Piezomagnetic effect

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. 132-136

Section 1.5.7.1. Piezomagnetic effect

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

1.5.7.1. Piezomagnetic effect

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The possibility of the existence of a piezomagnetic effect was first foreseen by Voigt (1928). However, he assumed that it is sufficient to consider only the crystallographic symmetry in order to predict this effect. In reality, the crystals that do not possess a magnetic structure are characterized by the transformation R being contained in the magnetic group as an independent element. The transformation R changes the sign of the magnetic vectors $[\bf H]$ , $[\bf L]$ , $[\bf M]$ . Hence, for such crystals all values of $[\Lambda_{ijk}]$ vanish and piezomagnetism is forbidden. The magnetic groups of magnetically ordered crystals (ferromagnets and antiferromagnets) contain R only in combination with other elements of symmetry, or do not contain this transformation at all. Hence the piezomagnetic effect may occur in such crystals. This statement was first made by Tavger & Zaitsev (1956). The most interesting manifestation of the piezomagnetic effect is observed in antiferromagnets, as there is no spontaneous magnetization in these materials.

From equation (1.5.7.1) it follows that $[\Lambda_{ijk}]$ is an axial tensor of third rank. Hence, apart from the restriction that piezomagnetism is forbidden for all para- and diamagnetic materials, it must be absent from the 21 magnetic point groups that contain the element $[C_{i}R = \overline{1}']$ (see Table 1.5.7.1). The stress tensor $[T_{jk}]$ is symmetrical ( $[T_{jk} = T_{kj}]$ ); see Section 1.3.2.4 . Thus the tensor $[\Lambda_{ijk}]$ is symmetrical in its last two indices. This is the reason why piezomagnetism is prohibited for three more magnetic point groups: $[{\bi O}=432]$ , $[{\bi T}_{d}=\bar{4}3m]$ and $[{\bi O}_{h}=m\bar{3}m]$ . The remaining 66 magnetic point groups were found by Tavger (1958), who also constructed the 16 corresponding forms of the piezomagnetic tensors appropriate to each point group. They are represented in Table 1.5.7.1. (See also Birss & Anderson, 1963; Birss, 1964.)

Table 1.5.7.1 | top | pdf |
The forms of the matrix characterizing the piezomagnetic effect

Magnetic crystal class		Matrix representation $[\Lambda_{i\alpha}]$ of the piezomagnetic tensor
Schoenflies	Hermann–Mauguin
$[{\bi C}_1]$		$[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} &\Lambda_{14} &\Lambda_{15} &\Lambda_{16}\cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} &\Lambda_{24} &\Lambda_{25} &\Lambda_{26}\cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} &\Lambda_{34} &\Lambda_{35} &\Lambda_{36} } \right]]$
$[{\bi C}_i]$	$[\bar{1}]$

