**2. Focused studies**

#### **2.1 Indialite/cordierite glass ceramics**

#### *2.1.1 Indialite Q-factor improved by Ni-substitution*

Cordierite (Mg2Al4Si5O18) has two polymorphs: cordierite and indialite, as shown in **Figure 1(a)** and **(c)**, respectively [30, 31]. Cordierite is of low symmetry form: orthorhombic crystal system *Cccm* (No. 66), which has Si4Al2O18 six-membered tetrahedron rings with ordered SiO4 and AlO4 tetrahedra as shown in **Figure 1(b)**. On the other hand, indialite is of high symmetry form: hexagonal crystal system *P*6/*mcc* (No. 192), which has disordered Si4Al2O18 equilateral hexagonal rings as shown in **Figure 1(c)**.

Cordierite shows a lower *ε*r of 6.19 which depends on the silicates and a near-zero *TCf* of −24 ppm/°C [32] as compared to other silicates as shown in **Figure 2(a)**. Based on these properties, Terada et al. carried out initiative research on these microwave dielectrics [16]. They reported an excellent *Qf* by substituting Ni for Mg as shown in **Figure 2(b)**. The *Qf* was improved from 40 × 103 GHz to 100 × 103 GHz by Ni substitution of *x* = 0.1 in (Mg1−*<sup>x</sup>*Ni*x*)2Al4Si5O18. The Ni substitution did not change the *ε*r value

**5**

*Dielectric Losses of Microwave Ceramics Based on Crystal Structure*

considerably, but the *TCf* was degraded from −24 to −30 ppm/°C [16]. For *x* > 0.1, the

*Crystal structure of Ni-substituted cordierite: (Mg1−xNix)2Al4Si5O18 with composition x = 0 (a), 0.05 (b), 0.1* 

*Schematic representation of cordierite (a), six-membered tetrahedron ring with ordered SiO4 and AlO4 (b)* 

*Cordierite with near zero ppm/°C deviated from other compounds (a). Ni-substituted cordierite Qf (b), volume of AlO4 and SiO4 (c) and covalencies of Si-O and Al-O as a function of composition x (d).*

Terada et al. also analysed the crystal structure by the Rietveld method [33] to clarify the origin of the improved *Qf* value. The X-ray powder diffraction (XRPD) pattern was obtained by a multi-detector system (MDS) [34] in the synchrotron radiation "Photon Factory" of the National Laboratory for High Energy Physics in Tsukuba, Japan. **Figure 3(a)–(d)** shows the crystal structures of Ni-substituted cordierite (Mg1−*<sup>x</sup>*Ni*x*)2Al4Si5O18 with *x* = 0, 0.05, 0.1 and 0.15. The crystal structure showed a tendency to deform to indialite with high symmetry on the hexagonal ring composed of corner-sharing of (Si, Al)O4 tetrahedra in the *a*-*b* plane. Ni-substituted cordierite (Mg1−*<sup>x</sup>*Ni*x*)2Al4Si5O18 with composition *x* = 0.1

properties were affected by the formation of the secondary phase of NiAl2O4.

*DOI: http://dx.doi.org/10.5772/intechopen.82483*

**Figure 1.**

**Figure 2.**

**Figure 3.**

*(c) and 0.15 (d).*

*and indialite (c).*

*Dielectric Losses of Microwave Ceramics Based on Crystal Structure DOI: http://dx.doi.org/10.5772/intechopen.82483*

**Figure 1.**

*Electromagnetic Materials and Devices*

nondistortion and without strain.

defects, inclusions, density and distortion from stress.

coefficient of the dielectric constant). Among the dielectric properties, the most essential property is *Q* , the inversion of the dielectric loss (tan*δ*); thus *Q* = 1/ tan*δ*. The dielectric losses of microwave dielectrics should be small. So, most of the microwave dielectrics are paraelectrics with inversion symmetry *i*, while most of the electronic materials are ferroelectrics with spontaneous polarity showing substantial dielectric losses [11–13]. The microwave dielectrics attract attention as a high potential material, which have an over-well-proportional rigid crystal structure with symmetry. That is, the structure should be without electric defects,

