International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.5, pp. 105-149
https://doi.org/10.1107/97809553602060000632 Chapter 1.5. Magnetic properties1
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 This chapter gives a short review of the structure and some properties of magnetic substances that depend mainly on the symmetry of these substances. Aspects related to the magnetic symmetry receive the most emphasis. The magnetic symmetry takes into account the fact that it is necessary to consider time inversion in addition to the usual spatial transformations in order to describe the invariance of the thermodynamic equilibrium states of a body. The first part of the chapter is devoted to a brief classification of disordered and ordered magnetics. The classification of ferromagnets according to the type of the magnetic structure is given in Section 1.5.1.2.1 Keywords: Bravais lattices; Gaussian system of units; Landau theory; S-domains; SI units; angular phase; anisotropy energy; antiferromagnetic ferroelectrics; antiferromagnetic helical structures; antiferromagnetic phases; antiferromagnetic structures; antiferromagnetic vectors; antiferromagnets; diamagnets; domains; easy-axis magnetics; easy-plane magnetics; exchange energy; exchange symmetry; ferrimagnets; ferroelectric antiferromagnets; ferroelectric materials; ferroic domains; ferromagnetic ferroelectrics; ferromagnetic materials; ferromagnetic vectors; ferromagnetism; ferromagnets; helical structures; incommensurate structures; magnetic Bravais lattices; magnetic anisotropy energy; magnetic birefringence; magnetic fields; magnetic induction; magnetic lattices; magnetic point groups; magnetic space groups; magnetic susceptibility; magnetic symmetry; magnetoelastic energy; magnetoelectric effect; magnetostriction; paramagnets; phase transitions; piezomagnetic effect; relativistic interactions; reorientation transitions; spin flip; spin flop; spontaneous magnetization; spontaneous magnetostriction; time inversion; twin domains; uniaxial antiferromagnets; uniaxial ferromagnets. |
In the present chapter, we shall give a short review of the structure and some properties of magnetic substances that depend mainly on the symmetry of these substances. Aspects related to the magnetic symmetry receive the most emphasis. The magnetic symmetry takes into account the fact that it is necessary to consider time inversion in addition to the usual spatial transformations in order to describe the invariance of the thermodynamic equilibrium states of a body.
The symmetry of magnetic materials depends not only on the mean charge density function , but also on the mean current density
and the mean spin density
. The symmetry of the function
is called the crystallographic or crystallochemical symmetry of a body. If the current density
in the crystal is not zero, an orbital magnetic moment is produced. It is obvious that there can be no macroscopic current in a substance which is in thermodynamic equilibrium and the integral
over the magnetic elementary cell is always equal to zero. The current
, however, may produce a macroscopic nonzero magnetic moment
. We shall consider below the function
, which determines the space distribution of the total (spin and orbital) magnetic moment density. The symmetry of the distribution of the magnetic moment density
may be considered as the symmetry of the arrangement and orientation of the mean atomic (or ionic) magnetic moments in the crystal (we shall not consider the magnetism of the conduction electrons in this chapter).
The first part of the chapter is devoted to a brief classification of magnetics. If at every point, the substance is a disordered magnetic. There are two types of such magnetics: diamagnets and paramagnets. The most important features of these magnetics are briefly outlined in Section 1.5.1.1
.
If , the substance possesses a magnetic structure. There are two cases to be considered: (1) The integral of
over the primitive cell is not zero (
); such a substance is called ferromagnetic. (2)
; such a substance is called antiferromagnetic. The integration is performed over the magnetic elementary cell, which may differ from the crystallographic one. This crude classification is extended in Section 1.5.1.2
.
The classification of ferromagnets according to the type of the magnetic structure is given in Section 1.5.1.2.1. The concept of the magnetic sublattice is introduced and the ferromagnets are divided into two groups: one-sublattice ferromagnets and multi-sublattice ferro- and ferrimagnets. Collinear and non-collinear ferromagnets are described.
In Section 1.5.1.2.2, the antiferromagnets are classified by the types of their magnetic structures: collinear, weakly non-collinear and strongly non-collinear antiferromagnets.
Incommensurate structures are briefly mentioned in Section 1.5.1.2.3.
The study of ordered magnetics has led to an extension of the theory of crystallographic symmetry. This extension is based on the fact that changes sign under a specific transformation R, which is equivalent to time inversion. The invariance of the equation of motion is preserved under R. The symmetry that admits the operation R along with ordinary crystallographic transformations (translations, rotations and reflections) is called the magnetic symmetry. Section 1.5.2
is devoted to magnetic symmetry. Different types of magnetic point (Section 1.5.2.1
) and magnetic space (Section 1.5.2.3
) groups are defined. The 22 magnetic Bravais lattices are displayed in Section 1.5.2.2
. All magnetic groups (both point and space) are categorized into three types: (1) The groups that possess R as an additional element. The crystals which belong to such space groups satisfy
at every point, hence
Such crystals are found to be paramagnetic or diamagnetic. Crystals with a point group that possesses R as an additional element may also be antiferromagnetic. This is the case if R appears in the space group multiplied by some translations but not as a separate element. (2) The groups that do not possess R at all. (3) The groups that contain the element R only in combination with some other elements (translations, rotations, reflections). The latter two types of space groups describe ordered magnetics.
The transition from the paramagnetic state into the magnetically ordered state entails a transition from one magnetic group into another. These transitions are considered in Section 1.5.3. Section 1.5.3.1
gives an example of the analysis of such transitions in terms of magnetic symmetry and introduces the concept of ferromagnetic and antiferromagnetic vectors, which characterize the magnetic structures. The phenomenological theory of magnetic transitions is based on the Landau theory of second-order transitions. Section 1.5.3.3
is dedicated to this theory (see also Section 3.1.2
). The Landau theory is based on the expansion of the thermodynamic potential into a series of the basic functions of irreducible representations of the space group of the crystal under consideration. It is essential to distinguish the exchange and relativistic terms in the expansion of the thermodynamic potential (see Section 1.5.3.2
).
The domain structure of ferromagnets and antiferromagnets is considered in Section 1.5.4, where 180° and T-domains are described. The change from a multidomain structure to a single-domain structure under the action of an applied magnetic field explains the magnetization process in ferro- and ferrimagnets. The existence of 180° domains in antiferromagnets was shown in experiments on piezomagnetism and the linear magnetoelectric effect.
Non-collinear antiferromagnetic structures (weakly ferromagnetic, non-collinear and non-coplanar antiferromagnetic structures) are described in Section 1.5.5. The existence of these structures is directly connected with the magnetic symmetry. Such a structure arises if the irreducible representation responsible for the phase transition into the ordered state is two- or three-dimensional. Correspondingly, the magnetic group allows the coexistence of two or three different ferro- or antiferromagnetic vectors.
Besides the magnetic phase transition from the disordered into the ordered state, there exist transitions from one magnetic structure into another. Those of these that are obtained by a rotation of the ferromagnetic or antiferromagnetic vector relative to the crystallographic axis are called reorientation transitions and are analysed in Section 1.5.6.
Sections 1.5.7 and 1.5.8
are devoted to phenomena that can be (and were) predicted only on the basis of magnetic symmetry. These are piezomagnetism (Section 1.5.7
) and the magnetoelectric effect (Section 1.5.8
). The reciprocal of the piezomagnetic effect (Section 1.5.7.1
) is linear magnetostriction (Section 1.5.7.2
). The magnetoelectric effect has been investigated far more than piezomagnetism. In addition to the linear magnetoelectric effect (Section 1.5.8.1
), effects of higher order (Section 1.5.8.2
) have also been observed. In connection with the magnetoelectric effect, ferromagnetic and antiferromagnetic ferroelectrics are also considered (Section 1.5.8.3
).
In Section 1.5.9, the magnetostriction in ferromagnets is discussed. Only fundamental points of this problem are considered.
As noted above, only those problems of magnetism that are closely connected with magnetic symmetry are considered in this chapter. However, these problems are only outlined briefly here because of the restrictions on the extent of this volume. For the same reason, it is impossible to give an exhaustive list of references. The references given here include selected publications on magnetic symmetry and those describing the first experimental work devoted to the properties connected with magnetic symmetry.
A crystal placed in a magnetic field is magnetized. The magnetized state is characterized by two vectors, the magnetization
(the magnetic moment per unit volume) and the magnetic induction
. The Gaussian system of units is used in this chapter (see Table 1.5.10.1
at the end of the chapter for a list of conversions from Gaussian to SI units). The magnetic induction is given by
This equation shows that the dimensions of B, H and M are the same in the Gaussian system. The unit for B, the gauss (G), and for H, the oersted (Oe), also coincide in magnitude, whereas the unit for M, usually called emu cm−3, is 4π times larger. These units are related to the corresponding SI units as follows: 1 G = 10−4 tesla (T), 1 Oe = 103/(4π) A m−1, 1 emu cm−3 = 103 A m−1.
In disordered magnetics, the vectors B and M are linear in the magnetic field. Using a Cartesian coordinate system, this can be expressed as where
is the dimensionless magnetic susceptibility per unit volume and
is the magnetic permeability. The susceptibility is frequently referred to 1 g or to one mole of substance. The mass susceptibility is written as
, the molar susceptibility as
.
All three vectors H, M and B are axial vectors (see Section 1.1.4.5.3
), the symmetry of which is
. Accordingly, the components of these vectors perpendicular to a mirror plane do not change sign on being reflected by this plane, whereas the components parallel to the plane do change sign. Consequently, these three vectors are invariant with respect to inversion. The quantities
and
are components of second-rank polar tensors. In principal axes, the tensors become diagonal and both the magnetic susceptibility and permeability of a crystal are characterized by the three values of the principal susceptibilities and principal permeabilities, respectively.
All disordered magnetics are divided into two types: diamagnets (χ < 0) and paramagnets (χ > 0).
Diamagnetism is a universal property of all materials. It is associated with the tendency of all the electrons to screen the applied external field according to the Lenz law. For materials in which the electron orbits are spherically symmetric, the relation for the diamagnetic susceptibility was calculated by Langevin. For monoatomic substances he obtainedwhere N is the number of atoms per unit volume, Z is the number of electrons per atom, e and m are the charge and the mass of the electron, respectively, and
are the mean squares of the radii of the electron orbits. In polyatomic substances, the summation must be done over all types of atoms. In most chemical compounds, the orbits are not spherical and the calculation of the diamagnetic susceptibility becomes more complicated. In metals, the conduction electrons contribute significantly to the diamagnetic susceptibility. The diamagnetic susceptibility of most substances is very small (
) and isotropic. Rare exceptions are bismuth and some organic compounds, in which the diamagnetism is strongly anisotropic.
Most paramagnetic materials contain ions (or free atoms) with a partly filled inner electronic shell. Examples are the transition metals and the rare-earth and actinide elements. Atoms, molecules and point defects possessing an odd number of electrons are also paramagnetic. Ions with a partly filled inner electronic shell possess orbital L and spin S angular momenta, which determine the total angular momentum J if the spin–orbit interaction is strong compared with the crystal field.
