International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.5, pp. 145148
Section 1.5.9. Magnetostriction^{a}P. L. Kapitza Institute for Physical Problems, Russian Academy of Sciences, Kosygin Street 2, 119334 Moscow, Russia,^{b}Labor für Neutronenstreuung, ETH Zurich, Switzerland, and ^{c}Paul Scherrer Institute, CH5232 Villigen PSI, Switzerland 
The transition to an ordered magnetic state is accompanied by a spontaneous distortion of the lattice, which is denoted spontaneous magnetostriction. The lattice distortion may be specified by the deformation (strain) components . The undeformed state is defined as the crystal structure that would be realized if the crystal remained in the paramagnetic state at the given temperature. This means that it is necessary to separate the magnetostrictive deformation from the ordinary thermal expansion of the crystal. This can be done by measurements of the magnetostriction in external magnetic fields applied in different directions (see Section 1.5.9.2). The magnetostriction arises because the first derivatives of the exchange and relativistic energies responsible for the magnetic order do not vanish at . Thus these energies depend linearly on the deformations around . That part of the magnetic energy which depends on the deformations (and consequently on the stresses) is called the magnetoelastic energy, . To find the equilibrium values of the spontaneous magnetostriction, one also has to take the elastic energy into account.
The magnetoelastic energy includes both an exchange and a relativistic part. In some ferromagnets that are cubic in the paramagnetic phase, the exchange interaction does not lower the cubic symmetry. Thus the exchange part of satisfies the relationsSuch a form of the magnetoelastic energy gives rise to an isotropic spontaneous magnetostriction or volume change (volume striction) which does not depend on the direction of magnetization. In what follows, we shall analyse mainly the anisotropic magnetostriction.
The spontaneous magnetostriction deformations are so small (about 10^{−5}) for some ferro and antiferromagnets that they cannot be observed by the usual Xray techniques. However, in materials with ions possessing strong spin–orbit interactions (like Co^{2+}), it may be as large as 10^{−4}. The magnetostriction in rareearth metals and their compounds with iron and cobalt are especially large (up to 10^{−3}).
Magnetostriction is observed experimentally as a change of the linear dimension along a direction specified by a unit vector : where are the deformation components, which are functions of the components of the unit vector aligned in the direction of the magnetization. Only the symmetric part of the deformation tensor has been taken into account, because the antisymmetric part represents a rotation of the crystal as a whole.
The magnetostriction that arises in an applied magnetic field will be discussed in Section 1.5.9.2; Section 1.5.9.1 is devoted to the spontaneous magnetostriction.
In this section, we shall assume that the crystal under consideration undergoes a phase transition from the paramagnetic state into a magnetically ordered state. The latter is a singledomain state with the magnetization (or the antiferromagnetic vector) aligned along the vector . As was mentioned above, to solve the problem of the spontaneous magnetostriction we have to minimize the sum of magnetoelastic and elastic energy.
Like the anisotropy energy, the anisotropic part of the magnetoelastic energy can be represented as a series in the components of the unit vector : As for every magnetically ordered crystal, this relation contains only even powers of the magnetization unit vector. The components of the tensors are called magnetostrictive or magnetoelastic coefficients. They are proportional to even powers of the magnetization M ( and ). The symmetry of the tensors and is defined by the crystallographic point group of the initial paramagnetic phase of the crystal.
It is convenient to consider the magnetoelastic energy as part of a general expansion of the free energy of a crystal into a series with respect to the deformation (as the magnetostrictive deformations are small): where all the expansion coefficients are functions of the components of the magnetization unit vector . The superscripts zero indicate that the expansion coefficients have been calculated relative to the undistorted lattice. Such a state in which, at a given temperature, there is no magnetic interaction to distort the crystal is not realizable practically. It will be shown below that the values of the coefficients may be obtained experimentally by observing the magnetostriction in a magnetic field (see Section 1.5.9.2).
The first term in (1.5.9.4) is the anisotropy energy at zero deformation : This expression has to be compared with the expression for the anisotropy at zero stress introduced in Section 1.5.3.2 [see (1.5.3.5)]. It is obvious that symmetry imposes the same restrictions on the tensors K in both expressions for the anisotropy. Later, we shall discuss these two relations for the anisotropy in more detail.
The second term in (1.5.9.4) is the magnetoelastic energy density, which is displayed in equation (1.5.9.3) and represents the energy of anisotropic deformation.
The third term in (1.5.9.4) is quadratic in and can be considered as an additional contribution to the elastic energy arising from the distortion of the lattice by spontaneous magnetostriction. This term is small compared with the main part of the elastic energy, and the effect it produces is called a morphic effect and is usually neglected.
