Symmetries of Magnetic Crystals
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Symmetries of Magnetic Crystals
From the book : Fundamentals of the Physics of Solids . Volume 1  Structure and Dynamics
Symmetries of Magnetic Crystals :
In the foregoing we have analyzed spatial transformations that take a point (x, y, z) of a crystal into an equivalent point. These form the group of spatial transformations. When ferroelectric or magnetic materials are studied, the transformation of the electric and magnetic dipole moments need to be taken into account as well. As a simple example, A. V. Shubnikov (1951) studied the symmetries of a system in which there is a twovalued variable s = ±1 at each site (x, y, z).
Because of the new variable, a new operation (R) appears, which is called antisymmetry. This takes the point characterized by the variables (x, y, z,s) into (x, y, z,−s). Adding this operation to the customary symmetry operations leads to magnetic point groups and magnetic space groups or Shubnikov (space) groups. The two values (s = ±1) are often referred to as “black” and “white” – and then the group is called “blackandwhite group”.
Magnetic point groups contain rotations, reﬂections, and the antisymmetry operation. In addition to the 32 ordinary point groups of crystals (that do not contain the antisymmetry operation), another 32 appear which contain the antisymmetry operator R as a symmetry element. This is possible only when the values of s are both present in each lattice point. In magnetic terms this means that upward and downward magnetic moments are equally probable at each individual lattice point, and thus the average vanishes. This corresponds to the paramagnetic phase. Using black and white, these are the socalled gray groups. There are 58 other magnetic point groups that do not contain R itself only its combination with a rotation. They are the true magnetic groups in the sense that they may appear as points groups of ordered magnetic structures.
When translations are also taken into account, one may study magnetic (or blackandwhite) lattices instead of ordinary Bravaislattice types, in which all lattice points are equivalent. Besides displacements through translation vectors tn, operations of the form {Rv} are also allowed. Obviously, this may be asymmetry only when v is half a translation vector. In addition to the 14 ordinary Bravaislattice types – in which lattice sites are of the same color–, 22 further Bravaislattice types are found, which contain both black and white lattice sites. The new magnetic (or blackand white) lattice types with tetragonal and cubic structures are shown in Figs. 5.33 and 5.34.
Figure 5.33 shows that blackandwhite Bravais lattices are built up of two interpenetrating Bravais lattices, a black and a white. In the case of a simple tetragonal lattice the relative displacement vector of the two sublattices is either onehalf of the primitive vector along the fourfold axis, or the vector to the center of the base, or the vector to the center of the primitive cell. In a bodycentered tetragonal lattice there is just one possibility: the relative displacement vector of the two sublattices must be in the direction of the fourfold axis, and its magnitude must be onehalf of the height. This is equivalent to a translation of the second sublattice through half the face diagonal.
In a simple cubic Bravais lattice the two interpenetrating sublattices are displaced by half the space diagonal. Each black atom is surrounded by eight white atoms, and vice versa. In a facecentered cubic lattice the relative displacement is the half of either edge vector. In this case each black atom is surrounded by six white ones.
The combination of magnetic Bravais lattices and magnetic point groups gives rise to 1651 magnetic (or blackandwhite) space groups. Just like for point groups, 230 of them are ordinary space groups that do not contain the antisymmetry operation at all. There are the same number of paramagnetic (or gray) space groups, in which the antisymmetry operation is a symmetry element in itself. In the remaining 1191 space groups antisymmetry appears only in combination with rotations, reﬂections, or translations.
The possible symmetries of rea lmagnetic systems are even more complicated than that. Blackandwhite groups may, at most, purport to give the symmetry groups of magnetic systems that can be described by the Ising model, in which the magnetic moment is represented by a twovalued variable.
In the general case the magnetic moment vector points in diﬀerent directions on diﬀerent magnetic atoms, giving rise to noncollinear magnetic structures. Moreover, account must be taken of the fact that magnetic moments are also transformed by rotations. It should also be borne in mind that the magnetic moment m is an axial vector, which does not transform as a true vector under reﬂections. (This last property is easily understood when the magnetic moment is considered to be produced by a current loop. Upon reﬂection, the component perpendicular to the mirror plane does not change sign but the component in the plane does.) All this leads to a great wealth of possibilities for symmetries in magnetic systems thatcannotbelistedhere.The issue of possible magnetic structures will be brieﬂy discussed in Chapter 14 on magnetically ordered systems.