The magnetic susceptibility of paramagnets follows the Curie–Weiss law in low magnetic fields ():
where N is the number of magnetic ions (or atoms) per cm3,
is the Bohr magneton, p is the effective number of Bohr magnetons,
is the Boltzmann factor and
is the Weiss constant. The Weiss constant is related to the interaction between the magnetic moments (mostly exchange interaction) and to the effect of the splitting of electron levels of the paramagnetic ion in the crystalline electric field. Many paramagnets that obey the Curie–Weiss law transform into ordered magnetics at a temperature
, which is of the order of
. The sign of
depends on the sign of the exchange constant J [see relation (1.5.1.7)
]. For the substances that at low temperatures become ferromagnets, we have
, for antiferromagnets
, and for ferrimagnets the temperature dependence of
is more complicated (see Fig. 1.5.1.1
). For those paramagnets that do not go over into an ordered state,
is close to zero and equation (1.5.1.4)
changes to the Curie law.
![]() |
Temperature dependence of |
The value of the effective number of Bohr magnetons p depends strongly on the type of the magnetic ions and their environment. For most rare-earth compounds at room temperature, the number p has the same value as for free ions: where g is the Landé g-factor or the spectroscopic splitting factor (
). In this case, the paramagnetic susceptibility is practically isotropic. Some anisotropy can arise from the anisotropy of the Weiss constant
.
The behaviour of the transition-metal ions is very different. In contrast to the rare-earth ions, the electrons of the partly filled shell in transition metals interact strongly with the electric field of the crystal. As a result, their energy levels are split and the orbital moments can be `quenched'. This means that relation (1.5.1.5) transforms to
Here the value of the effective spin S represents the degeneration of the lowest electronic energy level produced by the splitting in the crystalline field;
differs from the usual Landé g-factor. The values of its components lie between 0 and 10–20. The tensor
becomes diagonal in the principal axes. According to relation (1.5.1.6)
, the magnetic susceptibility also becomes a tensor. The anisotropy of
can be studied using electron paramagnetic resonance (EPR) techniques.
The Curie–Weiss law describes the behaviour of those paramagnets in which the magnetization results from the competition of two forces. One is connected with the reduction of the magnetic energy by orientation of the magnetic moments of ions in the applied magnetic field; the other arises from thermal fluctuations, which resist the tendency of the field to orient these moments. At low temperatures and in strong magnetic fields, the linear dependence of the magnetization versus magnetic field breaks down and the magnetization can be saturated in a sufficiently strong magnetic field. Most of the paramagnetic substances that obey the Curie–Weiss law ultimately transform to an ordered magnetic as the temperature is decreased.
The conduction electrons in metals possess paramagnetism in addition to diamagnetism. The paramagnetic susceptibility of the conduction electrons is small (of the same order of magnitude as the diamagnetic susceptibility) and does not depend on temperature. This is due to the fact that the conduction electrons are governed by the laws of Fermi–Dirac statistics.
As stated above, all ordered magnetics that possess a spontaneous magnetization Ms different from zero (a magnetization even in zero magnetic field) are called ferromagnets. The simplest type of ferromagnet is shown in Fig. 1.5.1.2(a). This type possesses only one kind of magnetic ion or atom. All their magnetic moments are aligned parallel to each other in the same direction. This magnetic structure is characterized by one vector M. It turns out that there are very few ferromagnets of this type in which only atoms or ions are responsible for the ferromagnetic magnetization (CrBr3, EuO etc.). The overwhelming majority of ferromagnets of this simplest type are metals, in which the magnetization is the sum of the magnetic moments of the localized ions and of the conduction electrons, which are partly polarized.
![]() |
Ordered arrangements of magnetic moments |
More complicated is the type of ferromagnet which is called a ferrimagnet. This name is derived from the name of the oxides of the elements of the iron group. As an example, Fig. 1.5.1.2(b) schematically represents the magnetic structure of magnetite (Fe3O4). It contains two types of magnetic ions and the number of Fe3+ ions (
and
) is twice the number of Fe2+ ions (
). The values of the magnetic moments of these two types of ions differ. The magnetic moments of all Fe2+ ions are aligned in one direction. The Fe3+ ions are divided into two parts: the magnetic moments of one half of these ions are aligned parallel to the magnetic moments of Fe2+ and the magnetic moments of the other half are aligned antiparallel. The array of all magnetic moments of identical ions oriented in one direction is called a magnetic sublattice. The magnetization vector of a given sublattice will be denoted by
. Hence the magnetic structure of Fe3O4 consists of three magnetic sublattices. The magnetizations of two of them are aligned in one direction, the magnetization of the third one is oriented in the opposite direction. The net ferromagnetic magnetization is
.
The special feature of ferrimagnets, as well as of many antiferromagnets, is that they consist of sublattices aligned antiparallel to each other. Such a structure is governed by the nature of the main interaction responsible for the formation of the ordered magnetic structures, the exchange interaction. The energy of the exchange interaction does not depend on the direction of the interacting magnetic moments (or spins S) relative to the crystallographic axes and is represented by the following relation: Here
,
are the spins of magnetic atoms (ions) and
is the exchange constant, which usually decreases fast when the distance between the atoms rises. Therefore, usually only the nearest neighbour interaction needs to be taken into account. Hence, according to (1.5.1.7)
, the exchange energy is a minimum for the state in which neighbouring spins are parallel (if
) or antiparallel (if
). If the nearest neighbour exchange interaction were the only interaction responsible for the magnetic ordering, only collinear magnetic structures would exist (except in triangle lattices). Together with the exchange interaction, there is also a magnetic dipole interaction between the magnetic moments of the atoms as well as an interaction of the atomic magnetic electrons with the crystalline electric field. These interactions are much smaller than the exchange interaction. They are often called relativistic interactions. The relativistic interactions and the exchange interaction between next-nearest atoms bring about the formation of non-collinear magnetic structures.
A simple non-collinear structure is the magnetic structure of a weak ferromagnet. It contains identical magnetic ions divided in equal amounts between an even number of sublattices. In the first approximation, the magnetizations of these sublattices are antiparallel, as in usual antiferromagnets. In fact, the magnetizations are not strictly antiparallel but are slightly canted, i.e. non-collinear, as shown in Fig. 1.5.1.2(c). There results a ferromagnetic moment
, which is small compared with the sublattice magnetization
. The magnetic properties of weak ferromagnets combine the properties of both ferromagnets and antiferromagnets. They will be discussed in detail in Section 1.5.5.1
.
As discussed above, the exchange interaction, which is of prime importance in the formation of magnetic order, can lead to a parallel alignment of the neighbouring magnetic moments as well as to an antiparallel one. In the latter case, the simplest magnetic structure is the collinear antiferromagnet, schematically shown in Fig. 1.5.1.3(a). Such an antiferromagnet consists of one or several pairs of magnetic sublattices of identical magnetic ions located in equivalent crystallographic positions. The magnetizations of the sublattices are oriented opposite to each other.
Fig. 1.5.1.3(b) shows a weakly non-collinear antiferromagnet, in which the vectors of magnetization of four equivalent sublattices form a cross with a small tilting angle
. Such a structure can be considered as an admixture of `weak antiferromagnetism'
with easy axis Ox to an ordinary antiferromagnet
with easy axis Oy. This weak antiferromagnetism is of the same origin as weak ferromagnetism. Its nature will be discussed in detail in Section 1.5.5.2
.
The minimum of the exchange interaction energy of three spins located at the corners of a triangle corresponds to a structure in which the angles between two adjacent spins are 120°. Correspondingly, many hexagonal crystals possess a triangular antiferromagnetic structure like the one shown in Fig. 1.5.1.3(c). The sum of the magnetizations of the three sublattices in this structure equals zero. In tetragonal crystals, there is a possibility of the existence of a 90° antiferromagnetic structure, which consists of four equivalent sublattices with magnetizations oriented along the positive and negative directions of the x and y axes.
Finally, it is worth noting that in addition to the electronic magnetically ordered substances, there exist nuclear ferro- and antiferromagnets (below 1 mK for some insulators and below 1 µK for metals).
There are many more complicated non-collinear magnetic structures. Fig. 1.5.1.4(a) shows an antiferromagnetic helical structure. It consists of planes perpendicular to the z axis in which all the magnetic moments are parallel to each other and are perpendicular to z. The polar angle of the direction of the moments changes from plane to plane by some constant
. Thus the magnetization vectors describe a spiral along the axis of the crystal. Such structures were observed in hexagonal rare-earth metals. A specific feature is that they often are incommensurate structures. This means that
is not a rational number and that the period of the magnetic spiral is not a multiple of the period of the lattice.
![]() |
Helical and sinusoidal magnetic structures. (a) An antiferromagnetic helix; (b) a cone spiral; (c) a cycloidal spiral; (d) a longitudinal spin-density wave; (e) a transverse spin-density wave. |
Similar to the antiferromagnetic helix, ferromagnetic helical or spiral structures exist [see Fig. 1.5.1.4(b)] in which the magnetizations of the layers are tilted to the axis at an angle
. As a result, the vectors of the magnetization of the layers are arranged on the surface of a cone. The ferromagnetic magnetization is aligned along the z axis. This structure is called a ferromagnetic helix. It usually belongs to the incommensurate magnetic structures.
More complicated antiferromagnetic structures also exist: sinusoidal structures, which also consist of layers in which all the magnetic moments are parallel to each other. Fig. 1.5.1.4(c) displays the cycloidal spiral and Figs. 1.5.1.4
(d) and (e) display longitudinal and transverse spin density waves, respectively.
As discussed in Section 1.5.1, in studies of the symmetry of magnetics one should take into account not only the crystallographic elements of symmetry (rotations, reflections and translations) but also the time-inversion element, which causes the reversal of the magnetic moment density vector
. Following Landau & Lifshitz (1957
), we shall denote this element by R. If combined with any crystallographic symmetry element G we get a product
, which some authors call the space-time symmetry operator. We shall not use this terminology in the following.
To describe the symmetry properties of magnetics, one should use magnetic point and space groups instead of crystallographic ones. (See also Section 1.2.5
.)
By investigating the `four-dimensional groups of three-dimensional space', Heesch (1930) found not only the 122 groups that now are known as magnetic point groups but also the seven triclinic and 91 monoclinic magnetic space groups. He also recognized that these groups can be used to describe the symmetry of spin arrangements. The present interest in magnetic symmetry was much stimulated by Shubnikov (1951
), who considered the symmetry groups of figures with black and white faces, which he called antisymmetry groups. The change of colour of the faces in antisymmetry (black–white symmetry, see also Section 3.3.5
) corresponds to the element R. These antisymmetry classes were derived as magnetic symmetry point groups by Tavger & Zaitsev (1956
). Beside antisymmetry, the concept of colour (or generalized) symmetry also was developed, in which the number of colours is not 2 but 3, 4 or 6 (see Belov et al., 1964
; Koptsik & Kuzhukeev, 1972
). A different generalization to more than two colours was proposed by van der Waerden & Burckhardt (1961
). The various approaches have been compared by Schwarzenberger (1984
).
