The Dielectric Constant of Ba6−3x(Sm1−yNdy)8+2xTi18O54 (x = 2/3) Ceramics for Microwave Communication by Linear Regression Analysis

The electronics related to the fifth generation mobile communication technology (5G) are projected to possess significant market potential. High dielectric constant microwave ceramics used as filters and resonators in 5G have thus attracted great attention. The Ba6−3x(Sm1−yNdy)8+2xTi18O54 (x = 2/3) ceramic system has aroused people’s interest due to its underlying excellent microwave dielectric properties. In this paper, the relationships between the dielectric constant, Nd-doped content, sintering temperature and the density of Ba6−3x(Sm1−yNdy)8+2xTi18O54 (x = 2/3) ceramics were studied. The linear regression equation was established by statistical product and service solution (SPSS) data analysis software, and the factors affecting the dielectric constant have been analyzed by using the enter and stepwise methods, respectively. It is found that the model established by the stepwise method is practically significant with Y = −71.168 + 6.946x1 + 25.799x3, where Y, x1 and x3 represent the dielectric constant, Nd content and the density, respectively. According to this model, the influence of density on the dielectric constant is greater than that of Nd doping concentration. We bring the linear regression analysis method into the research field of microwave dielectric ceramics, hoping to provide an instructive for the optimization of ceramic technology.


Introduction
With the development of mobile communication equipment and portable terminals towards miniaturization and multifunctionality, the demand for filters and resonators have been increasing steadily. Microwave dielectric ceramics are the key materials of filters and resonators for telecommunications [1][2][3][4][5]. The re are three important parameters to evaluate the ceramics: dielectric constant ε r , quality factor Qf and temperature coefficient of resonate frequency τ f . High dielectric constant is beneficial for microwave devices because the size of dielectrics is inversely proportional to √ ε r . Currently, BaO-Ln 2 O 3 -TiO 2 (Ln: lanthanide) based systems exhibit superior properties in the high dielectric materials, which have aroused considerable attention [6][7][8][9]. For the solid solution of Ba 6−3x (Sm 1−y Nd y ) 8+2x Ti 18 O 54 , it has a like-perovskite tungsten bronze structure with TiO 6 octahedra. According to the atomic occupation, the structural formula of the solid solution can be written into [Ln 8+2x Ba 2−3x V x ] A1 [Ba 4 ] A2 Ti 18 O 54 (0 ≤ x ≤ 2/3, V:Vacancy). A1 is the rhombic site and A2 is the pentagonal site [10,11]. Suzuki et al. attributed a significant drop in Qf to the occupation of large Ba by Sr at the A1-site [12]. Amaral et al. demonstrated that there exists evident crystal orientation in thick 1350 • C and 1400 • C for 2 h respectively with 10 • C/min of heating rate. When the heating schedule was finished, it was cooled with the furnace.

Ceramics Characterization
The crystal structures of the ceramics were characterized by X-ray diffraction using a Bruker AXS D8-Focus diffractometer (Bruker Corporation, Karlsruhe, Germany) with Cu Kα radiation (λ = 1.540598 Å). The step size was 0.02 • and the scan rate was 5 • /min. The densities were tested by the DE-200M apparatus made by HONGTUO Instrument (Dongguan, China). The measurement method of ceramic density was based on the Archimedes principle. The immersion liquid was the water. The formula for calculating the bulk density of the sample is according to Equation (1).
ρ is bulk density of the ceramic, g/cm 3 ; ρ w is the density of the water at the temperature of the test, g/cm 3 ; m 0 is the weight of the ceramic after drying at 105 • C for 3 h, g; m 1 is the weight of the ceramic in air with water saturation after vacuum-pumping, g, and m 2 is the weight of the ceramic in water with water saturation after vacuum-pumping, g. The morphologies of the ceramics were observed with a scanning electron microscope (SEM, SU8010, Hitachi, Tokyo, Japan). The microwave dielectric properties were measured using the Hakki and Coleman method by the Agilent E8362B network vector analyzer (Agilent, Palo Alto, USA). The spectroscopy frequency range is from 10 MHz to 20 GHz, and the resonant frequencies of all samples made in our study were lower than 10 GHz. The temperature range was from 20 to 80 • C for determining the temperature coefficient of the resonance frequency τ f .