Under the influence of an electric field, four types of polarisation mechanisms can occur in dielectric ceramics, that is, interfacial, dipolar, ionic and electronic. In general, the microwave dielectric properties such as *ε*r and *Q* are mostly influenced by ionic or electronic polarisation. The dielectric polarisation generates the dielectric losses in the presence of an electromagnetic wave. When the frequency is increased to millimetre-wave values, the dielectric losses may be increased or decreased depending on the polarisation mechanism. There are two kinds of losses: those depending on crystal structure and losses due to external factors. It was believed that the intrinsic losses are due to the ordering/disordering, symmetry and phonon vibration, while extrinsic losses are due to factors such as grain size,

In this chapter, the origins of high *Q* are discussed based on the intrinsic factors related to the crystal structure, such as symmetry, compositional ordering and compositional density. Although it has previously been believed that ordering based on the order-disorder phase transition brings high *Q* [14], the authors propose that it is primarily a high symmetry that leads to high *Q* [15]. The following focused studies relate to specific examples; indialite with high symmetry showing higher *Q* than cordierite with an ordered structure [16–18]; pseudo tungsten-bronze solid solutions without phase transition showing high *Q* based on the compositional ordering [19–21]; complex perovskite compounds with order-disorder transitions depending on density and grain size [22, 23] and complex perovskites with composition deviated from the stoichiometric depending on the compositional density showing

Cordierite (Mg2Al4Si5O18) has two polymorphs: cordierite and indialite, as shown in **Figure 1(a)** and **(c)**, respectively [30, 31]. Cordierite is of low symmetry form: orthorhombic crystal system *Cccm* (No. 66), which has Si4Al2O18 six-membered tetrahedron rings with ordered SiO4 and AlO4 tetrahedra as shown in **Figure 1(b)**. On the other hand, indialite is of high symmetry form: hexagonal crystal system *P*6/*mcc* (No. 192), which has disordered Si4Al2O18 equilateral hexagonal rings as

Cordierite shows a lower *ε*r of 6.19 which depends on the silicates and a near-zero

tution of *x* = 0.1 in (Mg1−*<sup>x</sup>*Ni*x*)2Al4Si5O18. The Ni substitution did not change the *ε*r value

GHz to 100 × 103

GHz by Ni substi-

*TCf* of −24 ppm/°C [32] as compared to other silicates as shown in **Figure 2(a)**. Based on these properties, Terada et al. carried out initiative research on these microwave dielectrics [16]. They reported an excellent *Qf* by substituting Ni for Mg as shown

**4**

a high *Q* [24–29].

**2. Focused studies**

shown in **Figure 1(c)**.

**2.1 Indialite/cordierite glass ceramics**

*2.1.1 Indialite Q-factor improved by Ni-substitution*

in **Figure 2(b)**. The *Qf* was improved from 40 × 103

*Schematic representation of cordierite (a), six-membered tetrahedron ring with ordered SiO4 and AlO4 (b) and indialite (c).*

**Figure 2.**

*Cordierite with near zero ppm/°C deviated from other compounds (a). Ni-substituted cordierite Qf (b), volume of AlO4 and SiO4 (c) and covalencies of Si-O and Al-O as a function of composition x (d).*

#### **Figure 3.**

*Crystal structure of Ni-substituted cordierite: (Mg1−xNix)2Al4Si5O18 with composition x = 0 (a), 0.05 (b), 0.1 (c) and 0.15 (d).*

considerably, but the *TCf* was degraded from −24 to −30 ppm/°C [16]. For *x* > 0.1, the properties were affected by the formation of the secondary phase of NiAl2O4.

Terada et al. also analysed the crystal structure by the Rietveld method [33] to clarify the origin of the improved *Qf* value. The X-ray powder diffraction (XRPD) pattern was obtained by a multi-detector system (MDS) [34] in the synchrotron radiation "Photon Factory" of the National Laboratory for High Energy Physics in Tsukuba, Japan. **Figure 3(a)–(d)** shows the crystal structures of Ni-substituted cordierite (Mg1−*<sup>x</sup>*Ni*x*)2Al4Si5O18 with *x* = 0, 0.05, 0.1 and 0.15. The crystal structure showed a tendency to deform to indialite with high symmetry on the hexagonal ring composed of corner-sharing of (Si, Al)O4 tetrahedra in the *a*-*b* plane. Ni-substituted cordierite (Mg1−*<sup>x</sup>*Ni*x*)2Al4Si5O18 with composition *x* = 0.1

(**Figure 3(c)**) was obviously closer to equilateral hexagonal rings compared to (Mg0. 95Ni0.05)2Al4Si5O18 (**Figure 3(b)**) and Mg2Al4Si5O18 (**Figure 3(a)**).