As the theories of antisymmetry and of magnetic symmetry evolved often independently, different authors denote the operation of time inversion (black–white exchange) by different symbols. Of the four frequently used symbols () we shall use in this article only two: R and
.
Magnetic point groups may contain rotations, reflections, the element R and their combinations. A set of such elements that satisfies the group properties is called a magnetic point group. It is obvious that there are 32 trivial magnetic point groups; these are the ordinary crystallographic point groups supplemented by the element R. Each of these point groups contains all the elements of the ordinary point group and also all the elements of this group
multiplied by R. This type of magnetic point group
can be represented by
These groups are sometimes called `grey' magnetic point groups. As pointed out above, all dia- and paramagnets belong to this type of point group. To this type belong also antiferromagnets with a magnetic space group that contains translations multiplied by R (space groups of type IIIb).
The second type of magnetic point group, which is also trivial in some sense, contains all the 32 crystallographic point groups without the element R in any form. For this type . Thirteen of these point groups allow ferromagnetic spontaneous magnetization (ferromagnetism, ferrimagnetism, weak ferromagnetism). They are listed in Table 1.5.2.4
. The remaining 19 point groups describe antiferromagnets. The groups
are often called `white' magnetic point groups.
The third type of magnetic point group , `black and white' groups (which are the only nontrivial ones), contains those point groups in which R enters only in combination with rotations or reflections. There are 58 point groups of this type. Eighteen of them describe different types of ferromagnetism (see Table 1.5.2.4
) and the others represent antiferromagnets.
Replacing R by the identity element E in the magnetic point groups of the third type does not change the number of elements in the point group. Thus each group of the third type is isomorphic to a group
of the second type.
The method of derivation of the nontrivial magnetic groups given below was proposed by Indenbom (1959). Let
denote the set of those elements of the group
which enter into the associated magnetic group
not multiplied by R. The set
contains the identity element E, for each element H also its inverse
, and for each pair
,
also its products
and
. Thus the set
forms a group. It is a subgroup of the crystallographic group
. Let
denote an element of
. All these elements enter
in the form of products
because
and
. Multiplying the elements of
by a fixed element
corresponds to a permutation of the elements of
. This permutation maps each element of the subgroup
on an element of
that does not belong to
and vice versa. It follows that one half of the elements of
are elements of
multiplied by R and the other half belong to
. The relation for the magnetic point groups of the third type may therefore be written as
is therefore a subgroup of index 2 of
. The subgroups of index 2 of
can easily be found using the tables of irreducible representations of the point groups. Every real non-unit one-dimensional representation of
contains equal numbers of characters
and
. In the corresponding magnetic point group
, the elements of
with character
are multiplied by R and those with character
remain unchanged. The latter form the subgroup
. This rule can be stated as a theorem: every real non-unit one-dimensional representation
of a point group of symmetry
produces an isomorphic mapping of this group upon a magnetic group
(Indenbom, 1959
). This concept will be developed in Section 1.5.3
.
Using the Schoenflies symbols and the method described above, the point groups of magnetic symmetry (magnetic point groups) can be denoted by , where
is the symbol of the original crystallographic point group and
is the symbol of that subgroup the elements of which are not multiplied by R. This notation is often used in the physics literature. In the crystallographic literature, the magnetic groups are defined by Hermann–Mauguin or Shubnikov symbols. In this type of designation, the symbols of elements multiplied by R are primed or underlined. The primed symbols are used in most of the recent publications. The Hermann–Mauguin and Shubnikov definitions differ slightly, as in the case of crystallographic groups. In Table 1.5.2.1
, different symbols of magnetic point groups (trivial and nontrivial ones) are compared. This is done for the family that belongs to the crystallographic point group
. The symbols of the symmetry elements of these four magnetic point groups are compared in Table 1.5.2.2
.
|
|
Table 1.5.2.3 gives a list of the 90 magnetic point groups belonging to types 2 and 3. The Schoenflies, Shubnikov and Hermann–Mauguin symbols of the point groups are given in the table. The entries in the Hermann–Mauguin symbol refer to symmetry directions, as explained in Section 2.2.4
of International Tables for Crystallography, Vol. A (2005
). The elements of symmetry of each point group are displayed using the Hermann–Mauguin symbols. The symbol
denotes N 180° rotations with axes perpendicular to the principal symmetry axis;
denotes N mirror planes with normals perpendicular to the principal symmetry axis. Similar definitions hold for the primed symbols
and
. The point groups are arranged in families. The part of the Schoenflies symbol before the bracket is the same for each member of a family. Each family begins with a trivial magnetic point group. It contains the same elements as the corresponding crystallographic point group; its Schoenflies symbol contains no brackets. For each nontrivial point group, the list of the elements of symmetry begins with the non-primed elements, which belong to a subgroup
of the head of the family
. The number of the primed elements is equal to the number of non-primed ones and the total number of the elements is the same for all point groups of one family.
|
The overall number of the magnetic point groups of all three types is 122. There are two general statements concerning the magnetic point groups. The element does not appear in any of the magnetic point groups of type 3. Only trivial magnetic point groups (of both first and second type) belong to the families containing the point groups
,
and
.
Only 31 magnetic point groups allow ferromagnetism. The different types of ferromagnetism (one-sublattice ferromagnet, ferrimagnet, weak ferromagnet, any magnetic order with nonzero magnetization) cannot be distinguished by their magnetic symmetry. Ferromagnetism is not admitted in any point group of type 1. For the magnetic point groups of the second type, ferromagnetism is not allowed if the point group contains more than one symmetry axis, more than one mirror plane or a mirror plane that is parallel to the axis. The same restrictions are valid for the point groups of type 3 (if the corresponding elements are not multiplied by R). If the point group contains , ferromagnetic order is also forbidden. There are the following rules for the orientation of the axial vector of ferromagnetic magnetization
:
. Table 1.5.2.4
lists those magnetic point groups that admit ferromagnetic order (Tavger, 1958
). The allowed direction of the magnetization vector is given for every point group. Ferromagnetic order is allowed in 13 point groups of the second type and 18 point groups of the third type.
|
All 31 point groups of magnetic symmetry allowing ferromagnetism are subgroups of the infinite noncrystallographic group The transition from a paramagnetic to a ferromagnetic state is always accompanied by a change of the magnetic symmetry.
If the point group of symmetry describes the macroscopic properties of a crystal, its microscopic structure is determined by the space group, which contains the group of translations as a subgroup. The elements
of
are defined by the following relation:
where
,
,
are basic primitive translation vectors and
,
,
are arbitrary integers. The set of points
obtained by applying all the translations of the group
to any point
defines a lattice. All sites of the crystallographic lattice are equivalent.
The structure of the ordered magnetics is described by the magnetic lattices and corresponding magnetic translation groups . In the magnetic translation groups
, some of the elements
may be multiplied by R (we shall call them primed translations). The magnetic lattices then have two types of sites, which are not equivalent. One set is obtained by non-primed translations and the other set by the primed ones. The magnetic translation group
is isometric to the crystallographic one
that is obtained by replacing R by E in
.
There are trivial magnetic translation groups, in which none of the translation elements is multiplied by R. The magnetic lattices of these groups coincide with crystallographic lattices.
Nontrivial magnetic translation groups can be constructed in analogy to relation (1.5.2.2). Zamorzaev (1957
) showed that every translation group
has seven subgroups of index 2. If the basic primitive translations of the group
are
,
,
, then the basic primitive translations of the seven subgroups
can be chosen as follows (see also Opechowski & Guccione, 1965
)
As an example, let us consider the case (1.5.2.5). In this case, the subgroup
consists of the following translations:
Therefore the elements
of
[which corresponds to
in relation (1.5.2.2)
] must have the following form:
The corresponding magnetic translation group consists of the elements (1.5.2.12)
multiplied by R and the elements (1.5.2.11)
.
The crystallographic lattices are classified into Bravais types or Bravais lattices. The magnetic lattices are classified into Bravais types of magnetic lattices. It turns out that there are 22 nontrivial magnetic Bravais types. Together with the trivial ones, there are 36 magnetic Bravais lattices.
Two types of smallest translation-invariant cells are in common use for the description of magnetically ordered structures: the crystallographic cell obtained if the magnetic order is neglected and the magnetic cell, which takes the magnetic order into account. The list of the basic translations of all the magnetic Bravais lattices was given by Zamorzaev (1957). The diagrams of the magnetic unit cells were obtained by Belov et al. (1957
).
In Figs. 1.5.2.1–1.5.2.7
, the diagrams of the magnetic unit cells of all 36 Bravais types are sketched in such a way that it is clear to which family the given cell belongs. All the cells of one family are displayed in one row. Such a row begins with the cell of the trivial magnetic lattice. All nontrivial cells of a family change into the trivial one of this family if R is replaced by E (to draw these diagrams we used those published by Opechowski & Guccione, 1965
). Open and full circles are used to show the primed and unprimed translations. A line connecting two circles of the same type is an unprimed translation; a line connecting two circles of different types is a primed translation. The arrows in the trivial magnetic cell represent the primitive (primed or unprimed) translations for all the magnetic lattices of the family. The arrows in the nontrivial cells are primitive translations of the magnetic unit cell. The magnetic unit cell of a nontrivial magnetic lattice is generated by unprimed translations only. Its volume is twice the volume of the smallest cell generated by all (primed and unprimed) translations. The reason for this is that one of the primitive translations of the magnetic cell is twice a primitive primed translation. The crystallographic cell of many simple collinear or weakly non-collinear structures coincides with the smallest cell generated by the primed and unprimed translations. However, there are also magnetic structures with more complicated transformations from the crystallographic to the magnetic unit cell. The second line after each part of Figs. 1.5.2.1
–1.5.2.7
gives, between braces, an extended vector basis of the magnetic translation group (Shubnikov & Koptsik, 1972
). The first line gives two symbols for each Bravais type: the symbol to the right was introduced by Opechowski & Guccione (1965
). The symbol to the left starts with a lower-case letter giving the crystal system followed by a capital letter giving the centring type of the cell defined by the unprimed translations (P: primitive; C, A, B: C-, A-, B-centred; I: body-centred; F: all-face-centred). The subscript, which appears for the nontrivial Bravais types, indicates the translations that are multiplied by time inversion R.
