*2.2.2.2 Tilting*

*Electromagnetic Materials and Devices*

*2.2.2.1 Structure*

Ohsato et al. and Negas et al. reported the microwave dielectric properties for the Sm, Nd, Pr and La systems as a function of composition *x* as shown in the **Figure 8(a)** [20, 43, 45] and Fukuda et al. reported the Pr system [46]. On the solid solutions, the composition with *x* = 2/3 was found by Ohsato et al. [42], at which the *Qf* value becomes the highest due to the ordering in the rhombic and pentagonal sites. The dielectric constants *ε*r and *TCf* (**Figure 8(b)** and **(c)**) are decreased as a function of the composition *x* and are affected by volume and tilting angle of the TiO6 octahedra and the polarizabilities of *R* and Ba ions [20]. The Clausius-Mosotti equation determined the temperature coefficient of the dielectric constant *TCε*r as a function of the ratio of the mean radii (*r*a*/r*b) of *A*- and *B*-site ions by Valant et al. [47]. Hither, ra/rb is connected to the tilting of the TiO6 octahedra. In this study, on the system without order-disorder phase transition that is without symmetry change, it is discussed that

The crystal structure of the pseudo tungsten-bronze Ba6−3xR8+2xTi18O54 (*R* = rare earth) solid solutions [48–51] includes the perovskite blocks of 2 × 2 unit cells with rhombic (*A*1) sites and pentagonal (*A*2) sites, as shown in **Figure 9**, which are named after the tetragonal tungsten-bronze structure with 1 × 1 perovskite blocks and pentagonal sites [20, 48, 50]. On this compound, two large sites including Baand *R*-ions are placed such as *A*1 and *A*2. The Ba-ions engaged on the pentagonal *A*2-sites and *R*-ions *A*1-sites on the perovskite blocks. Two more sites, *B* and *C* are positioned on the tungsten-bronze crystal structure. The *B*-site is the same as the TiO6 octahedral place in the perovskite, and the *C*-site is a triangular site which is usually empty. This crystal structure of this compound has a special relationship with the perovskite structure. If the two ions are the same size, the structure will change to perovskite with only an *A*1-site owing to the combination of the

*Crystal structure of the pseudo tungsten-bronze solid solutions. Rhombic (A1) sites located in 2 × 2 unit cells of* 

*perovskite blocks, and pentagonal (A2) and trigonal sites (C).*

the ordering especially compositional ordering brings high *Qf*.

*2.2.2 Crystal structure of pseudo tungsten-bronze solid solutions*

**10**

**Figure 9.**

The structure has a super lattice along the *c*-axis with a double lattice of perovskite as shown in **Figure 7(b)** of an oscillation photograph with super diffraction lines [53, 54]. The crystal data of the fundamental lattice are as follows: orthorhombic crystal system of space group *Pbam* (No. 55), point group *mmm* and lattice parameters *a* = 12.13, *b* = 22.27, *c* = 3.82 Å, *Z* = 1 and *D*x = 5.91 g/cm3 . The super lattice is depending on the tilting of TiO6 octahedra as shown in **Figure 7(d)**. The tilting was endowed in the density map (**Figure 7(c)**) which is of the fundamental lattice superimposed on a superstructure framework. The top oxygen ions (O(1), O(4), O(6), O(8) and O(14)) of octahedra are separated into two along the *c*-axis. The left figure of **Figure 7(d)** shows the reason for splitting of the top oxygen [20]. However, this super lattice is not depending on the order-disorder phase transition as complex perovskite as explained at 2.3 section. The tilting of octahedra might be depending on the size of *A*-ion in the perovskite block.