$[{\bi C}_2]$	$[2\,(=121)]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 &\Lambda_{16}\cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} & 0 &\Lambda_{25} & 0 \cr 0 & 0 & 0 &\Lambda_{34} & 0 &\Lambda_{36} } \right]]$
$[{\bi C}_s]$	$[m\,(=1m1)]$
$[{\bi C}_{2h}]$	$[2/m\,(=1\,2/m\,1)]$
	(unique axis y)
$[{\bi C}_2({\bi C}_1)]$	$[2'\,(=12'1)]$	$[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} & 0 &\Lambda_{26}\cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 &\Lambda_{35} & 0 } \right]]$
$[{\bi C}_s({\bi C}_1)]$	$[m'\,(=1m'1)]$
$[{\bi C}_{2h}({\bi C}_i)]$	$[2'/m'\,(=1\,2'/m'\,1)]$
	(unique axis y)
$[{\bi D}_2]$		$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{25} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{36} } \right]]$
$[{\bi C}_{2v}]$	$[mm2\,[2mm,m2m]]$
$[{\bi D}_{2h}]$
$[{\bi D}_2({\bi C}_2)]$		$[\left[\matrix{ 0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} & 0 & 0 \cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 & 0 & 0 }\right]]$
$[{\bi C}_{2v}({\bi C}_2)]$
$[{\bi C}_{2v}({\bi C}_s)]$	$[m'2'm\,[2'm'm]]$
$[{\bi D}_{2h}({\bi C}_{2h})]$
$[{\bi C}_4, \,{\bi C}_6]$	$[4,\, 6]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & \Lambda_{15}& 0 \cr 0 & 0 & 0 &\Lambda_{15} &-\Lambda_{14}& 0 \cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi S}_4,\,{\bi C}_{3h}]$	$[\bar{4},\,\bar{6}]$
$[{\bi C}_{4h},\,{\bi C}_{6h}]$	$[4/m,\,6/m]$
$[{\bi C}_4({\bi C}_2)]$		$[\left[\matrix{ 0 & 0 & 0 & \Lambda_{14}&\Lambda_{15} & 0 \cr 0 & 0 & 0 &-\Lambda_{15}&\Lambda_{14} & 0 \cr \Lambda_{31}&-\Lambda_{31} & 0 & 0 & 0 &\Lambda_{36} } \right]]$
$[{\bi S}_4({\bi C}_2)]$	$[\bar{4}']$
$[{\bi C}_{4h}({\bi C}_{2h})]$
$[{\bi D}_4,\,{\bi D}_6 ]$	$[422,\,622]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &-\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{4v},\,{\bi C}_{6v}]$	$[4mm,\,6mm]$
$[{\bi D}_{2d},\,{\bi D}_{3h}]$	$[\bar{4}2m\,[\bar{4}m2],\,\bar{6}m2\,[\bar{6}2m]]$
$[{\bi D}_{4h},\,{\bi D}_{6h}]$	$[4/mmm,\,6/mmm]$
$[{\bi D}_4({\bi C}_4),\,{\bi D}_6({\bi C}_6)]$	$[42'2',\,62'2']$	$[\left[\matrix{ 0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{15} & 0 & 0 \cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi C}_{4v}({\bi C}_4),\,{\bi C}_{6v}({\bi C}_6)]$	$[4m'm',\,6m'm']$
$[{\bi D}_{2d}({\bi S}_4),\,{\bi D}_{3h}({\bi C}_{3h})]$	$[\bar{4}2'm'\,[\bar{4}m'2'],\,\bar{6}m'2'\,[\bar{6}2'm']]$
$[{\bi D}_{4h}({\bi C}_{4h}),\,{\bi D}_{6h}({\bi C}_{6h})]$	$[4/mm'm',\,6/mm'm']$
$[{\bi D}_4({\bi D}_2)]$		$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{36} }\right]]$
$[{\bi C}_{4v}({\bi C}_{2v})]$
$[{\bi D}_{2d}({\bi D}_2),\,{\bi D}_{2d}({\bi C}_{2v})]$	$[\bar{4}'2m',\,\bar{4}'m2']$
$[{\bi D}_{4h}({\bi D}_{2h})]$
$[{\bi C}_3]$		$[\left[\matrix{ \Lambda_{11}&-\Lambda_{11} &0 &\Lambda_{14} & \Lambda_{15} &-2\Lambda_{22} \cr -\Lambda_{22} & \Lambda_{22} &0 &\Lambda_{15} &-\Lambda_{14} &-2\Lambda_{11} \cr \Lambda_{31} & \Lambda_{31} &\Lambda_{33}& 0 & 0 & 0 } \right]]$
$[{\bi S}_6]$	$[\bar{3}]$