Ferromagnetism is allowed only in trivial magnetic Bravais lattices. All nontrivial magnetic lattices represent antiferromagnetic order. There are only two magnetic sublattices in the simplest antiferromagnetic structures; one sublattice consists of the magnetic ions located in the black sites and the other of the ions located in the white sites. All the magnetic moments of one sublattice are oriented in one direction and those of the other sublattice in the opposite direction. However, antiferromagnetism is allowed also in trivial lattices if the (trivial) magnetic cell contains more than one magnetic ion. The magnetic point group must be nontrivial in this case. The situation is more complicated in case of strongly non-collinear structures. In such structures (triangle, 90° etc.), the magnetic lattice can differ from the crystallographic one despite the fact that none of the translations is multiplied by R. The magnetic elementary cell will possess three or four magnetic ions although the crystallographic cell possesses only one. An example of such a situation is shown in Fig. 1.5.1.3(c). More complicated structures in which the magnetic lattice is incommensurate with the crystallographic one also exist. We shall not discuss the problems of such systems in this chapter.
There are 1651 magnetic space groups , which can be divided into three types. Type I,
, consists of the 230 crystallographic space groups to which R is added. Crystals belonging to these trivial magnetic space groups show no magnetic order; they are para- or diamagnetic.
Type II, , consists of the same 230 crystallographic groups which do not include R in any form. In the ordered magnetics, which belong to the magnetic space groups of this type, the magnetic unit cell coincides with the classical one. Forty-four groups of type II describe different ferromagnetic crystals; the remaining antiferromagnets.
The nontrivial magnetic space groups belong to type III, . This consists of 1191 groups, in which R enters only in combination with rotations, reflections or translations. These groups have the structure described by relation (1.5.2.2)
. The magnetic space groups of this type are divided into two subtypes.
Subtype IIIa contains those magnetic space groups in which R is not combined with translations. In these groups, the magnetic translation group is trivial. To these space groups correspond magnetic point groups of type
. There are 674 magnetic space groups of subtype IIIa; 231 of them admit ferromagnetism, the remaining 443 describe antiferromagnets.
In the magnetic space groups of the subtype IIIb, R is combined with translations and the corresponding point groups are of type . They have a nontrivial magnetic Bravais lattice. There are 517 magnetic space groups of this subtype; they describe antiferromagnets.
In summary, the 230 magnetic space groups that describe dia- and paramagnets are of type I, the 275 that admit spontaneous magnetization are of types II and IIIa; the remaining 1146 magnetic space groups (types II, IIIa and IIIb) describe antiferromagnets.
The classification of magnetic structures on the basis of the magnetic (point and space) groups is an exact classification. However, it neglects the fundamental role of the exchange energy, which is responsible for the magnetic order (see Sections 1.5.1.2 and 1.5.3.2
). To describe the symmetry of the magnetically ordered crystals only by the magnetic space groups means the loss of significant information concerning those properties of these materials that are connected with the higher symmetry of the exchange forces. Andreev & Marchenko (1976
, 1980
) have introduced the concept of exchange symmetry.
The exchange forces do not depend on the directions of the spins (magnetic moments) of the ions relative to the crystallographic axes and planes. They depend only on the relative directions of the spins. Thus the exchange group contains an infinite number of rotations U of spin space, i.e. rotations of all the spins (magnetic moments) through the same angle about the same axis. The components of the magnetic moment density
transform like scalars under all rotations of spin space. The exchange symmetry group
contains those combinations of the space transformation elements, the rotations U of spin space and the element R with respect to which the values
are invariant. Setting all the elements U and R equal to the identity transformation, we obtain one of the ordinary crystallographic space groups
. This space group defines the symmetry of the charge density
and of all the magnetic scalars in the crystal. However, the vectors
may not be invariant with respect to
.
The concept of exchange symmetry makes it possible to classify all the magnetic structures (including the incommensurate ones) with the help of not more than three orthogonal magnetic vectors. We shall discuss this in more detail in Section 1.5.3.3.
More information about magnetic symmetry can be found in Birss (1964), Cracknell (1975
), Joshua (1991
), Koptsik (1966
), Landau & Lifshitz (1957
), Opechowski & Guccione (1965
), and in Sirotin & Shaskol'skaya (1979
).
Most transitions from a paramagnetic into an ordered magnetic state are second-order phase transitions. A crystal with a given crystallographic symmetry can undergo transitions to different ordered states with different magnetic symmetry. In Section 1.5.3.3, we shall give a short review of the theory of magnetic second-order phase transitions. As was shown by Landau (1937
), such a transition causes a change in the magnetic symmetry. The magnetic symmetry group of the ordered state is a subgroup of the magnetic group of the material in the paramagnetic state. But first we shall give a simple qualitative analysis of such transitions.
To find out what ordered magnetic structures may be obtained in a given material and to which magnetic group they belong, one has to start by considering the crystallographic space group of the crystal under consideration. It is obvious that a crystal in which the unit cell contains only one magnetic ion can change only into a ferromagnetic state if the magnetic unit cell of the ordered state coincides with the crystallographic one. If a transition into an antiferromagnetic state occurs, then the magnetic cell in the ordered state will be larger than the crystallographic one if the latter contains only one magnetic ion. Such antiferromagnets usually belong to the subtype IIIb described in Section 1.5.2.3
. In Section 1.5.3.1
, we shall consider crystals that transform into an antiferromagnetic state without change of the unit cell. This is possible only if the unit cell possesses two or more magnetic ions. To find the possible magnetic structures in this case, one has to consider those elements of symmetry which interchange the positions of the ions inside the unit cell (especially glide planes and rotation axes). Some of these elements displace the magnetic ion without changing its magnetic moment, and others change the moment of the ion. It is also essential to know the positions of all these elements in the unit cell. All this information is contained in the space group
. If the magnetic ordering occurs without change of the unit cell, the translation group
in the ordered state does not contain primed elements. Therefore, there is no need to consider the whole crystal space group
. It will suffice to consider the cosets of
in
. Such a coset consists of all elements of
that differ only by a translation. From each coset, a representative with minimum translative component is chosen. We denote a set of such representatives by
; it can be made into a group by defining
(
) as the representative of the coset that contains
. Obviously,
is then isomorphic to the factor group
and therefore to the point group
of
.
Once more, we should like to stress that to construct the magnetic structures and the magnetic groups of a given crystal it is not enough to consider only the point group of the crystal, but it is necessary to perform the analysis with the help of its space group in the paramagnetic state or the corresponding group of coset representatives. An example of such an analysis will be given in the following section.
Following Dzyaloshinskii (1957a), we consider crystals belonging to the crystallographic space group
. To this group belong
-Fe2O3 and the carbonates of Mn2+, Co2+ and Ni2+. Weak ferromagnetism was first observed in these materials. Cr2O3, in which the magnetoelectric effect was discovered, also belongs to this group. The magnetic ordering in these materials occurs without change of the unit cell.
The representatives of the cosets form the group
. Its symmetry operations are shown in Fig. 1.5.3.1
. Directed along the z axis is the threefold axis
and the sixfold roto-inversion axis
. Three twofold axes
run through the points
at right angles to the z axis. One of these axes is directed along the x axis. Arranged normal to each of the
axes are three glide planes
. The y axis is directed along one of these planes. The centre of inversion
is located at the point
, lying on the z axis halfway between two points
. The sign ~ means that the corresponding operation is accompanied by a translation along the z axis through half the period of the crystal (
means that the inversion centre is shifted from the point
to the point
). In Fig. 1.5.3.1
, the elementary period of translation along the z axis is marked by
. Thus the crystallographic group
has the following elements:
In two types of crystals, considered below, the magnetic ions are arranged on the z axis. If we place the magnetic ion at point 1 located between points and
(see Fig. 1.5.3.2
), then using symmetry operations (1.5.3.1)
we obtain three additional positions for other magnetic ions (points 2, 3, 4). Thus, the elementary cell will contain four magnetic ions. This is the structure of oxides of trivalent ions of iron and chromium (Fe2O3, Cr2O3). The structure of these oxides is shown in Fig. 1.5.3.2
. If the positions of the magnetic ions coincide with the positions of the inversion centre
, we obtain the structure of the carbonates of the transition metals (MnCO3, CoCO3, NiCO3, FeCO3), which is shown in Fig. 1.5.3.3
.
Evidently, the formation of a magnetic structure in the crystal does not result in the appearance of new elements of symmetry. The magnetic groups of magnetically ordered crystals may lack some elements contained in the crystallographic group and some of the remaining elements may happen to be multiplied by R (primed). Let us find the groups of symmetry that correspond to all possible collinear magnetic structures in rhombohedral crystals with four magnetic ions in the elementary cell. We shall assume that the magnetic moments are located at the points of the ion positions 1–4; they will be marked . The symmetry transformations cannot change the length of the vectors of the magnetic moments but they can change the direction of these vectors and interchange the positions of the sites 1
2, 3
4 and 1
3, 2
4. This interchange of the vectors
,
,
,
means that these vectors form a basis of a reducible representation of the group
. The following linear combinations of
form irreducible representations2 of
:
Vectors
characterize the antiferromagnetic states and are called antiferromagnetic vectors. The ferromagnetic vector
gives the total magnetic moment of the elementary cell. These vectors describe the four possible collinear magnetic structures. Three are antiferromagnetic structures:
(
,
),
(
,
),
(
,
) and one is a ferromagnetic structure,
(
). All these types are presented schematically in Fig. 1.5.3.4
.
![]() |
Four types of magnetic structures of rhombohedral oxides of transition metals. The direction of |
In the description of the structures of orthoferrites, other symbols were introduced to define the linear combinations of and to denote the antiferromagnetic structures under consideration (see Bertaut, 1963
). The two types of symbols are compared in Table 1.5.3.1
.
|
It should be borne in mind that in each of these types of magnetic ordering the respective vectors and
may be directed along any direction. There are 12 types of such structures in which
or
are directed along one of the axes or planes of symmetry. To find out to which group of magnetic symmetry each of these structures belongs, one needs to investigate how each element of the crystallographic symmetry transforms the Cartesian components of the four vectors. This is shown in Table 1.5.3.2
for the group
. If the component keeps its direction, it is marked by the
sign; the − sign corresponds to reversal of the component direction. In some cases, the transformation results in a change of the direction of the components
or
through an angle other than 0 or
. This is marked by 0. With the help of Table 1.5.3.2
, we can easily describe all the elements of symmetry of the magnetic group that corresponds to each structure (
or
) with the aid of the following rule. All the elements that yield the
sign are included in the magnetic group as they stand, while the elements yielding the − sign must be multiplied by R; the elements which are marked by the sign 0 are not included in the magnetic group.3 With the aid of this rule, Table 1.5.3.3
of the elements of the magnetic groups for the structures under consideration was compiled. In Table 1.5.3.4
, the symbols of the magnetic point groups of all the 12 magnetic structures considered are listed. The crystals with two ions in the elementary cell have only two sublattices and their antiferromagnetic structures belong to the same groups as the structures
.
|
|
|
One can see from Tables 1.5.3.3 and 1.5.3.4
that, in accordance with general theory, the magnetic point groups of the crystals under consideration are subgroups of the trivial magnetic point group
to which they belong in the paramagnetic state. In the example considered, the translation group does not change in going from the paramagnetic to the ordered state. Thus the same statement made for the point groups is also true for the space groups. Putting
gives a subgroup of the crystallographic group of the crystal. For the magnetic structures with the ferromagnetic or antiferromagnetic vector directed along the z axis, it turns out that the magnetic group is isomorphic to the crystallographic group. This rule is obeyed by all (optically) uniaxial crystals if the transition occurs without change of the elementary cell. (Optically uniaxial are the non-cubic crystals with a point group possessing a threefold, fourfold or sixfold axis.)