The equilibrium deformation components may be found by minimization of the sum of the magnetoelastic and elastic energies. The latter, , is given bywhere are the elastic stiffnesses. The minimization leads to We shall replace the elastic stiffnesses in this equation by the elastic compliances , taking into account that Hooke's law can be written in two forms (see Section 1.3.3 ): Thus the relation for the equilibrium components of the strain becomes Combining the relations (1.5.9.9) and (1.5.9.3), we get the following equation for the magnetostrictive strain components as a function of the magnitude and direction of the magnetization :
Let us denote the spontaneous magnetostriction by ( defines the direction of the magnetostriction relative to the crystallographic axes). According to (1.5.9.2), we obtain Relation (1.5.9.11) shows that can be chosen as symmetric in its first two indices and symmetric in its last four indices. It can therefore be represented by a matrix , where and . Table 1.5.9.1 lists the pairs that correspond to and the quadruples that correspond to A.

Similarly, can be chosen as symmetric in its first two and in its last two indices. It can therefore be represented by a matrix , where . The correspondence between the numbers 1 to 6 and pairs or is given in Table 1.5.9.1.
The tensors and must satisfy the symmetry of the paramagnetic state of the crystal under consideration. In the case of cubic crystals with fourfold axes (paramagnetic point groups , or ), the two matrices and possess instead of the 36 and 90 independent components only 3 and 6, i.e. , , and , , , , , , respectively. The exact form of the two matrices will be given in the following.
There are three reasons for the magnetostriction arising in a magnetic field: (a) the transfer of the crystal into a singledomain 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 singledomain 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 singledomain state. This transformation is shown schematically in Fig. 1.5.9.1.

Diagram explaining the occurrence of magnetostrictive strains in the demagnetized and saturated states of a cubeshaped 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: where 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 directions as in nickel or along the directions as in iron. Averaging the strains of all eight possible easyaxis directions of the domains in the type ferromagnet we obtain from (1.5.9.12) the following expression for the spontaneous magnetostriction of the demagnetized crystal: In the case of the type ferromagnet, the averaging over the six groups of domains leads to
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 and types of magnetic crystals:
Both types of crystals with a cubic prototype are described by a twoconstant equation if the terms of fourth power are neglected. This equation was obtained by Akulov (1928) in the formwhere the constants and correspond to the magnetostrictive deformation of a `cubic' ferromagnet along the direction of the magnetic field that is applied along the directions and , respectively. Let us denote by and the following equal coefficients in the equation for the magnetoelastic energy (1.5.9.3): According to (1.5.9.9), the coefficients and may be written as the following fractions of and the elastic stiffnesses :
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
Some data for magnetostriction of ferromagnets with prototype symmetry are presented in Table 1.5.9.2.
^{†}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 easyaxis 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 (, ).
Thus the magnetostriction of the demagnetized state is described byThe 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 (, ), the saturation magnetostrictions for three directions of the vector : areIf 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 Using the constants , , and introduced above, the general relation for the magnetostriction caused by magnetization to saturation can be presented in the formA typical hexagonal ferromagnet is cobalt. The magnetostriction constants introduced above have the following values for Co at room temperature:
A more sophisticated treatment of the symmetry of the magnetostriction constants is given in the monograph of Birss (1964) and in Zalessky (1981).
The spontaneous magnetostriction makes a contribution to the magnetic anisotropy (especially in crystals with a cubic prototype). Therefore, to find the full expression for the anisotropy energy, one has to sum up the magnetic [see (1.5.9.5)], the magnetoelastic [see (1.5.9.3)] and the elastic [see (1.5.9.6)] energies. At zero strain (), only . At zero stress Here we used the modified equation (1.5.9.7):
Substituting the values for the spontaneous magnetostriction, the final equation for the anisotropy energy measured at atmospheric pressure may be written as
As an example, for the ferromagnets with a cubic prototype this equation may be written asThe coefficients and may be expressed in terms of the saturation magnetostriction constants [see (1.5.9.12)] and the elastic stiffnesses :
For cubic crystals, and are of the same magnitude. As an example, for Ni one has = 80 000 erg cm^{−3} = 8000 J m^{−3} and = −139 000 erg cm^{−3} = −13 900 J m^{−3}.
References
Akulov, N. (1928). Über die Magnetostriktion der Eisenkristalle. Z. Phys. 52, 389–405.Becker, R. & Döring, W. (1939). Ferromagnetismus. Berlin: Springer.
Birss, R. R. (1964). Symmetry and Magnetism. Amsterdam: NorthHolland.
Mason, W. P. (1951). A phenomenological derivation of the first and secondorder magnetostriction and morphic effects for a nickel crystal. Phys. Rev. 82, 715–723.
Mason, W. P. (1952). A phenomenological derivation of the first and secondorder magnetostriction and morphic effects for a nickel crystal. Erratum. Phys. Rev. 85, 1065.
Mason, W. P. (1954). Derivation of magnetostriction and anisotropic energies for hexagonal, tetragonal, and orthorhombic crystals. Phys. Rev. 96, 302–310.
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.]