Noncollinear spin structure

For the previous examples it was assumed that the magnetization of the system is oriented either parallel or antiparallel to a common axis. There are many cases where 

Apart from the disordered noncollinear spin structures there are also ordered ones. A prominent example of these is the transition-metal compound y-FeMn, which has a magnetic unit cell that is not much larger than the chemical one. On the other hand, magnetic spiral structures, which are quite common in rare earth metals, can possess very large or even infinite magnetic unit cells. 

Most electronic structure calculations for noncollinear spin structures have so far been done assuming spherical symmetric potential terms within an atomic cell. During the last few years a number of computational schemes have been developed that take the noncollinearity of the intra-atomic magnetization in atoms (Eschrig and Servedio 

Due to the inclusion of the spin-orbit coupling there is no need to distinguish between rotations in real and spin space. As can be seen in Table 5.2, the operations c r can be accompanied by the time reversal T. Because the operator T reverses the direction of the magnetic moments, time reversal T itself is of course no symmetry operation. 

Ifeble 5.4 Deviation angles of spin, orbital and total moment calculated with the MASW method. Hie last column gives experimental values (Gukasov et al. 1996 Wisniewski et al. 1999). 

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The noncollinear spin structures due to the pinning of the surface spins... 

If the direction of the spins is not uniform in space, we are dealing with noncollinear magnetism. Noncollinear spin structures appear, e.g., as canted or helical spin configurations in rare-earth compounds, as helical spin-density waves, or as domain walls in fer-romagnets. To describe these, one requires a formulation of SDFT in which the spin magnetization is not a scalar, as above, but a three-component vector m(r). Different proposals for extending SDFT to this situation are available. 

Defects in ferrimagnetic structures often lead to noncollinear (canted) spin structures. For example, a diamagnetic substitution or a cation vacancy can result in magnetic frustration which leads to spin-canting such that a spin may form an angle 6c with the collinear spins in the sample [80, 81]. Similarly, the reduced number of neighbor ions at the surface can also lead to spin-canting [80-83]. 

Figure 1. Spin structures (schematic) (a) ferromagnetism, (b-c) antiferromagnetism, and (d) noncollinear structure. The shown structure of the Ll0 type the small atoms (with the large magnetization arrows) the iron-series transition-metal atoms, as compared to the bigger 4d/4f atoms. Examples of Ll0 magnets are CoPt and FePt.

Antiferromagnetic sheets, but coupling between sheets gives noncollinear, canted spin structure. The two spin directions alternate in successive sheets along the c axis. Spin vectors lie in y-z plane at alternately clockwise and counterclockwise angles of 25 =fc 2° from the b axis. 

Mossbauer spectroscopy is also able to give local moment orientations, with respect to the crystalline lattice, or the correlations between moment orientations and local distortion axis orientations in a chemically disordered or amorphous material. This arises from the interplay between the structural (electric field gradient) hyperfine parameters and the magnetic hyperfine parameters. In this way, the spin flop Morin transition of hematite, for example, is easily detected and characterized (e.g., Dang et al. 1998). The noncollinear magnetic structures of nanoparticles can also be characterized. 

Classic spin structures corresponding to ferromagnetic (FM) anti-ferromagnetic (AF) ferrimagnetic (FI) weak ferromagnetic (WF) noncollinear antiferromagnetic (NC-AF) spin glass (SG)... 

F.6.3.3. Spin Canting Effects. Mossbauer spectroscopy is appropriate for detecting a noncollinear magnetic structure. If an external field is applied parallel to the y-ray beam, and if all the spins are collinear with this field, then the Am = 0 transitions are forbidden (see Ap.5). The second and fifth lines of a six-line pattern will be absent. Conversely, nonzero second and fifth lines show that a noncollinearity is present in the sample. 

Often, this takes the form of noncollinear spin ordering. The best known example is that for the triangular lattice where the so-called 120° structure is often found. This is shown in Figure 2.5 where a chiral degeneracy is indicated. There is one report of the corresponding 109° structure. Figure 2.6, in a pyrochlore lattice material, the metastable, pyr-FeFs-f " Other more extreme examples will be discussed in sections which follow. 

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