Tables 1.5.3.3 and 1.5.3.4
show that different types of collinear structures may belong to the same point group (and also to the same space group). For the antiferromagnetic structure
and the ferromagnetic
the group is
, and for the structures
and
it is
. Thus the symmetry allows a phase to be simultaneously ferromagnetic and antiferromagnetic. That is not ferrimagnetic order because all the ions in the four sublattices are identical and their numbers are equal. The ferromagnetic vector
and the antiferromagnetic one
are perpendicular and
. This phenomenon is called weak ferromagnetism and will be discussed in detail in Section 1.5.5.1
. Like weak ferromagnetism, the symmetry also allows the coexistence of two orthogonal antiferromagnetic structures
and
. This gives rise to weakly non-collinear antiferromagnetic structures.
The strongly non-collinear structures are described by another set of basis vectors for the irreducible representations of the group . If the magnetic ions
in the crystal form triangular planes one gets instead of (1.5.3.2)
the relations for the basis vectors:
It is pertinent to compare the different kinds of interactions that are responsible for magnetic ordering. In general, all these interactions are much smaller than the electrostatic interactions between the atoms that determine the chemical bonds in the material. Therefore, if a crystal undergoes a transition into a magnetically ordered state, the deformations of the crystal that give rise to the change of its crystallographic symmetry are comparatively small. It means that most of the non-magnetic properties do not change drastically. As an example, the anisotropic deformation of the crystal that accompanies the transition into the ordered state (see Section 1.5.9.1) is mostly not larger than 10−4.
The formation of the ordered magnetic structures is due mainly to the exchange interaction between the spins (and corresponding magnetic moments
of the atoms or ions). The expression for the exchange energy can contain the following terms [see formula (1.5.1.7)
]:
The exchange interaction decreases rapidly as the distance between the atoms rises. Thus, it is usually sufficient to consider the interaction only between nearest neighbours. The exchange interaction depends only on the relative alignment of the spin moments and does not depend on their alignment relative to the crystal lattice. Therefore, being responsible for the magnetic ordering in the crystal, it cannot define the direction of the spontaneous magnetization in ferromagnets or of the antiferromagnetic vector. This direction is determined by the spin–orbit and magnetic spin–spin interactions, which are often called relativistic interactions as they are small, of the order of
, where v is the velocity of atomic electrons and c is the speed of light. The relativistic interactions are responsible for the magnetic anisotropy energy, which depends on the direction of the magnetic moments of the ions with regard to the crystal lattice. The value of the exchange energy can be represented by the effective exchange field He. For an ordered magnetic with a transition temperature of 100 K, He ≃ 1000 kOe. Thus the external magnetic field hardly changes the value of the magnetization
or of the antiferromagnetic vector
; they are conserved quantities to a good approximation. The effective anisotropy field
in cubic crystals is very small: 1–10 Oe. In most non-cubic materials,
is not larger than 1–10 kOe. This means that by applying an external magnetic field we can change only the direction of
, or sometimes of
, but not their magnitudes.
The magnetic anisotropy energy can be represented as an expansion in the powers of the components of the vectors
or
. The dependence of
on the direction of the magnetization is essential. Therefore, one usually considers the expansion of the spontaneous magnetization or antiferromagnetic vector in powers of the unit vector
. The anisotropy energy is invariant under time reversal. Therefore, the general expression for this energy has the form
where
,
,
are tensors, the components of which have the dimension of an energy density. The forms of the tensors depend on the symmetry of the crystal. There are at most two independent components in
. For a uniaxial crystal, the second-order term in the anisotropy energy expansion is determined by one anisotropy constant, K. Instead of using the components of the unit vector
, its direction can be described by two angles: polar
and azimuthal
. Correspondingly, the anisotropy energy for a uniaxial crystal can be written as
This relation is equivalent to
The direction of the magnetization vector in a ferromagnet or of the antiferromagnetic vector
in an antiferromagnet is called the direction or the axis of easy magnetization. The crystals in which this axis is aligned with a threefold, fourfold or sixfold axis of the magnetic point group are called easy-axis magnetics. The magnetic crystals with the main axis higher than twofold in the paramagnetic state in which, in the ordered state,
(or
) is perpendicular to this axis are often called easy-plane magnetics. The anisotropy in this plane is usually extremely small. In this case, the crystal possesses more then one axis of easy magnetization and the crystal is usually in a multidomain state (see Section 1.5.4
).
If the anisotropy constant K is positive, then the vector is aligned along the z axis, and such a magnetic is an easy-axis one. For an easy-plane magnetic, K is negative. It is convenient to use equation (1.5.3.6)
for easy-axis magnetics and equation (1.5.3.7)
for easy-plane magnetics. In the latter case, the quantity K is included in the isotropic part of the thermodynamic potential
, and (1.5.3.7)
becomes
. Instead we shall write
in the following, so that K becomes positive for easy-plane ferromagnetics as well.
Apart from the second-order term, terms of higher order must be taken into account. For tetragonal crystals, the symmetry allows the following invariant terms in the anisotropy energy: the azimuthal angle
is measured from the twofold axis x in the basal plane and the constant
determines the anisotropy in the basal plane.
Trigonal symmetry also allows second- and fourth-order invariants: where
is measured from the x axis, which is chosen parallel to one of the twofold axes. For easy-plane magnetics and
, the vector
is directed along one of the twofold axes in the basal plane. If
is negative, then
lies in a vertical mirror plane directed at a small angle to the basal plane. For the complete solution of this problem, the sixth-order term must be taken into account. This term is similar to the one that characterizes the anisotropy of hexagonal crystals. The expression for the latter is of the following form:
where x and
have the same meaning as in (1.5.3.9)
.
The symmetry of cubic crystals does not allow any second-order terms in the expansion of the anisotropy energy. The expression for the anisotropy energy of cubic crystals contains the following invariants:
In considering the anisotropy energy, one has to take into account spontaneous magnetostriction and magnetoelastic energy (see Section 1.5.9). This is especially important in cubic crystals. Any collinear cubic magnetic (being brought into a single domain state) ceases to possess cubic crystallochemical symmetry as a result of spontaneous magnetostriction. If
is positive, the easy axis is aligned along one of the edges of the cube and the crystal becomes tetragonal (like Fe). If
is negative, the crystal becomes rhombohedral and can be an easy-axis magnetic with vector
parallel to one of the spatial diagonals (like Ni) or an easy-plane magnetic with
perpendicular to a spatial diagonal. We shall discuss this topic in more detail in Section 1.5.9.3
.
The considerations presented above can be applied to all crystals belonging in the paramagnetic state to the tetragonal, trigonal or hexagonal system that become easy-plane magnetics in the ordered state. All of them, including the cubic crystals, may possess more than one allowed direction of easy magnetization. In the example considered in the previous section, these directions can be aligned along the three twofold axes for the structures and can be parallel to the three mirror planes for
.
It is worth noting that in some applications it is more convenient to use an expansion of the anisotropy energy in terms of surface spherical harmonics. This problem has been considered in detail by Birss (1964).
According to Landau (1937) (see also Landau & Lifshitz, 1951
), a phase transition of the second kind can be described by an order parameter
, which varies smoothly in the neighbourhood of the transition temperature
. The order parameter
when
and rises continuously as the temperature is decreased below
, but the symmetry of the crystal changes suddenly. The order parameter can be a scalar, a vector or a tensor.
Consider a crystal with known space group in the paramagnetic state. In this section, we show how the Landau theory allows us to determine the magnetic space groups that are possible after a second-kind phase transition into an ordered state. The application of the Landau theory to the magnetic transitions into different types of antiferromagnets was made by Dzyaloshinskii (1957a,c
; 1964
). In these cases, the order parameter is the magnetic moment density
. To determine the equilibrium form of this function, it is necessary to find the minimum of the thermodynamic potential
, which is a functional of
. Since the transition is continuous and
for
, the value of
must be very small in the neighbourhood below the transition point. In this region, the thermodynamic potential
will be expanded into a power series of
. To find the proper form of this expansion, it is convenient to represent
as a linear combination of functions that form bases of the irreducible representations of the space group of the paramagnetic phase
:
where
are functions that transform under the representation n (
is the number of the function in the representation) and
. In this expansion, the quantities
are independent of
and transform with respect to i as the components of an axial vector. The functions
are transformed into combinations of one another by the elements of the group
. Instead, these elements can be regarded as transforming the coefficients
and leaving the functions
invariant. In this case, the quantities
transform according to the direct product of the representation n of
and the representation formed by the components of the pseudovector. This representation is reducible in the general case. Irreducible representations
can be obtained by forming linear combinations of the
. Let us denote these combinations by
. These variables can be considered as components of the order parameter, and the thermodynamic potential can be expanded into a power series of
. The terms of this expansion must be invariant under the transformations of the magnetic space group of the crystal in the paramagnetic state
. This group possesses R as a separate element. Therefore the expansion can contain only even terms. For each irreducible representation, there is only one invariant of second order – the sum of the squares. Consequently, retaining only the square terms, the expansion of the thermodynamic potential
has the form:
To minimize
, it is necessary to add the terms of the fourth power. All the coefficients
in the relation (1.5.3.13)
depend on the temperature. At
all
. This solution corresponds to the minimum of
if all
are positive. The transition into the ordered state occurs if one of the quantities
changes its sign. This means that the transition temperature
is the temperature at which one of the coefficients
. This coefficient has the form:
Accordingly, the corresponding magnetic structure is defined by the order parameters
and belongs to the representation p.
The representation of the space group is realized by a set of functions of the following type: where the values of the vectors
are confined to the Brillouin zone in the reciprocal lattice and the function
is periodic in the real lattice. The irreducible representation defined by the vector
contains the functions with all the vectors
that belong to the same star. The star is the set of the vectors
obtained by applying all the transformations
of the corresponding point group to any vector of the star (see also Section 1.2.3.3
). If we denote it as
, then the set of the vectors of the star consists of all inequivalent vectors of the form
.
There are three types of transition we have to consider: (1) the magnetic lattice is commensurate with the crystallographic one and ; (2) the magnetic lattice is incommensurate with the crystallographic one; (3)
and the magnetic lattice coincides with the crystallographic lattice. Below we shall discuss in detail only the first and the third type of transition.
|
The temperature of transition from the paramagnetic to the ferromagnetic state is called the Curie temperature. The thermodynamic treatment of the behaviour of uniaxial ferromagnets in the neighbourhood of the Curie temperature is given below.