The transformation from cordierite to indialite, represented by the ratio of disordering between the SiO4 and AlO4 tetrahedra, is based on the volumes and covalencies of the SiO4 and AlO4 tetrahedra [35]. The volume was calculated using atomic coordinates obtained by Rietveld crystal structural analysis as shown above. The covalency (*fc*) of the cation-oxygen bond was estimated from the following equation [36].

$$f\_c = a\mathbf{s}^M\tag{1}$$

The empirical constants *a* and *M* depending on the inner-shell electron number 10 are 0.54 v.u. and 1.64, respectively [37], where *s* is the bond length obtaining from the following equation:

$$\mathfrak{s} = \left(\mathbb{R}/R\mathbb{1}\right)^{-N} \tag{2}$$

where, *R* is defined as the bond length, and *R*1 and N are the measured parameter reliant on the cation site and each cation-anion pair, respectively.

**Figure 2(c)** and **(d)** depicts the calculated volume and covalency of SiO4 and AlO4 octahedra, respectively. These figures show the phase changing from cordierite to indialite as substitution of Ni in the Mg site. In the cordierite Mg2Al4Si5O18 (**Figure 1(a)**), Si/Al ions in the tetrahedra are ordered. Therefore, the volume and covalency of tetrahedra are different values, but the values are becoming similar to the substitution of Ni in the Mg site. This is due to the disordering of Si/Al ion phase transition in the cordierite (**Figure 1(a)**) to indialite (**Figure 1(c)**). In the indialite, the disordered Si4Al2O18 equilateral hexagonal rings with 6-ford axis are the main framework as analysed by the Rietveld method as shown in **Figure 3(d)**. The improvement of *Qf* as shown in **Figure 2(b)** should be based on the disordering due to high symmetry instead of an ordering of SiO4 and AlO4 tetrahedra by orderdisorder transition. It is one example of high symmetry bringing a higher *Q* than ordering by the order-disorder transition [18].

#### *2.1.2 Indialite glass ceramics with high Q*

As described in the previous section, the *Qf* value of indialite derived by substituting Ni for Mg was improved to three times that of cordierite. Based on the new knowledge, Ohsato et al. proposed the synthesis of indialite with superior microwave dielectric properties [17]. The indialite, being a high-temperature form, could not be synthesised by the solid-state reaction because the order-disorder phase transition is hindered by the incongruent melting to form mullite and liquid. On the other hand, indialite is an intermediate phase during the crystallisation process from glass with a cordierite composition to cordierite, as shown in **Figure 4**.

Therefore, fabrication of indialite glass ceramics has been attempted [17, 39]. Although the indialite is a metastable phase transforming to cordierite at higher temperatures, it is a relatively stable phase which occurs in nature formed by the crystallisation of natural glass. As this occurrence is in India, the mineral was named indialite. Another phase of *μ*-cordierite precipitating in the early stage of the crystallisation of cordierite glass is *β*-quartz solid solutions. The naming of *μ*-cordierite is not correct because of the different crystal structure, so the name that should be used is *β*-quartz solid solutions [38].

The cordierite composition was melted at 1550°C and was cast into a cylindrical rod with the diameter *φ* = 10 mm and *l* = 30 mm in a graphite mould. In order to avoid fracture due to internal strain, the cast glass rod was annealed at 760°C below

**7**

**Figure 5.**

*sensitive test plate (b) and (c).*

*Dielectric Losses of Microwave Ceramics Based on Crystal Structure*

the glass transition point of 778°C [38]. The 10 mm diameter glass rod was cut to form a resonator with a height of 6 mm. The glass pellets were crystallised at temperatures in the range 1200–1470°C/10 and 20 h. The crystallised pellets had two problems: deformation by the formation of glass phase and cracking by anisotropic crystal growth from the surface (**Figure 5(a)**) [39]. **Figure 5(b)** and **(c)** shows photographs taken by a polarising microscope of a thin section of the crystallised samples. The needle-like crystals grown from the surface had an orientation with *c*-axis elongation. The microwave dielectric properties of the sample with cracking

*Polymorphism of cordierite: indialite is the high temperature/high symmetry form, and cordierite is the low temperature/low symmetry form. In addition, indialite is an intermediate phase during crystallisation from* 

