Bethe–Slater curve

The Bethe–Slater curve is a heuristic explanation for why certain metals are ferromagnetic and others are antiferromagnetic. It assumes a Heisenberg model of magnetism, and explains the differences in exchange energy of transition metals as due to the ratio of the interatomic distance a to the radius r of the 3d electron shell.[1] When the magnetically important 3d electrons of adjacent atoms are relatively close to each other, the exchange interaction, , is negative, but when they are further away, the exchange interaction becomes positive, before slowly dropping off.

Bethe–Slater curve: elements above the horizontal axis are ferromagnetic, and those below are antiferromagnetic

The idea of relating exchange energy to inter-atomic distance was first proposed by John C. Slater in 1930,[2][3] and illustrated as a curve on a graph in a review by Sommerfeld and Bethe in 1933.[4]

For a pair of atoms, the exchange interaction wij (responsible for the energy E) is calculated as:[5]

where: = exchange integral; S = electron spins; i and j = indices of the two atoms.

The Slater curve does produce realistic results, predicting Iron, Cobalt and Nickel to be the elements with ferromagnetic ordering. The curve is of practical use as a simple way of estimating based on the average atomic separation.[6] However, more recent evaluations with realistic calculations of the exchange interactions show significantly more complex physics when treating the interactions of different atomic orbitals in an atom separately, rather than as a single unit.[7]

References
  1. http://www.nitt.edu/home/academics/departments/physics/faculty/lecturers/justin/students/magnetic/exchange/
  2. Slater, J. C. (1930). "Cohesion in Monovalent Metals". Physical Review. 35 (5): 509–529. doi:10.1103/PhysRev.35.509. ISSN 0031-899X.
  3. Slater, J. C. (1930). "Atomic Shielding Constants" (PDF). Physical Review. 36 (1): 57–64. doi:10.1103/PhysRev.36.57. ISSN 0031-899X.
  4. Sommerfeld, A.; Bethe, H. (1933). Geiger, H.; Scheel, K. (eds.). Elektronentheorie der Metalle. Handbuch der Physik (in German). 24/2. Springer Berlin, Heidelberg. p. 595. doi:10.1007/978-3-642-91116-3_3. ISBN 978-3-642-89260-8.
  5. Soshin Chikazumi, Physics of Ferromagnetism, Oxford University Press, New York, 1997, p. 125, ISBN 0-19-851776-9
  6. Gallagher, K. A.; Willard, M. A.; Zabenkin, V. N.; Laughlin, D. E.; McHenry, M. E. (1999). "Distributed exchange interactions and temperature dependent magnetization in amorphous Fe88−xCoxZr7B4Cu1 alloys". Journal of Applied Physics. 85 (8): 5130–5132. doi:10.1063/1.369100. ISSN 0021-8979.
  7. Cardias, R.; Szilva, A.; Bergman, A.; Marco, I. Di; Katsnelson, M. I.; Lichtenstein, A. I.; Nordström, L.; Klautau, A. B.; Eriksson, O.; Kvashnin, Y. O. (2017). "The Bethe-Slater curve revisited; new insights from electronic structure theory". Scientific Reports. 7 (1). doi:10.1038/s41598-017-04427-9. ISSN 2045-2322.
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