Establishment of the Mathematical Models Based on the SPSS Multiple Linear Regression Analysis
The resonate frequency f, dielectric constant εr and quality factor Qf were measured at microwave frequencies by the Hakki and Coleman method. For the specific ceramics, the resonate frequency is constant, so the dielectric constant value is very reliable. The mathematical models of the dielectric constant were established, in which Y, x1, x2 and x3 denote the dielectric constant, Nd content relative to Sm, sintering temperature (°C) and the density (g/cm 3 ), respectively. Table 1 summarizes the densities and dielectric constants of the ceramics with different Nd contents and sintered at different temperatures. The Nd contents x1 varied from 0 to 0.4, while the sintering temperatures x2 from 1300 to 1400 °C at which the samples were fired into ceramics. According to Table 1, the multiple linear regression mathematical models were established by using SPSS software.

Establishment of the Mathematical Models Based on the SPSS Multiple Linear Regression Analysis
The resonate frequency f, dielectric constant ε r and quality factor Qf were measured at microwave frequencies by the Hakki and Coleman method. For the specific ceramics, the resonate frequency is constant, so the dielectric constant value is very reliable. The mathematical models of the dielectric constant were established, in which Y, x 1 , x 2 and x 3 denote the dielectric constant, Nd content relative to Sm, sintering temperature ( • C) and the density (g/cm 3 ), respectively. Table 1 summarizes the densities and dielectric constants of the ceramics with different Nd contents and sintered at different temperatures. The Nd contents x 1 varied from 0 to 0.4, while the sintering temperatures x 2 from 1300 to 1400 • C at which the samples were fired into ceramics. According to Table 1, the multiple linear regression mathematical models were established by using SPSS software.