**Figure 6(a)** shows the volume of indialite/cordierite examined by the Rietveld method [40], which is estimated with two phases such as indialite and cordierite. Hereabout, the residual % is compared to that of cordierite. At 1200°C, the precipitated phase of indialite was about 96.7%. The volume of indialite reduced as the temperature and to 17.1% (82.9% for cordierite) at 1400°C. **Figure 6(b)** and **(c)** shows the microwave dielectric properties of indialite/cordierite glass ceramics and

with Ni using the conventional solid-state reaction as previously described (**Figure 2(b)**) and is feasible for millimetre-wave dielectrics. The *Qf* values decreased as crystallisation temperature. In comparison with the amount of indialite as shown in **Figure 6(a)**

*Cracking of crystallised pellets (a) and anisotropic crystal growth of the pellets under the crossed polars with a* 

GHz at 1300°C/20 h [17]. This is

GHz obtained by substitution

*DOI: http://dx.doi.org/10.5772/intechopen.82483*

**Figure 4.**

*glass to cordierite.*

had a wide scattering range of the data [17, 39].

remarkably high *Qf* value of more than 200 × 103

much better than the highest *Qf* value of 100 × 103

*Dielectric Losses of Microwave Ceramics Based on Crystal Structure DOI: http://dx.doi.org/10.5772/intechopen.82483*

**Figure 4.**

*Electromagnetic Materials and Devices*

from the following equation:

(**Figure 3(c)**) was obviously closer to equilateral hexagonal rings compared to (Mg0.

The transformation from cordierite to indialite, represented by the ratio of disordering between the SiO4 and AlO4 tetrahedra, is based on the volumes and covalencies of the SiO4 and AlO4 tetrahedra [35]. The volume was calculated using atomic coordinates obtained by Rietveld crystal structural analysis as shown above. The covalency (*fc*) of the cation-oxygen bond was estimated from the following equation [36].

*fc* = *as<sup>M</sup>* (1)

The empirical constants *a* and *M* depending on the inner-shell electron number 10 are 0.54 v.u. and 1.64, respectively [37], where *s* is the bond length obtaining

*s* = (*R*/*R*1)<sup>−</sup>*<sup>N</sup>* (2)

where, *R* is defined as the bond length, and *R*1 and N are the measured param-

**Figure 2(c)** and **(d)** depicts the calculated volume and covalency of SiO4 and AlO4 octahedra, respectively. These figures show the phase changing from cordierite to indialite as substitution of Ni in the Mg site. In the cordierite Mg2Al4Si5O18 (**Figure 1(a)**), Si/Al ions in the tetrahedra are ordered. Therefore, the volume and covalency of tetrahedra are different values, but the values are becoming similar to the substitution of Ni in the Mg site. This is due to the disordering of Si/Al ion phase transition in the cordierite (**Figure 1(a)**) to indialite (**Figure 1(c)**). In the indialite, the disordered Si4Al2O18 equilateral hexagonal rings with 6-ford axis are the main framework as analysed by the Rietveld method as shown in **Figure 3(d)**. The improvement of *Qf* as shown in **Figure 2(b)** should be based on the disordering due to high symmetry instead of an ordering of SiO4 and AlO4 tetrahedra by orderdisorder transition. It is one example of high symmetry bringing a higher *Q* than

As described in the previous section, the *Qf* value of indialite derived by substituting Ni for Mg was improved to three times that of cordierite. Based on the new knowledge, Ohsato et al. proposed the synthesis of indialite with superior microwave dielectric properties [17]. The indialite, being a high-temperature form, could not be synthesised by the solid-state reaction because the order-disorder phase transition is hindered by the incongruent melting to form mullite and liquid. On the other hand, indialite is an intermediate phase during the crystallisation process from glass with a

Therefore, fabrication of indialite glass ceramics has been attempted [17, 39]. Although the indialite is a metastable phase transforming to cordierite at higher temperatures, it is a relatively stable phase which occurs in nature formed by the crystallisation of natural glass. As this occurrence is in India, the mineral was named indialite. Another phase of *μ*-cordierite precipitating in the early stage of the crystallisation of cordierite glass is *β*-quartz solid solutions. The naming of *μ*-cordierite is not correct because of the different crystal structure, so the name that should be used is *β*-quartz solid solutions [38]. The cordierite composition was melted at 1550°C and was cast into a cylindrical rod with the diameter *φ* = 10 mm and *l* = 30 mm in a graphite mould. In order to avoid fracture due to internal strain, the cast glass rod was annealed at 760°C below

eter reliant on the cation site and each cation-anion pair, respectively.

ordering by the order-disorder transition [18].

cordierite composition to cordierite, as shown in **Figure 4**.