In the case of a ferromagnet , the thermodynamic potential (1.5.3.27)
near
including the magnetic energy
is given by (see 1.5.3.25)
where
is used to designate the thermodynamic potential in variables
[instead of
]; at the given field,
should be a minimum. The equilibrium value of the magnetization
is found by minimizing the thermodynamic potential.
First consider the ferromagnet in the absence of the external field . The system of equations
has three solutions:
In the whole range of temperatures when
, the minimum of the potential is determined by solution (I) (i.e. absence of a spontaneous magnetization). The realization of the second or third state depends on the sign of the coefficient b. If
, then the third state is realized, the magnetization
being directed along the axis. In this case, the transition from the paramagnetic into the ferromagnetic state will take place at
(when
). If
, the magnetization is directed perpendicular to the axis. In this case, the Curie temperature is
(when
). In the absence of a magnetic field, the difference between the two values of
has no physical meaning, since it only means another value of the coefficient B [see (1.5.3.25)
]. In a magnetic field, both temperatures may be determined experimentally, i.e. when B becomes zero and when
becomes zero.
If a magnetic field is applied parallel to the z axis and
, the minimization of the thermodynamic potential
leads to
This relation has been verified in many experiments and the corresponding graphical representations are known in the literature as Arrott–Belov–Kouvel plots (see Kouvel & Fisher, 1964
). Putting
according to (1.5.3.14)
, equations (1.5.3.32)
and (1.5.3.33)
may be used to derive expressions for the initial magnetic susceptibilities (for
):
where
.
The Landau theory of phase transitions does not take account of fluctuations of the order parameter. It gives qualitative predictions of all the possible magnetic structures that are allowed for a given crystal if it undergoes a second-order transition. The theory also explains which of the coefficients in the expression for the thermodynamic potential is responsible for the corresponding magnetic structure. It describes also quantitative relations for the magnetic properties of the material if where
is the coefficient in the term which describes the gradient energy. In this chapter, we shall not discuss the behaviour of the material in the fluctuation region. It should be pointed out that, in this region,
in relations (1.5.3.34)
and (1.5.3.35)
depends on the dimensionality of the structure n and equals 1.24 for
, 1.31 for
and 1.39 for
. Similar considerations are relevant to the relations (1.5.3.31)
and (1.5.3.32)
, which describe the temperature dependence of spontaneous magnetization.
The relations (1.5.3.31) and (1.5.3.32)
describe the behaviour of the ferromagnet in the `saturated' state when the applied magnetic field is strong enough to destroy the domain structure. The problem of the domains will be discussed later (see Section 1.5.4
).
The transition from the paramagnetic to the ferromagnetic state is a second-order transition, provided that there is no magnetic field. In the presence of a magnetic field that is parallel to the easy axis of magnetization, the magnetic symmetry of the crystal is the same () both above and below
. From the point of view of symmetry, no transition occurs in this case.
Now let us proceed to the uniaxial antiferromagnet with two ions in the primitive cell. The thermodynamic potential for such an antiferromagnet is given in accordance with (1.5.3.26)
and (1.5.3.27)
by (Landau, 1933
)
If the magnetic field is absent , then
because B, D and
. Then three possible magnetic states are obtained by minimizing the potential with respect to
only:
When , state (II) with
is thermodynamically stable. When
, state (III) is stable and the antiferromagnetic vector is directed along the axis. This means that the term with the coefficient a is responsible for the anisotropy of the uniaxial antiferromagnet. We introduce the effective anisotropy field:
where
is the sublattice magnetization.
Formulas (1.5.3.39) and (1.5.3.14)
in the form
yield the expression for the temperature dependence of the sublattice magnetization:
where
is the Néel temperature. The assertions relating to formulas (1.5.3.34)
and (1.5.3.35)
concerning the fluctuation region are also valid for the temperature dependence of the sublattice magnetization.
The minimization of the potential with respect to
for given
when
yields the following relation for the magnetization:
where
. Thus the magnetization of an antiferromagnet is linear with the magnetic field, as for a paramagnet, if the magnetic field is not too strong. The main difference is in the anisotropy and temperature dependence of the susceptibility. The parallel susceptibility
decreases when the temperature is lowered, and
does not depend on temperature (
) (see Fig. 1.5.3.6
). The coefficient B belongs to the exchange term and defines the effective exchange field
![]() |
Temperature dependence of the mass susceptibility |
As seen from Fig. 1.5.3.6,
. Therefore, when the magnetic field applied parallel to the axis of a uniaxial antiferromagnet reaches the critical value
(
is the value of L at
), a flopping of the sublattices from the direction along the axis to some direction in the plane perpendicular to the axis occurs. In this spin-flop transition (which is a first-order transition into a new magnetic structure), the magnetization jumps as shown in Fig. 1.5.3.7
.
![]() |
Dependence of the relative magnetization |
A second-order transition into a saturated paramagnetic state takes place in a much stronger magnetic field . This transition is called a spin-flip transition. Fig. 1.5.3.7
shows the magnetic field dependence of the magnetization of a uniaxial antiferromagnet. Fig. 1.5.3.8
shows the temperature dependence of both critical fields.
The quantitative behaviour of the critical magnetic fields in the neighbourhood of for both directions of the magnetic field (
and
) can be determined from the theory of second-order phase transitions starting from the thermodynamic potential
and taking into account that L is small and
close to
.
In the presence of the magnetic field ,
is parallel to
,
, the coefficient A at
is replaced by
and the latter is zero at the new transition point. The critical field is given by the relation
If the field is applied parallel to the z axis, then remains parallel to
if
(
in the neighbourhood of
). Therefore,
If
,
becomes perpendicular to the z axis and the anisotropy term has to be taken into account:
Formulas (1.5.3.46)–(1.5.3.48)
show that the transition temperature is reduced by applying the magnetic field. The displacement of the transition point is directly proportional to the square of the applied field. Fig. 1.5.3.9
shows the phase diagram of an antiferromagnet in the neighbourhood of
. Unlike ferromagnets, antiferromagnets maintain the second-order phase transition when a magnetic field is applied because the symmetry of the crystal in the antiferromagnetic state differs essentially from that in the paramagnetic state also if the crystal is placed into a magnetic field.
![]() |
Phase diagram for a uniaxial antiferromagnet in the proximity of |
Formula (1.5.3.43) describes the magnetization process only in easy-axis antiferromagnets. For easy-plane antiferromagnets, the anisotropy in the plane is usually extremely small and the antiferromagnetic vector rotates freely in the basic plane. Therefore, for any direction of the magnetic field, the vector
becomes aligned perpendicular to the applied magnetic field. Correspondingly the magnetization becomes
where
and
are unit vectors parallel and perpendicular to the axis.
Neither symmetry nor energy considerations can determine the alignment of the magnetization vector in a non-chiral easy-axis magnetic (of ferro- or antiferromagnetic type). The vector
may be aligned parallel or antiparallel to the positive direction of the z axis. Therefore, specimens of any magnetic are usually split into separate regions called domains. In each domain of an easy-axis magnetic, the vector
has one of its two possible directions. Such domains are called 180° domains. Adjacent domains are separated by a domain wall, in which the magnetic moments are no longer strictly parallel (or antiparallel). As a result of this, both the exchange and the anisotropy energy rise inside the volume of the domain wall.
In ferromagnets (and ferrimagnets), the loss in the exchange and anisotropy energy in a multidomain sample is compensated by the gain in the magnetostatic energy. The existence of the domain structure is responsible for the behaviour of a ferromagnet in an applied magnetic field. There are two kinds of magnetization processes that one has to distinguish: the displacement of the domain walls and the rotation of the spontaneous magnetization vector from the easy direction to the direction of the applied magnetic field. The magnetization process will first be considered without taking the demagnetizing field into account. If the magnetic field is applied parallel to the axis of an easy-axis ferromagnet, the displacement of the domain wall will completely determine the magnetization process. If the sample contains no impurities and crystal defects, such a displacement must take place in an infinitely small magnetic field [see curve (1) in Fig. 1.5.4.1 and Fig. 1.5.4.3
a]. If the magnetic field is applied perpendicular to the easy axis, the size of the domains does not change but their magnetization vectors rotate. Let us denote the spontaneous magnetization by
. Then the sample magnetization M rises linearly with respect to the applied magnetic field:
where
is defined by relations (1.5.3.8)
–(1.5.3.10)
. Some nonlinearity in H can arise from the fourth-order term with
[see curve (2) in Fig. 1.5.4.1
and Fig. 1.5.4.3
c]. When
, the magnetizations of all the domains are rotated by 90° and the magnetization of the sample becomes oriented along the magnetic field; its value is saturated and is equal to the spontaneous magnetization
. If
K, there is an additional rise in magnetization with the magnetic field. This rise, which is called true magnetization, is relatively very small at all temperatures except for the temperature region close to the transition temperature. If the magnetic field is applied at an arbitrary angle
to the easy axis, the magnetization process occurs in two steps [see curves (2) in Fig. 1.5.4.2
and Fig. 1.5.4.3
b]. First, as a result of the wall displacement, the magnetization jumps to the value
in a small magnetic field. Next, the rotation process follows and at
the sample becomes saturated [see curves (2) in Fig. 1.5.4.2
]. It is essential to take the shape of the sample into account in considering the problem of the magnetization processes in ferromagnets, as the demagnetizing field can be up to
. In real materials, the displacement process is partly (at low fields) reversible and partly (at higher fields) irreversible. Therefore, complicated hysteresis processes arise in magnetizing ferromagnets.
![]() |
Schematic display of the magnetization: (a) along the easy axis; (b) at an arbitrary angle to the easy axis; (c) perpendicular to the easy axis. |
The problem of 180° domains in antiferromagnets is not as clear. These domains differ in the sign of the antiferromagnetic vector . This vector was defined as the difference of the vectors of sublattice magnetizations in a two-sublattice antiferromagnet, i.e.
. Thus two such antiferromagnetic domains differ only by the numbering of the sites in the sublattices. Antiferromagnetic 180° domains are also called S-domains. The wall between two S-domains is schematically represented in Fig. 1.5.4.4
.
The origin of the antiferromagnetic S-domains cannot be explained from the point of view of energy balance as in a ferromagnet. These domains give rise to additional exchange and anisotropy energies which are not compensated by a decrease of any other kind of energy. Thus the S-domain structure is thermodynamically not stable. However, experiments show that S-domains exist in most easy-axis antiferromagnets.
The formation of S-domains can be explained by assuming that when the material is cooled down to the Néel temperature, antiferromagnetic ordering arises in different independent regions. The direction of the vector in these regions is accidental. When growing regions with different directions of
meet, the regular alternation of the directions of magnetic moments of the ions is broken on the border between these regions. Domain walls are created on such borders. Such domain structures can be metastable.
The existence of S-domains in easy-axis antiferromagnets was first proved in experiments in which effects that depend on the sign of were investigated. These are piezomagnetism, linear magnetostriction and the linear magnetoelectric effect. The sign of these effects depends on the sign of
. We shall discuss this problem in detail in Sections 1.5.7
and 1.5.8
. Later, 180° domain walls were observed in neutron scattering experiments (Schlenker & Baruchel, 1978
), and the domains themselves in magneto-optical experiments (see Kharchenko et al., 1979
; Kharchenko & Gnatchenko, 1981
).