$[{\bi D}_3]$	$[32\,(=321)]$	$[\left[\matrix{\Lambda_{11} &-\Lambda_{11}& 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &-\Lambda_{14} &-2\Lambda_{11}\cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3v}]$	$[3m\,(=3m1)]$
$[{\bi D}_{3d}]$	$[\bar{3}m\,(=\bar{3}m1)]$
$[{\bi D}_3({\bi C}_3)]$	$[32'\,(=32'1)]$	$[\left[\matrix{ 0 & 0 & 0 & 0 & \Lambda_{15}&-2\Lambda_{22}\cr -\Lambda_{22} & \Lambda_{22} & 0 &\Lambda_{15} & 0 & 0 \cr \Lambda_{31} & \Lambda_{31} &\Lambda_{33} & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3v}({\bi C}_3)]$	$[3m'\,(=3m'1)]$
$[{\bi D}_{3d}({\bi S}_6)]$	$[\bar{3}m'\,(=\bar{3}m'1)]$
$[{\bi C}_6({\bi C}_3) ]$		$[\left[\matrix{\Lambda_{11} &-\Lambda_{11} & 0 & 0 & 0 &-2\Lambda_{22} \cr -\Lambda_{22} & \Lambda_{22} & 0 & 0 & 0 &-2\Lambda_{11} \cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{3h}({\bi C}_3)]$	$[\bar{6}']$
$[{\bi C}_{6h}({\bi S}_6)]$
$[{\bi D}_6({\bi D}_3)]$		$[\left[\matrix{\Lambda_{11} &-\Lambda_{11} & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 &-2\Lambda_{11}\cr 0 & 0 & 0 & 0 & 0 & 0 } \right]]$
$[{\bi C}_{6v}({\bi C}_{3v})]$
$[{\bi D}_{3h}({\bi D}_3),\,{\bi D}_{3h}({\bi C}_{3v})]$	$[\bar{6}'2m',\,\bar{6}'m2']$
$[{\bi D}_{6h}({\bi D}_{3d})]$
$[{\bi T},\,{\bi T}_h ]$	$[23,\,m\bar{3} ]$	$[\left[\matrix{ 0 & 0 & 0 &\Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 &\Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{14} }\right]]$
$[{\bi O}({\bi T})]$
$[{\bi T}_d({\bi T})]$	$[\bar{4}'3m']$
$[{\bi O}_h({\bi T}_h)]$	$[m\bar{3}m']$

Since the stress tensor $[T_{jk}]$ is symmetrical, it has only six independent components. Therefore the notation of its components can be replaced by a matrix notation (Voigt's notation, see Section 1.3.2.5 ) in the following manner: $[\matrix{\hbox{Tensor notation}\hfill&\hbox{Matrix notation}\hfill\cr T_{11}\hfill & T_{1}\hfill\cr T_{22}\hfill & T_{2}\hfill\cr T_{33}\hfill & T_{3}\hfill\cr T_{23}, T_{32}\hfill & T_{4}\hfill\cr T_{31}, T_{13}\hfill & T_{5}\hfill\cr T_{12}, T_{21}\hfill & T_{6}\hfill\cr&\cr}]$ In matrix notation, equation (1.5.7.2) may be written in the form $[M_{i} = \Lambda_{i\alpha}T_{\alpha}, \eqno(1.5.7.4)]$ where [i = 1, 2, 3] and $[\alpha = 1, 2, 3, 4, 5, 6]$ . These notations are used in Table 1.5.7.1. Notice that $[\Lambda_{ij}=\Lambda_{ijj}]$ for [j=1,2,3] , $[\Lambda_{i4}=2\Lambda_{i23}]$ , $[\Lambda_{i5}=2\Lambda_{i31}]$ , and $[\Lambda_{i6}=2\Lambda_{i12}]$ .

The form of the matrix $[\Lambda_{i\alpha}]$ depends on the orientation of the axes of the Cartesian coordinate system (CCS) with respect to the symmetry axes of the point group of the crystal under consideration. These symmetry axes may be rotation axes, rotoinversion axes or mirror-plane normals, all possibly combined with time reversal. The usual orientations of the CCS with respect to the symmetry axes can be expressed by the order of the entries in the Hermann–Mauguin symbol. An entry consists (apart from possible primes and bars) of a number [N=1] , 2, 3, 4 or 6 or the letter m or [N/m] ( $[={\textstyle{N \over m}}]$ ). The conventional rules will be followed: in the monoclinic and orthorhombic crystal systems the x, y and z axes of the CCS are parallel to the symmetry axes given in the first, second and third entries, respectively. In the monoclinic system, there is only one symmetry axis, which is usually chosen parallel to the y axis, and a short Hermann–Mauguin symbol with only one entry is usually used, e.g. [2/m] instead of $[1\,2/m\,1]$ . In the trigonal and hexagonal systems, the z, x and y axes are parallel to the symmetry axes given in the first, second and third entries, respectively. In the tetragonal system, the z axis is parallel to the symmetry axis given in the first entry, and the x and y axes are parallel to the symmetry axes given in the second entry, which appear in two mutually perpendicular directions. In the cubic system, the symmetry axes given in the first entry appear in three mutually perpendicular directions; the x, y and z axes of the CCS are chosen parallel to these directions. Alternative orientations of the same point group that give rise to the same form of $[\Lambda_{i\alpha}]$ have been added between square brackets [] in Table 1.5.7.1. Notice that the Schoenflies notation does not allow us to distinguish different orientations of the CCS with respect to the symmetry axes.

