Commensurate magnetic structures

The matrices Mjg and phases can be deduced from the relations between the Fourier coefficients and atomic basis functions [Equation (54)]. The matrices Mp correspond, in the case of commensurate magnetic structures, to the rotational parts of the magnetic Shubnikov group acting on magnetic moments. 

Fig. 14.41. Th low temperature commensurate magnetic structure of TbAui below 42.5 K. The larger and smaller circles represent Tb and Au, respectively. The pertinent interatomic distances are given in Angstroms (Atoji, 1968a).

The effect of commensurate magnetic structures on the elastic properties... 

SC — superconducting, AFM — commensurate antifenomagnet structure, SDW — incommensurate antiferro-magnet order (spin density wave), WFM — weak ferromagnetism Tn — magnetic ordering temperature, Tc — superconducting transition temperature and N(Ep) —density of states at the Fermi level. 

Fig. 26. Different types of magnetic structures in the ground stale of ftNijI C compounds, (a) For R = Pr, Dy or Ho commensurate antiferromagnelic structure, (b, c and d) for R = Er, Tb and Tm incommensurate antiferromagnetic structures (spin density waves) with a propagation vector q in the (a, 6)-plane, (b) Moments in the (a, b) plane and X to q. (c) Moments in the (a, b) plane and q. (d) Moments c and X to q (after Lynn et al.

Fig. 39. The different magnetic structures of HoNi2 B2C as determined by neutron scattering, (a) Commensurate antiferromagnetic, (b) incommensurate restructure (spiral) with the modulation vector T2 = (0.0,0.916) and (c) proposal how the incommensurate a -structure looks like (Loewenhaupt et al. 1997). Its modulation vector is...

It is obvious that the commensurate antiferromagnetic structure of fig. 39a coexists with superconductivity in HoNi2B2C, similar as in DyNi2B2C. On the other hand, as can be seen in fig. 43(a and c) the superconductivity is suppressed in the small temperature range where the two incommensurate magnetic structures of fig. 39(b and c) occur. Now the question is which of these two structures is more relevant for the near-reentrant behaviour. In Y().i5Hoo.85Ni2B2C the situation is totally different (fig. 43(b and d)). Here the a ... 

Amici and Thalmeier (1998) used the quasi one-dimensional model mentioned in Section 4.9.1. In their approach the presence of ferromagnetically ordered Flo layers with the magnetic moments oriented perpendicular to the tetragonal c-axis is adopted and the competition of the RKKY interaction along the c-axis with the crystalline electric field is analyzed in order to determine the transition between the commensurate antiferromagnetic structure and the incommensurate c spiral shown in Figure 39. 

Figure 1. Crystal structure of CUB2O4. Positions of Cu(A), Cu(B), B(I) and B(II) sites are indicated. Arrows in the figure denote the magnetic structure in the commensurate phase reported in ref. [7],...

Contrary to most crystal structures, many magnetic structures are incommensurate the periodicity of the orientation of the magnetic moments is not commensurate with the underlying crystal structure. This is a consequence of... 

We may have defined Equation (44) in a slightly different manner as is usual in the literature. Instead of writing R in the argument of the exponential function, one can write R/ = R/ + [similarly to f,- = x vectors in Equation (29) for atom displacements]. In such a case the Fourier coefficients, Tiy-, of the new expression are related to those of Equation (44) by a phase factor, Sk/ = Tiyexp(—27rikxy), that depends on the atom positions inside the unit cell. We shall see that the convention we have adopted is more convenient for a unified description of commensurate and incommensurate magnetic structures. 

If the canting is not the same for each magnetic cation but varies in a regular way, then a number of commensurate or incommensurate spin structures can arise, including helical, helicoidal, cycloidal and sinusoidal. The helicoidal and sinusoidal ordering patterns are illustrated by the magnetic structure of TbMnOj (Section 7.10). 

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