*2.1.2 Indialite glass ceramics with high Q*

95Ni0.05)2Al4Si5O18 (**Figure 3(b)**) and Mg2Al4Si5O18 (**Figure 3(a)**).

**6**

*Polymorphism of cordierite: indialite is the high temperature/high symmetry form, and cordierite is the low temperature/low symmetry form. In addition, indialite is an intermediate phase during crystallisation from glass to cordierite.*

the glass transition point of 778°C [38]. The 10 mm diameter glass rod was cut to form a resonator with a height of 6 mm. The glass pellets were crystallised at temperatures in the range 1200–1470°C/10 and 20 h. The crystallised pellets had two problems: deformation by the formation of glass phase and cracking by anisotropic crystal growth from the surface (**Figure 5(a)**) [39]. **Figure 5(b)** and **(c)** shows photographs taken by a polarising microscope of a thin section of the crystallised samples. The needle-like crystals grown from the surface had an orientation with *c*-axis elongation. The microwave dielectric properties of the sample with cracking had a wide scattering range of the data [17, 39].

**Figure 6(a)** shows the volume of indialite/cordierite examined by the Rietveld method [40], which is estimated with two phases such as indialite and cordierite. Hereabout, the residual % is compared to that of cordierite. At 1200°C, the precipitated phase of indialite was about 96.7%. The volume of indialite reduced as the temperature and to 17.1% (82.9% for cordierite) at 1400°C. **Figure 6(b)** and **(c)** shows the microwave dielectric properties of indialite/cordierite glass ceramics and remarkably high *Qf* value of more than 200 × 103 GHz at 1300°C/20 h [17]. This is much better than the highest *Qf* value of 100 × 103 GHz obtained by substitution with Ni using the conventional solid-state reaction as previously described (**Figure 2(b)**) and is feasible for millimetre-wave dielectrics. The *Qf* values decreased as crystallisation temperature. In comparison with the amount of indialite as shown in **Figure 6(a)**

**Figure 5.**

*Cracking of crystallised pellets (a) and anisotropic crystal growth of the pellets under the crossed polars with a sensitive test plate (b) and (c).*

**Figure 6.**

*Amount of indialite (a) and microwave dielectric properties of crystallised indialite at 1200–1440°C for 10 (b) and 20 (c) hours.*

[39] and its *Qf* values as shown in **Figure 6(b)** and **(c)** [17], it is clear that the indialite glass ceramics present a higher *Qf* than that of cordierite. The *ε*r was the lowest among the silicates, about 4.7 as shown in **Figure 6(b)** and **(c)**, and the *TCf* was −27 ppm/°C as shown in **Figure 6(b)**. Therefore, from these figures, indialite shows a higher *Qf* than cordierite. This *TCf* value of −27 ppm/°C is better than that of other silicates having a low *TCf* of approximately −60 ppm/°C [39].

#### *2.1.3 Conclusions for indialite/cordierite glass ceramics*


### **2.2 Pseudo tungsten-bronze solid solutions: compositional ordering bringing high** *Q*

#### *2.2.1 Introduction*

The pseudo tungsten-bronze solid solutions Ba6−<sup>3</sup>*xR*8+2*x*Ti18O54 (*R* = rare earth) located on the tie-line of BaTiO3-*R*2Ti3O9 are shown in **Figure 7(a)** and have been utilised in mobile phones because of their high dielectric constant of 80–90 [20, 21]. This solid solution was first reported by Varfolomeev et al. [41], based on Nd and Sm systems. The composition ranges 0.0 < < *x* < < 0.7 for *R* = Nd and 0.3 < < *x* < < 0.7 for

**9**

**Figure 8.**

**Figure 7.**

*Dielectric Losses of Microwave Ceramics Based on Crystal Structure*

Sm [42] were reported by Ohsato et al. [19] and Negas et al. [43]. The composition range of the solid solutions becomes narrower with the decrease in the ionic radius of

*A part of the BaO-R2O3-TiO2 ternary phase diagram with pseudo tungsten-bronze type solid solutions (a). Oscillation photograph along c-axis of pseudo tungsten-bronze type solid solutions (b). Electron density map (Fourier map) of the fundamental structure superimposed on a superstructure framework (c) and TiO6 tilting octahedra along the c-axis on the super-lattice (d) deduced from the splitting of oxygen in the fundamental structure (c), and the splitting of oxygen atoms based on the tilting of octahedra as shown in left side figure of the fundamental lattice (d). Right side schematic figure: super structure produced by tilting octahedral (d).*

the *R*-ion, and Ga and Eu form only BaO·*R*2O3·4TiO2 composition [44].