The forms of $[\Lambda_{i\alpha}]$ for frequently encountered orientations of the CCS other than those given in Table 1.5.7.1 are

(1) $[112, 11m, 11\,2/m]$ (unique axis z): $[\left[\matrix{0 & 0 & 0 &\Lambda_{14} &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{24} &\Lambda_{25} & 0 \cr \Lambda_{31} &\Lambda_{32} &\Lambda_{33} & 0 & 0 &\Lambda_{36}}\right]\semi]$
(2) $[112', 11m', 11\,2'/m']$ (unique axis z): $[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} & 0 & 0 &\Lambda_{16} \cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} & 0 & 0 &\Lambda_{26} \cr 0 & 0 & 0 &\Lambda_{34} &\Lambda_{35} & 0}\right]\semi]$
(3) $[22'2', 2m'm', mm'2'\,[m2'm'], mm'm']$ : $[\left[\matrix{\Lambda_{11} &\Lambda_{12} &\Lambda_{13} &0 &0 & 0 \cr 0 & 0 & 0 & 0 & 0 &\Lambda_{26} \cr 0 & 0 & 0 & 0 &\Lambda_{35} & 0 }\right]\semi]$
(4) $[2'22', m'2m', 2'mm'\,[m'm2'], m'mm']$ : $[\left[\matrix{0 & 0 & 0 & 0 & 0 &\Lambda_{16} \cr \Lambda_{21} &\Lambda_{22} &\Lambda_{23} & 0 & 0 & 0 \cr 0 & 0 & 0 &\Lambda_{34} & 0 & 0 }\right]\semi]$
(5) $[4'2'2, 4'm'm, \overline{4}\,'m'2, \overline{4}\,'2'm, 4'/mm'm]$ : $[\left[\matrix{0 & 0 & 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &-\Lambda_{15}& 0 & 0 \cr \Lambda_{31} &-\Lambda_{31}& 0 & 0 & 0 & 0}\right]\semi]$
(6) $[6'2'2, 6'm'm, \overline{6}\,'m'2, \overline{6}\,'2'm, 6'/m'm'm]$ : $[\left[\matrix{0 & 0 & 0 & 0 & 0 &-2\Lambda_{22}\cr -\Lambda_{22} &\Lambda_{22} & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 }\right]\semi]$
(7) $[312, 31m, \overline{3}1m]$ : $[\left[\matrix{0 & 0 & 0 &\Lambda_{14} & 0 &-2\Lambda_{22}\cr -\Lambda_{22} &\Lambda_{22} & 0 & 0 &-\Lambda_{14}& 0 \cr 0 & 0 & 0 & 0 & 0 & 0 }\right]\semi]$
(8) $[312', 31m', \overline{3}1m']$ : $[\left[\matrix{\Lambda_{11} &-\Lambda_{11}& 0 & 0 &\Lambda_{15} & 0 \cr 0 & 0 & 0 &\Lambda_{15} & 0 &-2\Lambda_{11}\cr \Lambda_{31} &\Lambda_{31} &\Lambda_{33} & 0 & 0 & 0}\right].]$

Many connections between the different forms of $[\Lambda_{i\alpha}]$ given above and in Table 1.5.7.1 have been derived by Kopský (1979a,b) and Grimmer (1991). These connections between the forms that the matrix can assume for the various magnetic or crystallographic point groups hold for all matrices and tensors that describe properties of materials, not just for the special case of piezomagnetism.

Dzyaloshinskii (1957b) pointed out a number of antiferromagnets that may display the piezomagnetic effect. These include the fluorides of the transition metals, in which the piezomagnetic effect was first observed experimentally (see Fig. 1.5.7.1) (Borovik-Romanov, 1959b). Below we shall discuss the origin of the piezomagnetic effect in fluorides in more detail.

Figure 1.5.7.1 | top | pdf |

The dependence of the magnetic moment of CoF₂ on the magnetic field. (1) Without stress; (2) under the stress T_xz = 33.3 MPa (Borovik-Romanov, 1960).

