Modulation vector

In the paramagnetic regime, the evolution of the EPR line width and g value show the presence of two transitions, observed at 142 and 61 K in the Mo salt, and at 222 and 46 K in the W salt. Based on detailed X-ray diffraction experiments performed on the Mo salt, the high temperature transition has been attributed to a structural second-order phase transition to a triclinic unit cell with apparition of a superstructure with a modulation vector q = (0,1/2, 1/2). Because of a twinning of the crystals at this transition, it has not been possible to determine the microscopic features of the transition, which is probably associated to an ordering of the anions, which are disordered at room temperature, an original feature for such centrosymmetric anions. This superstructure remains present down to the Neel... 

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...

In the case of semi-commensurability in one or two directions it is likely that, structurally, the two layer sets are not quite independent. Semi-coherent structural and/or compositional modulation is then present i.e. a cooperative periodic variation in the size and/or content of the component subcells. Each modulation vector (one only in Fig. 2a, two in Fig. 2 b) will be equal to (or a multiple or sub-multiple of) that of the coincidence net. For each modulated layer set, A or B, a true-structure (component) lattice and unit cell can then be defined, based on its modulation period or periods plus the basic vectors or vector in the direction(s) in which there is no visible modulation of the basic structure. The longer-range modulation pattern of the two layer sets is imposed on the short range approximate periodicities which, in turn, describe sub-motifs manifested as a subnet (or subcell) of each layer set. If, as may be the case, the separate component unit cells of each of the two sets are identical, then they are also the coincidence cell of the two sets (Fig. 2 b). In the more general case, when this is not so, the vectors of the coincidence net will be multiples of the identity vectors of the unit nets of the two layer sets (or some simple summations of them). 

In addition to one-dimensional modulations considered above, both two-and three-dimensional modulations are possible. Furthermore, atomic parameters affected by modulations may be one or several of the following positional (as shown in Figure 1.52 and Figure 1.53), occupancy, thermal displacement, and orientation of magnetic moments. The latter, i.e. commensurately or incommensurately modulated orientations of magnetic moments are quite common in various magnetically ordered structures (e.g. pure lanthanide metals such as Er and Ho), and both the value of the modulation vector and the amplitude of the modulation function often vary with temperature. 

Conventional lattices may be perturbed by functions with different periodicity, e.g. by sinusoidal or saw-tooth-like modulations, see section 1.21 in Chapter 1. In the simplest case (one-dimensional modulation), Eq. 5.1 becomes d / = h + kb + k + wq assuming that the perturbation function is periodic and has the modulation vector q. In a case of three-dimensional modulation, a total of six indices (k, k, I, m, n, and p) are required to identify every point observed in reciprocal space = ha + kb + /c +... 

Let us consider the simplest case. Suppose that we have a pure sinusoidal (Cq = 0) wave modulation with a single modulation vector q. The expression of the structure factor for the satellite reflection h = H + mq is reduced to ... 

Hence, the method of MD simulations is able to reproduce the characteristic modulations that occur in DCPS under specific temperature conditions. The modulation vector obtained by MD exceeds only slightly the experimental value measured by x-ray diffraction. This method was also successfully applied to other organic crystals results," reinforcing the results obtained by diffraction and confirming them by an independent method. 

In cases where Cj and fail to come into register at all, that is, when piq is irrational, or at least when q is large, the stmctures are called incommensurate modulated composite structures. In order to analyse these incoimnensurate structures, a more complex superspace description needs to be adopted, which involves the introduction of a modulation vector q =yc, where c is a vector corresponding to the basic repeat along the c-axis, corresponding to Cj. 

A number of other copper-containing and non-copper-containing phases have structures related to these cuprate phases. The oxide Sr (Feg Co )Ojj g has a complex modulated structure in which the modulation vector is dependent on the... 

Neutron diffraction measurements have, in fact, identified two incommensurate antiferromagnetic structures, one with a long modulation vector gj = (0.28, 0.28,... 

Figure 15. Interlayer match in cylindrite, FePb n Sb. f4 (Makovicky, 1974). The two component layers, Q and H, are indicated by their submesh (pseudotetragonai and orthohexagonal denoted by the two grids with their centrations partly omitted) and by the two unit cells. The bx c (incommensurate x semicommensurate) coincidence mesh is stippled modulation vector is [001].

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