Magnetostriction in an external magnetic field

Borovik-Romanov, A. S.; Grimmer, H.

doi:10.1107/97809553602060000632

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 144-145

Section 1.5.9.2. Magnetostriction in an external magnetic field

A. S. Borovik-Romanov^a ^‡ and H. Grimmer^b ^*

^a P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia, and ^bLabor für Neutronenstreuung, ETH Zurich, and Paul Scherrer Institute, CH-5234 Villigen PSI, Switzerland
Correspondence e-mail: hans.grimmer@psi.ch

1.5.9.2. Magnetostriction in an external magnetic field

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There are three reasons for the magnetostriction arising in a magnetic field: (a) the transfer of the crystal into a single-domain state if the magnetic field is directed along one of the easy axes; (b) the deflection of the magnetization (or antiferromagnetic vector) by the magnetic field from the easy axis in a single-domain crystal; (c) the change of the magnetization in a sufficiently strong magnetic field.

Let us begin with case (a) and consider a crystal with cubic symmetry in the paramagnetic state (i.e. with a cubic prototype ). We calculate the magnetostriction that occurs when the applied magnetic field transforms the crystal from the demagnetized multidomain state into the saturated single-domain state. This transformation is shown schematically in Fig. 1.5.9.1.

Figure 1.5.9.1 | top | pdf |

Diagram explaining the occurrence of magnetostrictive strains in the demagnetized and saturated states of a cube-shaped crystal with a cubic prototype.

Each domain in the demagnetized state is distorted by spontaneous magnetostriction . The number of domains in the sample is usually much larger than shown in the figure. Thus a sample of a crystal with a cubic prototype which in the paramagnetic state has the form of a cube will retain this form in the ordered state. Its linear dimension will be changed as a result of magnetostriction. Averaging these strains over all the domains, one gets the spontaneous magnetostrictive change of the linear dimension of the sample, which is equal for any direction x, y or z: $[(\delta l)_{\rm dem}/l_0 = \lambda^{\rm dem} = \overline{\lambda^0_{\beta}(n_k)}, \eqno(1.5.9.16)]$ where $[n_{k}]$ defines the directions parallel to all the easy axes of the crystal. For crystals with a cubic prototype, there are two principal ordered states: with the easy axis along the $[\langle 111\rangle]$ directions as in nickel or along the $[\langle 100\rangle]$ directions as in iron. Averaging the strains of all eight possible easy-axis directions of the domains in the $[\langle111 \rangle]$ -type ferromagnet we obtain from (1.5.9.12) the following expression for the spontaneous magnetostriction of the demagnetized crystal: $[ \lambda ^{\rm dem} = h_{0} +{\textstyle{1 \over 3}}(h_{1} + h_{3} + h_{4}). \eqno(1.5.9.17)]$ In the case of the $[\langle 100\rangle]$ -type ferromagnet, the averaging over the six groups of domains leads to $[\lambda ^{\rm dem} = h_{0} +{\textstyle{1 \over 3}}(h_{1} + h_{4}). \eqno(1.5.9.18)]$

In the saturated state, the sample loses its cubic form. It becomes longer parallel to the magnetic field and thinner perpendicular to it. By definition, the demagnetized state is taken as a reference state for the magnetostriction in the magnetic field. Subtracting from the general relation for spontaneous magnetostriction (1.5.9.12) the expressions (1.5.9.17) and (1.5.9.18) for the demagnetized sample, Becker & Döring (1939) obtained the equations that describe the anisotropy of the magnetostriction caused by saturation magnetization of the $[\langle 111\rangle]$ and $[\langle 100\rangle]$ types of magnetic crystals:

$[\langle 111\rangle]$ type: $[\eqalignno{\lambda^{\rm sat}_{\beta} &= h_1 [S(n_1^2\beta_1^2) - {\textstyle{1 \over 3}}] + 2h_2 S(n_1 n_2\beta_1\beta_2) + h_3 [S(n_1^2 n_2^2) - {\textstyle{1 \over 3}}] &\cr&\quad + h_4 [S(n_1^4\beta_1^2 + {\textstyle{2 \over 3}}n_1^2 n_2^2) - {\textstyle{1 \over 3}}] + 2h_5 S(n_1^2 n_2 n_3\beta_2\beta_3)\semi &\cr &&(1.5.9.19)}]$
$[\langle 100\rangle]$ type: $[\eqalignno{\lambda^{\rm sat}_{\beta} &= h_1 [S(n_1^2\beta_1^2) - {\textstyle{1 \over 3}}] + 2h_2 S(n_1 n_2\beta_1\beta_2) + h_3 S(n_1^2 n_2^2) &\cr&\quad + h_4 [S(n_1^4\beta_1^2 + {\textstyle{2 \over 3}}n_1^2 n_2^2) - {\textstyle{1 \over 3}}] + 2h_5 S(n_1^2 n_2 n_3\beta_2\beta_3). &\cr&&(1.5.9.20)}]$

Both types of magnetics with a cubic prototype are described by a two-constant equation if the terms of fourth power are neglected. This equation was obtained by Akulov (1928) in the form $[\eqalignno{{\lambda}^{\rm sat}_{\beta} &= {\textstyle{3 \over 2}}\lambda_{100} (n_1^2\beta_1^2 + n_2^2\beta_2^2 + n_3^2\beta_3^2 - {\textstyle{1 \over 3}}) &\cr&\quad +3\lambda_{111} (n_1 n_2\beta_1\beta_2 + n_2 n_3\beta_2\beta_3 + n_3 n_1\beta_3\beta_1), &\cr&&(1.5.9.21)}]$ where the constants $[\lambda_{100}]$ and $[\lambda_{111}]$ correspond to the magnetostrictive deformation of a `cubic' ferromagnet along the direction of the magnetic field that is applied along the directions $[\langle 100\rangle]$ and $[\langle 111\rangle]$ , respectively. Let us denote by $[Q_{1}]$ and $[Q_{2}]$ the following equal coefficients in the equation for the magnetoelastic energy (1.5.9.3): $[Q_{1} = Q_{xxxx} = Q_{yyyy} = Q_{zzzz}; \quad Q_{2} = Q_{xyxy} = Q_{yzyz} = Q_{zxzx}. \eqno(1.5.9.22)]$ According to (1.5.9.9), the coefficients $[\lambda_{100}]$ and $[\lambda_{111}]$ may be written as the following fractions of $[Q_{i}]$ and the elastic stiffnesses $[c_{\alpha \beta}]$ : $[\lambda _{100} = {{Q_{1}}\over{c_{12} - c_{11}}}, \quad \lambda _{111} = -{{1}\over{3}} {{Q_{2}}\over{c_{44}}}. \eqno(1.5.9.23)]$

If the magnetic field transforms the crystal from the demagnetized to the saturated state and if the linear dimension of the sample along the magnetic field increases, then its dimension perpendicular to the field will decrease (see Fig. 1.5.9.1). It follows from relation (1.5.9.21) that the magnetostriction perpendicular to the magnetic field is $[\lambda^{\perp}_{100} = -{\textstyle{1 \over 2}}\lambda_{100} \; \hbox{ and } \; \lambda^{\perp}_{111} = -{\textstyle{1 \over 2}}\lambda_{111}. \eqno(1.5.9.24)]$

Some data for magnetostriction of ferromagnets with prototype symmetry $[m\bar{3}m1']$ are presented in Table 1.5.9.2.

Table 1.5.9.2 | top | pdf |
Magnetostriction data for ferromagnets with prototype symmetry $[m\bar{3}m1']$

Compound	$[\lambda_{100} \times 10^{6}]$	$[\lambda_{111} \times 10^{6}]$	References^†
Fe	20.7	−21.2	(1)
Ni	−45.9	−24.3	(1)
Fe₃O₄	−20	78	(2)
YIG (T = 300 K)	−1.4	−2.4	(3)
DyIG (T = 300 K)	−12.5	−5.9	(3)
DyIG (T = 4.2 K)	−1400	−550	(4)