Symmetries of Magnetic Crystals :
In the foregoing we have analyzed spatial transformations that take a point (x, y, z) of a crystal into an equivalent point. These form the group of spatial transformations. When ferroelectric or magnetic materials are studied, the transformation of the electric and magnetic dipole moments need to be taken into account as well. As a simple example, A. V. Shubnikov (1951) studied the symmetries of a system in which there is a twovalued variable s = ±1 at each site (x, y, z).
Because of the new variable, a new operation (R) appears, which is called antisymmetry. This takes the point characterized by the variables (x, y, z,s) into (x, y, z,−s). Adding this operation to the customary symmetry operations leads to magnetic point groups and magnetic space groups or Shubnikov (space) groups. The two values (s = ±1) are often referred to as “black” and “white” – and then the group is called “blackandwhite group”.
Magnetic point groups contain rotations, reﬂections, and the antisymmetry operation. In addition to the 32 ordinary point groups of crystals (that do not contain the antisymmetry operation), another 32 appear which contain the antisymmetry operator R as a symmetry element. This is possible only when the values of s are both present in each lattice point. In magnetic terms this means that upward and downward magnetic moments are equally probable at each individual lattice point, and thus the average vanishes. This corresponds to the paramagnetic phase. Using black and white, these are the socalled gray groups. There are 58 other magnetic point groups that do not contain R itself only its combination with a rotation. They are the true magnetic groups in the sense that they may appear as points groups of ordered magnetic structures.
When translations are also taken into account, one may study magnetic (or blackandwhite) lattices instead of ordinary Bravaislattice types, in which all lattice points are equivalent. Besides displacements through translation vectors tn, operations of the form {Rv} are also allowed. Obviously, this may be asymmetry only when v is half a translation vector. In addition to the 14 ordinary Bravaislattice types – in which lattice sites are of the same color–, 22 further Bravaislattice types are found, which contain both black and white lattice sites. The new magnetic (or blackand white) lattice types with tetragonal and cubic structures are shown in Figs. 5.33 and 5.34.
Figure 5.33 shows that blackandwhite Bravais lattices are built up of two interpenetrating Bravais lattices, a black and a white. In the case of a simple tetragonal lattice the relative displacement vector of the two sublattices is either onehalf of the primitive vector along the fourfold axis, or the vector to the center of the base, or the vector to the center of the primitive cell. In a bodycentered tetragonal lattice there is just one possibility: the relative displacement vector of the two sublattices must be in the direction of the fourfold axis, and its magnitude must be onehalf of the height. This is equivalent to a translation of the second sublattice through half the face diagonal.
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In a simple cubic Bravais lattice the two interpenetrating sublattices are displaced by half the space diagonal. Each black atom is surrounded by eight white atoms, and vice versa. In a facecentered cubic lattice the relative displacement is the half of either edge vector. In this case each black atom is surrounded by six white ones.
The combination of magnetic Bravais lattices and magnetic point groups gives rise to 1651 magnetic (or blackandwhite) space groups. Just like for point groups, 230 of them are ordinary space groups that do not contain the antisymmetry operation at all. There are the same number of paramagnetic (or gray) space groups, in which the antisymmetry operation is a symmetry element in itself. In the remaining 1191 space groups antisymmetry appears only in combination with rotations, reﬂections, or translations.
The possible symmetries of rea lmagnetic systems are even more complicated than that. Blackandwhite groups may, at most, purport to give the symmetry groups of magnetic systems that can be described by the Ising model, in which the magnetic moment is represented by a twovalued variable.
In the general case the magnetic moment vector points in diﬀerent directions on diﬀerent magnetic atoms, giving rise to noncollinear magnetic structures. Moreover, account must be taken of the fact that magnetic moments are also transformed by rotations. It should also be borne in mind that the magnetic moment m is an axial vector, which does not transform as a true vector under reﬂections. (This last property is easily understood when the magnetic moment is considered to be produced by a current loop. Upon reﬂection, the component perpendicular to the mirror plane does not change sign but the component in the plane does.) All this leads to a great wealth of possibilities for symmetries in magnetic systems thatcannotbelistedhere.The issue of possible magnetic structures will be brieﬂy discussed in Chapter 14 on magnetically ordered systems.
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