The fluorides of transition metals MnF₂, CoF₂ and FeF₂ are tetragonal easy-axis antiferromagnets (see Fig. 1.5.5.3). It is easy to check that the expansion of the thermodynamic potential $[\tilde{\Phi}]$ up to terms that are linear in stress $[T_{ij}]$ and invariant relative to the transformations of the crystallographic space group $[{\bi D}^{14}_{4h} =]$ $[ P4_{2}/mnm]$ is represented by $[\eqalignno{\tilde{\Phi} &= \tilde{\Phi}_{0} + (A/2){\bf L}^{2} + (a/2)(L^{2}_{x} + L^{2}_{y}) &\cr&\quad+ (B/2){\bf M}^{2} + (b/2)(M^{2}_{x} + M^{2}_{y}) &\cr&\quad + d(L_{x}M_{y} + L_{y}M_{x}) &\cr&\quad + 2\lambda_{1}(M_{x}T_{yz} + M_{y}T_{xz})L_{z}&\cr&\quad + 2\eta _{1}(L_{y}T_{yz} + L_{x}T_{xz})L_{z} &\cr&\quad + 2\lambda_{2}M_{z}L_{z}T_{xy} + 2\eta_{2}L_{x}L_{y} T_{xy} - {\bf MH}. &(1.5.7.5)}]$ In this expression, the sums ( $[T_{ij} + T_{ji}]$ ) that appear in the magnetoelastic terms have been replaced by $[2T_{ij}]$ , as $[T_{ij} \equiv T_{ji}]$ .

The analysis of expression (1.5.7.5) in the absence of stresses proves that fluorides may possess weak ferromagnetism provided that $[a \,\lt\, 0]$ ( $[L_{z} = 0]$ ) (see Section 1.5.5.1). Here we shall discuss the easy-axis structure of the fluorides MnF₂, CoF₂, FeF₂ (see Fig. 1.5.5.3b). In the absence of magnetic fields and stresses only $[L_{z} \neq 0]$ for this structure. All other components of the vector $[\bf L]$ and the magnetization vector $[\bf M]$ are equal to zero. The magnetic point group is $[{\bi D}_{4h}({\bi D}_{2h})= 4^{\prime}/mmm^{\prime}]$ .

To transform the potential $[\tilde{\Phi}(L_{i}, M_{j}, T_{k\ell})]$ [(1.5.7.5)] into the form $[\Phi(T, {\bf T}, {\bf H})]$ [(1.5.7.1)], one has to insert into the magnetoelastic terms the dependence of the components of $[\bf L]$ and $[\bf M]$ on the magnetic field. The corresponding relations, obtained by minimization of (1.5.7.5) without the magnetoelastic terms, are $[\matrix{M_{x} = \displaystyle{{a}\over{a(B + b) - d^{2}}} H_{x}\semi \hfill& L_{x} = -\displaystyle{{d}\over{a(B + b) - d^{2}}} H_{y}\semi\hfill \cr M_{y} = \displaystyle{{a}\over{a(B + b) - d^{2}}} H_{y}\semi \hfill& L_{y} = -\displaystyle{{d}\over{a(B + b) - d^{2}}} H_{x}\semi\hfill\cr M_{z} = \displaystyle{{1}\over{B}} H_{z}\semi\hfill&L_{z} \simeq {\rm constant}.\hfill} \eqno(1.5.7.6)]$ To a first approximation, the component $[L_{z}]$ does not depend on the magnetic field.

