (1983). “Special points of the Brillouin zone and their use in the solid state theory,” Phys. Status Solidi B 119, 9.Google Scholar
Ewald, P. P. (1912). Dispersion und Doppelbrechung in Elektronengittern (Kristallen) [Dispersion and Double Refraction in Lattices of Electrons (Crystals)], PhD Thesis.Google Scholar
Ewald, P. P. (1913). “Zur Theorie der Interferenzen der Rontgenstrahlen in Kristallen,” Phys. Z. 14, 465–472.Google Scholar
Ewald, P. P. (1916a). “Zur Begrundung der Kristalloptik. I. Theorie der Dispersion,” Ann. Phys. (Leipzig) 49, 1–38.Google Scholar
Ewald, P. P. (1916b). “Zur Begrundung der Kristalloptik. II. Theorie der Reflexion und Brechung,” Ann. Phys. (Leipzig) 49, 117–143.Google Scholar
Ewald, P. P. (1917). “Zur Begrundung der Kristalloptik. III. Rontgenstrahlen,” Ann. Phys. (Leipzig) 54, 519–597.Google Scholar
Ewald, P. P. (1925). “Die Intensitaten der Rontgenreflexe und der Strukturfaktor,” Phys. Z. 26, 29–32. See correction (1926): Phys. Z. 27, 182.Google Scholar
Ewald, P. P. (1936). “Historisches und Systematisches zum Gebrauch des ‘Reziproken Gitters’ in der Kristallstrukturlehre.” Z. Kristallogr. 93, 396–398.Google Scholar
Ewald, P. P. (1937). “Zur Begrundung der Kristalloptik. IV. Aufstellung einer Allgemeinen Dispersions Bedingung, Inbesondere fur Rontgenfelder,” Z. Kristallogr. A 97, 1–27.Google Scholar
Ewald, P. P. (Ed.) (1962). Fifty Years of X-ray Diffraction, dedicated to the International Union of Crystallography on the occasion of the commemoration meeting in Munich, July (Springer).Google Scholar
Ewald, P. P. (1979). “A review of my papers on crystal optics 1912 to 1968”, Acta Crystallogr. A 35, 1–9.Google Scholar
Fathi, M. B. (2015). Crystallography (in Persian) (Nashr-e Ketab-e Daneshgahi), 1st ed.Google Scholar
Fourier, J. B. Joseph (1807). “Mémoire sur la Propagation de la Chaleur dans les Corps Solides” (Treatise on the Propagation of Heat in Solid Bodies), présenté le 21 Décembre 1807 à l'Institut national – Nouveau Bulletin des sciences par la Société philomatique de Paris. I. Paris: Bernard, March 1808, pp. 112–116. Reprinted in “Mémoire sur la propagation de la chaleur dans les corps solides”. Joseph Fourier – Œuvres complètes, tome 2. pp. 215–221. Archived from the original on 6 December 2008.Google Scholar
Fourier, J. B. Joseph (1822). Théorie analytique de la chaleur (Paris: Firmin Didot, père et fils, OCLC 2688081). Its English translation can be found in: Fourier, J. B. Joseph (1878). The Analytical Theory of Heat, translated by Alexander Freeman, (The University Press).Google Scholar
Gaskill, J. D. (1978). Linear Systems, Fourier Transforms, and Optics (John Wiley & Sons).Google Scholar
Gibbs, J. W., and Wilson, E. B. (1901). Vector Analysis: A Textbook for the use of Students of Mathematics and Physics, Founded upon the Lectures of JWG (Yale University Press).Google Scholar
Jones, H. (1960). The Theory of Brillouin Zones and Electronic States in Crystals (North-Holland publishing Co.).Google Scholar
Kittel, C. (2004). Introduction to Solid State Physics (John Wiley & Sons), 8th ed., p. 27.Google Scholar
Landau, Lev. D., and Lifshitz, E. M. (1963). Statistical Physics (Pergamon Press), Part 1, 3rd ed.Google Scholar
Laue, M. von (1914). “Die Interferenzerscheinungen an Rontgenstrahlen, Hervorgerufen durch das Raumgitter der Kristalle,” Jahrb. der Radioaktivität und Elektronik (Editor J. Stark) 11, 308–345.Google Scholar
Laue, M. von (1920). Nobel Lecture, http://www.nobelprize.org.Google Scholar
Ruddin, W. (1962). Fourier Analysis on Groups (New York: Interscience).Google Scholar
Shmueli, U. (Ed.) (2010). International Tables for Crystallography, Vol. B, Reciprocal Space (Kluwer Academic Publishers), 2nd ed., ch. 1.1, p. 2.Google Scholar
Sneddon, I. N. (1969). Fourier Series (Dover Publications Inc.), 4th impression.Google Scholar
Titchmarsh, E. C. (1967). Introduction to the Theory of Fourier Integrals (Oxford University Press), 3rd ed.Google Scholar