^† References: (1) Lee (1955

); (2) Bickford et al. (1955

); (3) Iida (1967

); (4) Clark et al. (1966

In a uniaxial crystal, the magnetostriction in the magnetic field arises mainly as a result of the rotation of the magnetization vector from the direction of the easy axis to the direction of the applied field. The magnetostriction in the magnetic field of an easy-axis hexagonal ferromagnet can be obtained from the relation for the spontaneous magnetostriction (1.5.9.14). In the demagnetized state, such a ferromagnet possesses only two types of antiparallel domains, in which the magnetization is aligned parallel or antiparallel to the hexagonal axis ( $[n_z = \pm 1]$ , [n_x = n_y = 0] ).

Thus the magnetostriction of the demagnetized state is described by $[\lambda^{\rm dem}_{\beta} = h_{0} + (h_{1} + h_{6})\beta ^{2}_{3}. \eqno(1.5.9.25)]$ The saturation magnetostriction can be calculated for different directions of the applied magnetic field using the equations (1.5.9.14), (1.5.9.15) and (1.5.9.25). If the magnetic field is applied along the x axis ( $[n_{x}= 1]$ , $[n_{y}=n_{z}=0]$ ), the saturation magnetostrictions for three directions of the vector $[\boldbeta]$ : $[\lambda ^{\rm sat}_{\beta} = \lambda _{A}, \lambda _{B}, \lambda _{C}]$ are $[\eqalignno{\boldbeta \parallel Ox \quad \lambda _{A} &= h_{2}, &\cr \boldbeta \parallel Oy \quad \lambda _{B} &= h_{3}, &\cr \boldbeta \parallel Oz \quad \lambda _{C} &=-h_{1}. &(1.5.9.26)}]$ If the magnetic field is applied at an angle of 45° to the hexagonal axis along the [101] direction, the saturation magnetostriction along the magnetic field is described by $[\lambda_{D} = \lambda^{\rm sat}_{101} = {\textstyle{1 \over 4}}(h_{2} - h_{1} + 2h_{5}). \eqno(1.5.9.27)]$ Using the constants $[\lambda_{A}]$ , $[\lambda_{B}]$ , $[\lambda_{C}]$ and $[\lambda _{D}]$ introduced above, the general relation for the magnetostriction caused by magnetization to saturation can be presented in the form $[\eqalignno{\lambda ^{\rm sat}_{\beta} &= \lambda_A [(n_1\beta_1 + n_2\beta_2)^2 - (n_1\beta_1 + n_2\beta_2)n_3\beta_3]&\cr&\quad + \lambda_B [(1 - n_3^2)(1 - \beta_3^2) - (n_1\beta_1 + n_2\beta_2)^2] &\cr&\quad+ \lambda_C [(1 - n_3^2)\beta_3^2 - (n_1\beta_1 + n_2\beta_2)n_3\beta_3] &\cr&\quad+ 4\lambda_D (n_1\beta_1 + n_2\beta_2)n_3\beta_3. &(1.5.9.28)}]$ A typical hexagonal ferromagnet is cobalt. The magnetostriction constants introduced above have the following values for Co at room temperature: $[\matrix{\lambda_A = - 45\times 10^{-6 }\hfill& \lambda_C = + 110\times 10^{-6} \hfill\cr \lambda_B = - 95\times 10^{-6} \hfill& \lambda_D = - 100\times 10^{-6}\hfill}]$

A more sophisticated treatment of the symmetry of the magnetostriction constants is given in the monograph of Birss (1964) and in Zalessky (1981).

References

Akulov, N. (1928). Über die Magnetostriktion der Eisenkristalle. Z. Phys. 52, 389–405.Google Scholar

Becker, R. & Döring, W. (1939). Ferromagnetismus. Berlin: Springer.Google Scholar

Birss, R. R. (1964). Symmetry and magnetism. Amsterdam: North-Holland.Google Scholar

Zalessky, A. V. (1981). Magnetic properties of crystals. In Modern crystallography, Vol. IV, edited by L. A. Shuvalov. (In Russian.) Moscow: Nauka. [English translation (1988): Berlin: Springer.]Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 144-145