As pointed out in Section 1.5.3, in tetragonal non-easy-axis magnetics, in easy-plane hexagonal and trigonal and in cubic magnetics there is more than one easy magnetization direction (3, 4 or 6). As a result, domains arise in which vectors
or
are directed to each other at 120, 109.5, 90, 70.5 and 60°. Such domains are called twin or T-domains. The formation of magnetic T-domains is accompanied by the formation of crystallographic domains as a result of spontaneous magnetostriction. But mostly this is very small. Each of the T-domains may split into 180° domains.
The magnetization process in ferromagnets possessing T-domains is similar to the previously described magnetization of an easy-axis ferromagnet in a magnetic field directed at an oblique angle. First the displacement process allows those 180° domains that are directed unfavourably in each T-domain to disappear, and then the rotation process follows.
In easy-plane antiferromagnets, the T-domain structure is destroyed by a small magnetic field and the antiferromagnetic vector in the whole specimen becomes directed perpendicular to the applied magnetic field, as was explained in Section 1.5.3.
There are four kinds of T-domains in cubic antiferromagnets, in which the vectors are directed parallel or perpendicular to the four
axes. Such a T-domain structure can be destroyed only when the applied magnetic field is so strong that the antiferromagnetic order is destroyed at a spin-flip transition.
Aizu (1970) gave a classification of domain formation when a crystal undergoes a transition from an unordered to a magnetically ordered state that has a lower point-group symmetry (see also Section 3.1.1
). The unordered state (called the prototype phase) has a grey point group. The number of elements in this group is equal to the product of the number of elements in the point group of the ordered state (called the ferroic state) times the number of domains. Aizu found that there are 773 possible combinations of the point-group symmetries of the prototype and the ferroic state, if crystallographically inequivalent orientations of the subgroup in the group of the prototype are distinguished. These 773 combinations are called ferroic species and are characterized by a symbol giving first the point group of the prototype, then the letter F, then the point group of the ferroic state and finally a letter between parentheses if different orientations are possible. As an example, the
axis of the ferroic state is parallel to the fourfold axis of the prototype in
and perpendicular to it in
.
Let us discuss the ferroic states of rhombohedral transition-metal oxides given in Table 1.5.3.4. The paramagnetic prototype has point group
. The four monoclinic ferroic species have six domains (`orientation states') each, which form three pairs of 180° domains (`time-conjugate orientation states'). All four species are `fully ferroelastic', i.e. the three pairs show different orientations of the spontaneous strain; two of the four species (
and
) are also `fully ferromagnetic' because all six domains have different orientations of the spontaneous magnetization. Switching a domain into another with a different orientation of the spontaneous strain can be achieved by applying mechanical stress. If the domain was spontaneously magnetized, the orientation of the magnetization is changed simultaneously. Similarly, a domain can be switched into another with a different orientation of the spontaneous magnetization by means of a magnetic field. If the two spontaneous magnetizations have different directions (not just opposite sign), the direction of the spontaneous strain will change at the same time.
The Aizu classification is of interest for technological applications because it gives an overall view not only of domain formation but also of the possibilities for domain switching.
As was indicated above (see Tables 1.5.3.3 and 1.5.3.6
), certain magnetic space groups allow the coexistence of two different types of magnetic ordering. Some magnetic structures can be described as a superposition of two antiferromagnetic structures with perpendicular antiferromagnetic vectors
. Such structures may be called weakly non-collinear antiferromagnets. There can also be a superposition of an antiferromagnetic structure
with a ferromagnetic one
(with
). This phenomenon is called weak ferromagnetism. We shall demonstrate in this section why one of the magnetic vectors has a much smaller value than the other in such mixed structures.
The theory of weak ferromagnetism was developed by Dzyaloshinskii (1957a). He showed that the expansion of the thermodynamic potential
may contain terms of the following type:
(
). Such terms are invariant with respect to the transformations of many crystallographic space groups (see Section 1.5.3.3
). If there is an antiferromagnetic ordering in the material (
) and the thermodynamic potential of the material contains such a term, the minimum of the potential will be obtained only if
as well. The term
is a relativistic one. Therefore this effect must be small.
We shall consider as an example the origin of weak ferromagnetism in the two-sublattice antiferromagnets MnCO3, CoCO3 and NiCO3, discussed in Section 1.5.3.1. The following analysis can be applied also to the four-sublattice antiferromagnet
-Fe2O3 (assuming
,
). All these rhombohedral crystals belong to the crystallographic space group
. The thermodynamic potential
for these crystals was derived in Section 1.5.3.3
. For the case of a two-sublattice antiferromagnet, one has to add to the expression (1.5.3.26)
the invariant (1.5.3.24)
:
The coefficients of the isotropic terms (A and B) are of exchange origin. They are much larger than the coefficients of the relativistic terms (
). Minimization of
for a fixed value of
and
gives two solutions:
Weak ferromagnetism was first observed in the following trigonal crystals: the high-temperature modification of haematite, -Fe2O3 (Townsend Smith, 1916
; Néel & Pauthenet, 1952
), MnCO3 (Borovik-Romanov & Orlova, 1956
) and later also in CoCO3, NiCO3 and FeBO3. In accordance with theory, weak ferromagnetism does not occur in trigonal crystals with a positive anisotropy coefficient a. Such crystals become easy-axis antiferromagnets. Of this type are FeCO3 and the low-temperature modification of
-Fe2O3. For four-sublattice antiferromagnets, the sequence of the directions of the magnetic moments of the sublattices is also essential. For example, the structures of the types
and
(see Fig. 1.5.3.4
and Table 1.5.3.3
) do not exhibit weak ferromagnetism.
The behaviour of weak ferromagnets in magnetic fields applied perpendicular and parallel
to the trigonal axis is described by the following relations:
where
An external magnetic field can freely rotate the ferromagnetic moment in the basal plane of the easy-plane weak ferromagnets under consideration because their anisotropy in the basal plane is extremely small. During such a rotation, both vectors and
move simultaneously as a rigid structure. On the other hand, it is impossible to deflect the vector
out of the basal plane, as this is forbidden by symmetry. This is illustrated by the magnetization curves plotted in Fig. 1.5.5.2
, which confirm the relations (1.5.5.3)
.
![]() |
Dependence of magnetization |
It is worth mentioning that when the weakly ferromagnetic structure is rotated in the basal plane, a change of the magnetic space groups occurs in the following order:
. Each of these symmetry transformations corresponds to a second-order phase transition. Such transitions are allowed because
is a subgroup of both groups
and
.
NiF2 was one of the first weak ferromagnets to be discovered (Matarrese & Stout, 1954). In the paramagnetic state, it is a tetragonal crystal. Its crystallographic space group is
. In the ordered state its magnetic point group is
and the vectors
and
are directed along two twofold axes (one of which is primed) in the plane perpendicular to the former fourfold axis (see Fig. 1.5.5.3
a). The invariant term responsible for the weak ferromagnetism in tetragonal fluorides has the form
The anisotropy of the crystals of NiF2 and the relation given above for the invariant lead to the same dependence on the magnetic field as for trigonal crystals. However, the anisotropy of the magnetic behaviour in the basal plane is much more complicated than for rhombohedral crystals (see Bazhan & Bazan, 1975
). The anisotropy constant
is positive for most other fluorides (MnF2, FeF2 and CoF2) and their magnetic structure is described by the magnetic point group
. They are easy-axis antiferromagnets without weak ferromagnetism.
![]() |
Magnetic structures of fluorides of transition metals. (a) The weak ferromagnet NiF2; (b) the easy-axis antiferromagnets MnF2, FeF2 and CoF2. |
The interaction described by the invariant in equation (1.5.5.1)
is called Dzyaloshinskii–Moriya interaction. It corresponds to the interaction between the spins of neighbouring ions, which can be represented in the form
where the vector
has the components (
). Terms of such type are allowed by symmetry for crystals that in the paramagnetic state belong to certain space groups of the trigonal, tetragonal and hexagonal systems. In some groups of the tetragonal system, weak ferromagnetism is governed by the term
(as for NiF2) and in the orthorhombic system by
. In the monoclinic system, the latter sum contains four terms. The weak ferromagnetism in most groups of the hexagonal and cubic systems is governed by invariants of fourth and sixth order of
. Turov (1963
) determined for all crystallographic space groups the form of the invariants of lowest order that allow for collinear or weakly non-collinear antiferromagnetic structures a phase transition into a state with weak ferromagnetism. The corresponding list of the numbers of the space groups that allow the transition into an antiferromagnetic state with weak ferromagnetism is given in Table 1.5.5.1
. The form of the invariant responsible for weak ferromagnetism is also displayed in the table. Turov (1963
) showed that weak ferromagnetism is forbidden for the triclinic system, for the six trigonal groups with point groups
or
, and the 12 cubic groups with point groups
or
.
|
The microscopic theory of the origin of weak ferromagnetism was given by Moriya (1960a,b
, 1963
). In this chapter, however, we have restricted our consideration to the phenomenological approach to this problem.
A large number of orthorhombic orthoferrites and orthochromites with the formula RMO3 (where R is a trivalent rare-earth ion and M is Fe3+ or Cr3+) have been investigated in many laboratories (cf. Wijn, 1994). Some of them exhibit weak ferromagnetism. The space group of these compounds is
in the paramagnetic state. The primitive cell is the same in the paramagnetic and magnetically ordered states. It contains four magnetic transition-metal ions (see Fig. 1.5.5.4
). They determine to a large extent the properties of orthoferrites (outside the region of very low temperatures). For a four-sublattice antiferromagnet, there are four possible linear combinations of the sublattice vectors, which define three types of antiferromagnetic vectors
and one ferromagnetic vector
[see relations (1.5.3.2)
and Table 1.5.3.1
]. The exchange interaction in these compounds governs magnetic structures, which to a first approximation are described by the following antiferromagnetic vector (which is usually denoted by the symbol
):
In the case of orthoferrites, the other two antiferromagnetic vectors
and
[see relations (1.5.3.2)
] are named
and
, respectively.
![]() |
Magnetic structures of orthoferrites and orthochromites RMO3. (Only the transition-metal ions are shown; the setting Pbnm is used.) (a) |
The magnetic structure of the compounds under consideration is usually called the or
state. Depending on the signs and the values of the anisotropy constants, there are three possible magnetic states:
The magnetic structures (I) and (III) are weak ferromagnets. They are displayed schematically in Fig. 1.5.5.4
. Both are described by the same magnetic point group
yet in different orientations:
(i.e.
) for structure (I) and
(i.e.
) for structure (III). The magnetic point group of structure (II) is
.