*Qf (a), εr (b) and TCf (c) of Sm, Nd, Pr and La system as a function of composition x.*

*DOI: http://dx.doi.org/10.5772/intechopen.82483*

#### *Dielectric Losses of Microwave Ceramics Based on Crystal Structure DOI: http://dx.doi.org/10.5772/intechopen.82483*

#### **Figure 7.**

*Electromagnetic Materials and Devices*

**Figure 6.**

*and 20 (c) hours.*

[39] and its *Qf* values as shown in **Figure 6(b)** and **(c)** [17], it is clear that the indialite glass ceramics present a higher *Qf* than that of cordierite. The *ε*r was the lowest among the silicates, about 4.7 as shown in **Figure 6(b)** and **(c)**, and the *TCf* was −27 ppm/°C as shown in **Figure 6(b)**. Therefore, from these figures, indialite shows a higher *Qf* than cordierite. This *TCf* value of −27 ppm/°C is better than that

*Amount of indialite (a) and microwave dielectric properties of crystallised indialite at 1200–1440°C for 10 (b)* 

• Indialite/cordierite glass ceramics are one of the examples of high symmetry bringing a higher *Q* than ordering by order-disorder transition. Indialite glass ceramics with disordered high symmetry have higher *Qf* properties than

• Cordierite with substituted Ni for Mg synthesised by solid-state reaction

Rietveld crystal structure analysis showed that the cordierite was transformed

• A novel idea from glass ceramics suggested the fabrication of indialite as an intermediate phase. Glass ceramics crystallised at 1200°C were almost com-

• Indialite/cordierite crystallised from cordierite glass at 1300°C/20 h showed

The pseudo tungsten-bronze solid solutions Ba6−<sup>3</sup>*xR*8+2*x*Ti18O54 (*R* = rare earth) located on the tie-line of BaTiO3-*R*2Ti3O9 are shown in **Figure 7(a)** and have been utilised in mobile phones because of their high dielectric constant of 80–90 [20, 21]. This solid solution was first reported by Varfolomeev et al. [41], based on Nd and Sm systems. The composition ranges 0.0 < < *x* < < 0.7 for *R* = Nd and 0.3 < < *x* < < 0.7 for

lised at 1400°C were cordierite at 82.9% with a lower *Qf* of 80 × 103

good microwave dielectric properties of *ε*r = 4.7, *Qf* > 200 × 103

**2.2 Pseudo tungsten-bronze solid solutions: compositional ordering** 

to 100 × 103

GHz (**Figure 2(b)**).

GHz, and those crystal-

GHz.

GHz and

of other silicates having a low *TCf* of approximately −60 ppm/°C [39].

*2.1.3 Conclusions for indialite/cordierite glass ceramics*

cordierite with ordered low symmetry.

exhibited an improved *Qf* from 40 × 103

*TCf* = −27 ppm/°C (**Figure 6**) [17, 39].

pletely indialite at 96.7% with a high *Qf* of 150 × 103

to indialite [16].

(**Figure 6**) [17, 39].

**bringing high** *Q*

*2.2.1 Introduction*

**8**

*A part of the BaO-R2O3-TiO2 ternary phase diagram with pseudo tungsten-bronze type solid solutions (a). Oscillation photograph along c-axis of pseudo tungsten-bronze type solid solutions (b). Electron density map (Fourier map) of the fundamental structure superimposed on a superstructure framework (c) and TiO6 tilting octahedra along the c-axis on the super-lattice (d) deduced from the splitting of oxygen in the fundamental structure (c), and the splitting of oxygen atoms based on the tilting of octahedra as shown in left side figure of the fundamental lattice (d). Right side schematic figure: super structure produced by tilting octahedral (d).*

Sm [42] were reported by Ohsato et al. [19] and Negas et al. [43]. The composition range of the solid solutions becomes narrower with the decrease in the ionic radius of the *R*-ion, and Ga and Eu form only BaO·*R*2O3·4TiO2 composition [44].

**Figure 8.** *Qf (a), εr (b) and TCf (c) of Sm, Nd, Pr and La system as a function of composition x.*

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 ordering especially compositional ordering brings high *Qf*.
