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Magnetostriction and Magnetoelasticity

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Handbook of Magnetism and Magnetic Materials

Abstract

The physical concepts magnetostriction and magnetoelasticity are presented. Spontaneous volume magnetostriction and saturation linear magnetostriction are distinguished. Various magnetoelastic phenomena are introduced, but the emphasis is on magnetostriction in bulk samples and thin films. The equations for magnetostrictive and magnetoelastic coefficients are derived for cubic, hexagonal, and isotropic systems. Experiments on the measurement of the linear magnetostriction λi and magnetoelastic coupling coefficients Bj are discussed. Ab initio-based theory elucidates the physical origin of magnetostrictive effects in metals at the electronic level, although accurate calculations are often elusive. The magnetoelastic properties of nm thin films may deviate in magnitude and sign from the bulk values. Both experiment and theory identify substrate-induced lattice strain as a driving force for this deviation. Data on magnetostriction and magnetoelasticity are compiled, including those of highly magnetostrictive systems, such as (Tb,Dy)Fe2 (Terfenol) and (Fe,Ga) (Galfenol).

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Authors and Affiliations

  1. Max Planck Institute of Microstructure Physics, Halle, Germany

    Dirk Sander

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  1. Dirk Sander
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  1. School of Physics, Trinity College, Dublin, Ireland

    J. M. D. Coey

  2. Max Planck Institute of Microstructure Physics, Halle (Saale), Germany

    Stuart S.P. Parkin

Appendix

Appendix

Notations for Lattice Strain

There are different conventions for the definition of strain, and it is mandatory essential to distinguish between tensor strain 𝜖ij and engineering strain γij [49, 17]. The two conventions differ by a factor of two for the off-diagonal elements such that 2𝜖ij = γij for i ≠ j. Note that the use of the tensor strain is required if it is transformed according to the tensor transformation rules to account for rotated coordinate systems [49]. Throughout this contribution, all strain expressions are given by the tensor strain. For completeness, we note that also the contracted Voigt notation is often used. Starting from the tensor strain, the replacement rule given in Table 9 applies to combine two subscripts into one. Note that factors of two are inserted for the components with subscripts 4, 5, and6 to get 𝜖4 = 2𝜖23, 𝜖5 = 2𝜖13, and𝜖6 = 2𝜖12.

Table 9 Subscript replacement rule for the contracted Voigt notation. Note that factors of two are inserted for the components with subscripts 4, 5, and6 to get 𝜖4 = 2𝜖23, 𝜖5 = 2𝜖13, and𝜖6 = 2𝜖12
Full size table

Relation Between λ and B for the Hexagonal System

We present here the relation between λi and Bj from Eq. (21).

$$\displaystyle \begin{aligned} \lambda_{\mathrm{A}}&=\frac{B_2(c_{11}-c_{12})c_{13}+B_3(-c_{11}+c_{12})c_{33}+B_1(c_{13}^2-c_{11}c_{33})}{a} \end{aligned} $$
(37)
$$\displaystyle \begin{aligned} \lambda_{\mathrm{B}}&=\frac{B_2(c_{11}-c_{12})c_{13}+B_3(-c_{11}+c_{12})c_{33}+B_1(-c_{13}^2+c_{11}c_{33})}{a} \end{aligned} $$
(38)
$$\displaystyle \begin{aligned} \lambda_{\mathrm{C}}&=\frac{-B_2(c_{11}+c_{12})+(B_1+2B_3)c_{13}}{a} \end{aligned} $$
(39)
$$\displaystyle \begin{aligned} \lambda_{\mathrm{D}}&=\frac{-B_4 a+(-B_2(c_{11}-c_{12})(c_{11}+c_{12}-c_{13})}{4 a c_{44}}+\frac{B_3(c_{11}-c_{12})(2c_{13}-c_{33})}{4a c_{44}}\\ &\quad +\frac{B_1(c_{11}c_{13}-c_{12}c_{13}+c_{13}^2-c_{11}c_{33}))c_{44}}{4a c_{44}}\\ a&=(c_{11}-c_{12})(-2 c_{13}^2 + (c_{11}+c_{12})c_{13}) {} \end{aligned} $$
(40)

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Sander, D. (2021). Magnetostriction and Magnetoelasticity. In: Coey, J.M.D., Parkin, S.S. (eds) Handbook of Magnetism and Magnetic Materials. Springer, Cham. https://doi.org/10.1007/978-3-030-63210-6_11

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