Magnetic noncollinearity

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. 

Anton, J., Ericke, B. and Engel, E. (2004) Noncollinear and collinear relativistic density-functional program for electric and magnetic properties of molecules. Physical Review A, 69, 012505-1-012505-10. 

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]. 

If we refrain from such a restriction and consider a spin-operator-dependent Hamiltonian, such as the 4-component KS Hamiltonian or the Dirac-Coulomb Hamiltonian, the Hamiltonian does not commute with the square of the spin operator. The square of the spin operator and the Hamiltonian then do not share the same set of eigenfunctions, and hence spin is no longer a good quantum number. In this noncollinear framework we must therefore find a different solution and may define a spin density equal to the magnetization vector (32). 

As reported by Blanco et al. (1999), neutron diffraction patterns of powder and bulk polycrystalline samples of GdCu were obtained for both structures in the cubic CsCl type of structure, which orders antiferromagnetically at 7n 150 K, a propagation vector of ( j 0) has been found with the moments probably parallel to the c-axis (note that other noncollinear magnetic structures might give rise to the same neutron-diffraction pattern). In the orthorhombic low temperature phase (7n 45 K) the available diffraction patterns... 

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.

Fig. 34. Spontaneous magnetization vs. applied field for several spinels. Saturation attained by II = 20 kilo-oersteds for sample with collinear spins, top curve, but not attained by 140 kilo-oersteds for samples with noncollinear spins. (After pulsed-ficld data of Jacobs (295).)...

The A-B interactions are especially weak if the B-site eg orbitals are empty, and noncollincar spin configurations are anticipated if the B-site 3dn ions have n < 3. The common occurrence of noncollinear configurations is evident from Table XVI, where the saturation magnetizations of several spinels containing V4+ and Cr3+ are found to be incompatible with N el ordering. That the ratio JbhSbb/Ja b a... 

Figure 54 Noncollinear AF magnetic structures, 120° for the triangular lattice and 109° for the tetrahedral lattice. (Ref 110. Reproduced by permission of Royal Society of Chemistry)...

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. 

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