Inserting the relations (1.5.7.6) for $[M_{i}]$ and $[L_{i}]$ into the magnetoelastic terms of (1.5.7.5), one gets the following expression for the corresponding terms in $[\Phi(T, H_{i}, T_{jk})]$ : $[\eqalignno{\Phi(T, H_{i}, T_{jk}) &= \Phi_{0}(T, H_i) + 2L_{z}{{a\lambda_{1} - d\eta_{1}}\over{a(B + b) - d^{2}}}T_{yz}H_{x} &\cr&\quad +2L_{z}{{a\lambda_{1} - d\eta_{1}}\over{a(B + b) - d^{2}}}T_{xz}H_{y} + 2L_{z}{{\lambda_{2}}\over{B}}T_{xy}H_{z}. &\cr&&(1.5.7.7)}]$ In this case, the expression for the magnetoelastic energy contains only three components of the stress tensor: $[T_{yz}]$ , $[T_{xz}]$ and $[T_{xy}]$ . Using (1.5.7.2), we get formulas for the three main components of the piezomagnetic effect: $[\eqalignno{M_{x} &= 2L_{z} {{d\eta_{1} - a\lambda_{1}}\over{a(B + b) - d^{2}}}T_{yz} = 2\Lambda_{xyz}T_{yz} = \Lambda_{14}T_{4}, &(1.5.7.8)\cr M_{y} &= 2L_{z} {{d\eta_{1} - a\lambda_{1}}\over{a(B + b) - d^{2}}}T_{xz} = 2\Lambda_{yxz}T_{xz} = \Lambda_{25}T_{5}, &(1.5.7.9)\cr M_{z} &= -2L_{z} {{\lambda_{2}}\over{B}}T_{xy} = 2\Lambda_{zxy}T_{xy} = \Lambda_{36}T_{6}. &(1.5.7.10)\cr}%fd1.5.7.10]$ In all three cases, the piezomagnetic moment is produced in the direction perpendicular to the shear plane. Comparing (1.5.7.8) and (1.5.7.9), we see that $[\Lambda_{25} = \Lambda_{14}]$ . This is in agreement with the equivalence of the axes x and y in the tetragonal crystals. If the stress is applied in the plane xz (or yz), the vector $[\bf L]$ turns in the shear plane and a component $[L_{x}]$ (or $[L_{y}]$ ) is produced: $[L_{x} = 2L_{z}{{\eta_{1}(B+b) - d\lambda_{1}}\over{a(B+b) - d^{2}}}T_{xz}. \eqno(1.5.7.11)]$ For $[T _{xy}]$ stress, no rotation of the vector $[\bf L]$ occurs.

Formulas (1.5.7.8)–(1.5.7.10) show that in accordance with Table 1.5.7.1 the form of the matrix $[\Lambda _{i\alpha}]$ for the magnetic point group $[{\bi D}_{4h}({\bi D}_{2h})=4^{\prime}/mmm^{\prime}]$ is $[\boldLambda_{i\alpha} = \left[\matrix{0 & 0 & 0 & \Lambda_{14} & 0 & 0 \cr 0 & 0 & 0 & 0 & \Lambda_{14} & 0 \cr 0 & 0 & 0 & 0 & 0 & \Lambda_{36}} \right]. \eqno(1.5.7.12)]$

The relations (1.5.7.8)–(1.5.7.10) show that the components of the piezomagnetic tensor $[\Lambda_{ijk}]$ are proportional to the components of the antiferromagnetic vector $[\bf L]$ . Thus the sign of the piezomagnetic moment depends on the sign of the vector $[\bf L]$ and the value of the piezomagnetic effect depends on the domain structure of the sample (we are referring to S-domains ). The piezomagnetic moment may become equal to zero in a polydomain sample. On the other hand, piezomagnetism may be used to obtain single-domain antiferromagnetic samples by cooling them from the paramagnetic state in a magnetic field under suitably oriented external pressure.

There are relatively few publications devoted to experimental investigations of the piezomagnetic effect. As mentioned above, the first measurements of the values of the components of the tensor $[\Lambda_{ijk}]$ were performed on crystals of MnF₂ and CoF₂ (Borovik-Romanov, 1960). In agreement with theoretical prediction, three components were observed: $[\Lambda_{xyz} = \Lambda_{yxz}]$ and $[\Lambda_{zxy}]$ . The largest value obtained for these components was Λ₁₄ = 21 × 10⁻¹⁰ Oe⁻¹. The piezomagnetic effect was also observed for two modifications of α-Fe₂O₃ (Andratskii & Borovik-Romanov, 1966). The magnetic point group of the low-temperature modification of this compound is $[{\bi D}_{3d} = \bar{3}m]$ . In accordance with form (7) given above, the following nonzero components $[\Lambda_{ijk}]$ were found for the low-temperature state: $[\eqalignno{\Lambda_{xyz} &= - \Lambda_{yxz}, &(1.5.7.13)\cr \Lambda_{yyy} &= - \Lambda_{yxx} = -\Lambda_{xxy}. &(1.5.7.14)}%fd1.5.7.14]$ The values of these components are one order of magnitude smaller than for CoF₂.

The temperature dependence of the components is similar for the piezomagnetic tensor and the sublattice magnetization. This means that the magnetoelastic constants $[\lambda_{1}]$ and $[\lambda_{2}]$ (as well as the constants B and d) in the relations (1.5.7.7) and (1.5.7.8) depend only slightly on temperature.

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International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 132-136