Weak ferromagnetism is observed in boracites with chemical formula M3B7O13X (where M = Co, Ni and X = Br, Cl, I). These compounds are unique, being simultaneously antiferromagnets, weak ferromagnets and ferroelectrics. Section 1.5.8.3 is devoted to these ferromagnetoelectrics.
Concerning the magnetic groups that allow weak ferromagnetism, it should be noted that, as for any ferromagnetism, weak ferromagnetism is allowed only in those space groups that have a trivial magnetic Bravais lattice. There must be at least two magnetic ions in the primitive cell to get antiferromagnetic order. Among the 31 magnetic point groups that admit ferromagnetism (see Table 1.5.2.4), weak ferromagnetism is forbidden in the magnetic groups belonging to the tetragonal, trigonal and hexagonal systems. Twelve magnetic point groups that allow weak ferromagnetism remain. These groups are listed in Table 1.5.5.2
.
|
A material that becomes a weak ferromagnet below the Néel temperature differs from a collinear antiferromagnet in its behaviour above
. A magnetic field applied to such a material above
gives rise to an ordered antiferromagnetic state with vector
directed perpendicular and magnetization
parallel to the field. Thus, as in usual ferromagnets, the magnetic symmetry of a weak ferromagnet in a magnetic field is the same above and below
. As a result, the magnetic susceptibility has a maximum at
[like the relations (1.5.3.34)
and (1.5.3.35)
]. This is true only if the magnetic field is aligned along the easy axis for weak ferromagnetism. Fig. 1.5.5.5
shows the anomalous anisotropy of the temperature dependence of the magnetic susceptibility in the neighbourhood of
for weak ferromagnets.
Similar anomalies in the neighbourhood of are observed in materials with a symmetry allowing a transition into a weakly ferromagnetic state for which the sign of the anisotropy constant causes their transition into purely antiferromagnetic states.
A thermodynamic potential of the form (1.5.5.1)
may give rise not only to the weak ferromagnetism considered above but also to the reverse phenomenon. If the coefficient B (instead of A) changes its sign and
, the material will undergo a transition into a slightly canted ferromagnetic structure, in which
and the expression for
is
Experimental detection of such structures is a difficult problem and to date no-one has observed such a phenomenon.
The thermodynamic potential of a four-sublattice antiferromagnet may contain the mixed invariant [see (1.5.3.24)
]
Such a term gives rise to a structure in which all four vectors of sublattice magnetization
form a star, as shown in Fig. 1.5.5.6
(see also Fig. 1.5.1.3
b). The angle
between the vectors
and
(or
and
) is equal to
if the main antiferromagnetic structure is defined by the vector
[see relation (1.5.3.2)
]. Such a structure may occur in Cr2O3. In most orthoferrites discussed above, such non-collinear structures are observed for all three cases: purely antiferromagnetic (
) and weakly ferromagnetic (
and
). The structure
is not coplanar. Apart from the main antiferromagnetic vector
aligned along the y axis, it possesses two other antiferromagnetic vectors:
(aligned along the x axis) and
(aligned along the z axis). The weakly ferromagnetic structure
has an admixture of the
antiferromagnetic structure.
The helical (or spiral) structure described in Section 1.5.1.2.3 and depicted in Fig. 1.5.1.4
is also a weakly non-collinear antiferromagnetic structure. As mentioned above, this structure consists of atomic layers in which all the magnetic moments are parallel to each other and parallel to the layer. The magnetizations of neighbouring layers are antiparallel to a first approximation; but, more specifically, there is a small deviation from a strictly antiparallel alignment. The layers are perpendicular to a vector
, which is parallel to the axis of the helix. The two mutually perpendicular antiferromagnetic vectors
are both perpendicular to
. These vectors define the helical structure by the following relation for the density of the magnetization
in the layer with the coordinate
(Dzyaloshinskii, 1964
; Andreev & Marchenko, 1980
):
Most helical structures are incommensurate, which means that the representation defined by the vector
does not satisfy the Lifshitz condition (see Section 1.5.3.3
).
In many materials, the anisotropy constants change sign at some temperature below the critical temperature. As a result, the direction of the vector (or
) changes relative to the crystallographic axes. Correspondingly, the magnetic symmetry of the material also changes. Such phase transitions are called reorientation transitions.
Cobalt is a typical ferromagnet and experiences two such reorientation transitions. It is a hexagonal crystal, which at low temperatures behaves as an easy-axis ferromagnet; its magnetic point group is . If the anisotropy energy were described by the relations (1.5.3.6)
and (1.5.3.7)
with only one anisotropy constant
, the change of the sign of this constant would give rise to a first-order transition from an easy-axis to an easy-plane antiferromagnet. This transition would occur at the temperature
at which
. In fact, the polar angle
which determines the direction of the spontaneous magnetization increases progressively over a finite temperature interval. The behaviour of
during the process of this reorientation may be obtained by minimizing the expression of the anisotropy energy (1.5.3.10)
, which contains two anisotropy coefficients
and
. If
, the minimum of
corresponds to three magnetic phases, which belong to the following magnetic point groups:
The low-temperature phase is of the easy-axis type and the high-temperature phase is of the easy-plane type. The intermediate phase is called the angular phase. The two second-order phase transitions occur at temperatures which are the roots of the two equationsThe chain of these transitions (including the transition to the paramagnetic state at
) may be represented by the following chain of the corresponding magnetic point groups:
In Co and most of the other ferromagnets, the rotation of the spontaneous magnetization described above may be obtained by applying an external magnetic field in an appropriate direction. In many antiferromagnets, there occur similar reorientation transitions, which cannot be achieved by means of a magnetic field.
The first reorientation transition in antiferromagnets was observed in haematite (- Fe2O3), which at room temperature is a weak ferromagnet with magnetic structure
or
(see Tables 1.5.3.3
and 1.5.3.4
in Section 1.5.3.1
). Morin (1950
) found that the weak ferromagnetism in haematite disappears below
K. At low temperature, haematite becomes an easy-axis antiferromagnet with the structure
. Unlike in cobalt, the transition at
is a first-order transition in haematite. This is so because the anisotropy constant
is negative in haematite. As a result, there are only two solutions for the angle
that lead to a minimum of the anisotropy energy
[(1.5.3.9)
],
if
and
if
. The transition temperature
is defined by
There is the following change in the magnetic space groups at this transition:
Which of the two groups is realized at high temperatures depends on the sign of the anisotropy constant in equation (1.5.3.9)
. Neither of the high-temperature magnetic space groups is a subgroup of the low-temperature group. Therefore the transition under consideration cannot be a second-order transition.
Reorientation transitions have been observed in many orthoferrites and orthochromites. Orthoferrites of Ho, Er, Tm, Nd, Sm and Dy possess the structure [see (1.5.5.8)
] at room temperature. The first five of them undergo reorientation transitions to the structure
at lower temperatures. This reorientation occurs gradually, as in Co. Both vectors
and
rotate simultaneously, as shown in Fig. 1.5.6.1
. These vectors remain perpendicular to each other, but the value of
varies from
for
to
for
. The coefficients
and
belong to the terms
and
, respectively. The following magnetic point groups are observed when these transitions occur:
![]() |
Schematic representation of the rotation of the vectors |
Anomalies typical for second-order transitions were observed at the temperatures and
. The interval
varies from 10 to 100 K.
At low temperatures, DyFeO3 is an easy-axis antiferromagnet without weak ferromagnetism – . It belongs to the trivial magnetic point group
. At TM = 40 K, DyFeO3 transforms into a weak ferromagnet
. This is a first-order reorientation transition of the type
Reorientation transitions in antiferromagnets occur not only as a result of a sign change of the anisotropy constant. They can be governed by the applied magnetic field. In Section 1.5.3.3.2, we described the spin-flop first-order reorientation transition in an easy-axis antiferromagnet. This transition splits into two second-order transitions if the magnetic field is not strictly parallel to the axis of the crystal. There is a specific type of reorientation transition, which occurs in antiferromagnets that do not exhibit weak ferromagnetism, but would become weak ferromagnets if the antiferromagnetic vector was directed along another crystallographic direction. As an example, let us consider such a transition in CoF2. It is a tetragonal crystal with crystallographic space group
. Below
, CoF2 becomes an easy-axis antiferromagnet. The magnetic structure of this crystal is shown in Fig. 1.5.5.3
. Its magnetic point group is
. Let us apply the magnetic field H parallel to the twofold axis x (see Fig. 1.5.6.2
). In a typical antiferromagnet, the field stimulates a magnetization
. The structure
allows weak ferromagnetism if
is perpendicular to the z axis. As a result, if the vector
is deflected from the z axis by an angle
in the plane yz perpendicular to the x axis, the magnetization will rise according to the relation
where
[see (1.5.5.3)
and (1.5.5.4)
]. As a result, there is a gain in the magnetic energy, which compensates the loss in the anisotropy energy. The beginning of the deflection is a second-order transition. The balance of both energies determines the value of
:
The second second-order transition occurs when
becomes equal to
at the critical field
:
After the reorientation transition, CoF2 has the same magnetic point group as the weak ferromagnet NiF2, i.e.
.
As we have seen, the appearance of weak ferromagnetism in antiferromagnets is closely connected with their magnetic symmetry. If the magnetic point group of the antiferromagnetic crystal contains an axis of higher than twofold symmetry, the magnetic structure is purely antiferromagnetic. By applying an external force that disturbs the symmetry of the crystal and destroys the axis of high symmetry, one may create a structure possessing weak ferromagnetism. In the previous section, we considered such reduction of the symmetry with the aid of a magnetic field applied perpendicular to the main axis of the crystal. Another possibility for symmetry reduction is to apply an external pressure and to deform the crystal. Thus, in some antiferromagnetic crystals, a ferromagnetic moment may be produced on application of external stress. This phenomenon is called piezomagnetism.
To investigate the piezomagnetic effect from the phenomenological point of view, we have to add the terms of the magnetoelastic energy in the expansion of the thermodynamic potential. The magnetoelastic terms of the least degree in the expansion of the thermodynamic potential for a given stable magnetic structure will be of the type
(
are the components of the elastic stress tensor
). These terms must be invariant relative to the crystallographic group of the material under examination. If we consider the potential
, which is a function of T,
,
, the terms of the magnetoelastic energy that are responsible for piezomagnetism are of the form
. Thus, for the piezomagnetic crystals the expansion of the thermodynamic potential should be expressed by
If at least one term of this expansion remains invariant under the magnetic symmetry of the given crystal, then the corresponding component
will not be zero and hence
Thus, when a stress
is applied, a magnetic moment is produced which is linear with the stress.
It follows from expression (1.5.7.1) that the converse of the piezomagnetic effect also exists, i.e. linear magnetostriction:
where
are the components of the deformation tensor.
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
,
,
. Hence, for such crystals all values of
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
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
(see Table 1.5.7.1
). The stress tensor
is symmetrical (
); see Section 1.3.2.4
. Thus the tensor
is symmetrical in its last two indices. This is the reason why piezomagnetism is prohibited for three more magnetic point groups:
,
and
. 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
.)
|