Linear Regression Model Based on the Enter Method
The enter method is also named as the forced enter method, while all of the arguments are introduced into the model simultaneously. Generally, the enter method is suitable to find out the significance of the argument. By fitting the enter method, the arguments that are nonsignificant will be found. Table 2 is the summary of the regression model based on the enter model. The parameters were obtained with the correlation coefficient R = 0.973 and coefficient of determination R 2 = 0.947. R 2 refers to the fitting degree, which indicates the quality of the model. The closer R 2 is to 1, the more appropriate the model will be. The adjusted multiple correlation coefficient R a 2 is 0.943, which confirms that there are strong linear correlations between dielectric constants and Nd content, sintering temperature and the density of the ceramics [26]. The F-value (joint hypotheses test) was 243.722, and p-value (probability value under the corresponding F-value) was 0.000, which further confirmed that our model was suitable. The p-value was smaller than 0.05, which indicates that the linear regression equation had passed the 0.05 alpha-level significance test. From Table 2, the Durbin-Watson test value (DW) was 2.112, which means that the model does not have self-correlation [27]. The regression equation (Equation (2)) can be obtained by the regression coefficient shown in Table 3: Notes: B-Non standard regression coefficient; Beta-Standard regression coefficient; t-Hypothesis test of partial regression coefficient; p-value-Probability value under the corresponding F value; VIF-Variance inflation factor (when it is larger than 10, the re is serious multicollinearity).
According to the regression Equation (2), when the density increases by 1 g/cm 3 , the dielectric constant will increase by 23.568. When the sintering temperature increases by 100 • C, the dielectric constant will increase by 10.7. When Nd content is increased by 1% relative to that of Sm, the dielectric constant will enhance by 0.07347.
Both the arguments x 1 and x 3 have passed the significance test for the confidence level α = 0.05. However, the p-value for x 2 was 0.073, which is greater than 0.05. This suggests that x 2 should not be an argument in this linear regression model, which will be corroborated by the stepwise regression method as shown in our following discussion. From Equation (2), the most important impact factor on Y is x 3 , indicating that the density is the most important parameter affecting the dielectric constant. The relationship between densities and dielectric constants is also verified from Table 1. It is found that density plays a more important role than temperature. In fact, the density is related to sintering temperature. When the temperature was lower than the optimized sintering temperature range, the density gradually increased to the maximum value with increasing temperature. While the temperature was beyond the optimized sintering temperature range, the ceramics were over-fired, and consequently, the densities decreased. However, in this model, the re were no significance correlations between temperatures and dielectric constants. We thought that the impact of temperatures might be included in the density factor. If fired at lower temperatures, the ceramics will not be densified and deteriorate the densities. The refore, temperature and density should not be considered concurrently as influencing factors. From Table 3, the VIFs (variance inflation factor) of three arguments x 1 , x 2 and x 3 were less than 10, which suggests that there was no multicollinearity among them [28,29].
In Table 3, Beta is the standardization coefficient and B is the non standardization coefficient. In SPSS, the standardization is to eliminate the error caused by different units among arguments and dependent variable. The data standardization method is that the value of original data subtracts the mean of the corresponding variable and then divides by the standard deviation of the variable. The calculated regression equation is called the standardized regression equation, and the corresponding regression coefficient is the standardized regression coefficient. The relationship between the non standardized coefficient B and standardized coefficient Beta can be expressed by Equation (3): Beta is the standardized coefficient; B is the non standardized coefficient; σ x is the standard deviation of an argument and σ y is the standard deviation of the dependent variable.
The standardized coefficient can be used to evaluate which one is more important among all arguments. The non standardized coefficient reflected the absolute effect of the change of an argument on the dependent variable. From Table 3, the maximum Beta was 0.827 for x 3 , so among x 1 , x 2 and x 3 , x 3 was the most important for the dielectric constant. Figure 2 is the residual frequency distribution histogram and residual normal probability plot based on the enter method. Figure 2a is the normal distribution, and the dots were nearly on a straight line in Figure 2b, which shows that the mathematical model established by the enter method had passed the error test of normality. Through the Shapiro-Wilk test, the level of significance was more than 0.05, so this further proved the model was suitable.
Materials 2020, 13, x FOR PEER REVIEW 6 of 12 is found that density plays a more important role than temperature. In fact, the density is related to sintering temperature. When the temperature was lower than the optimized sintering temperature range, the density gradually increased to the maximum value with increasing temperature. While the temperature was beyond the optimized sintering temperature range, the ceramics were over-fired, and consequently, the densities decreased. However, in this model, there were no significance correlations between temperatures and dielectric constants. We thought that the impact of temperatures might be included in the density factor. If fired at lower temperatures, the ceramics will not be densified and deteriorate the densities. Therefore, temperature and density should not be considered concurrently as influencing factors. From Table 3, the VIFs (variance inflation factor) of three arguments x1, x2 and x3 were less than 10, which suggests that there was no multicollinearity among them [28,29].
In Table 3, Beta is the standardization coefficient and B is the non standardization coefficient. In SPSS, the standardization is to eliminate the error caused by different units among arguments and dependent variable. The data standardization method is that the value of original data subtracts the mean of the corresponding variable and then divides by the standard deviation of the variable. The calculated regression equation is called the standardized regression equation, and the corresponding regression coefficient is the standardized regression coefficient. The relationship between the non standardized coefficient B and standardized coefficient Beta can be expressed by Equation (3): Beta is the standardized coefficient; B is the non standardized coefficient; σx is the standard deviation of an argument and σy is the standard deviation of the dependent variable.
The standardized coefficient can be used to evaluate which one is more important among all arguments. The non standardized coefficient reflected the absolute effect of the change of an argument on the dependent variable. From Table 3, the maximum Beta was 0.827 for x3, so among x1, x2 and x3, x3 was the most important for the dielectric constant. Figure 2 is the residual frequency distribution histogram and residual normal probability plot based on the enter method. Figure 2a is the normal distribution, and the dots were nearly on a straight line in Figure 2b, which shows that the mathematical model established by the enter method had passed the error test of normality. Through the Shapiro-Wilk test, the level of significance was more than 0.05, so this further proved the model was suitable.

Linear Regression Model Based on the Stepwise Method
The stepwise method is totally different from the enter method. In the stepwise method, the arguments are introduced into the model one by one. The F-value test was performed at each step to

Linear Regression Model Based on the Stepwise Method
The stepwise method is totally different from the enter method. In the stepwise method, the arguments are introduced into the model one by one. The F-value test was performed at each step to ensure that only significant arguments were included in the regression equation before the new argument was introduced. Table 4 is the summary of the regression model based on the stepwise method. Model 1 is the transition model and Model 2 is the final model. In the process of simulation, the x 3 argument was firstly chosen by the software, leading to the strongest linear correlation between x 3 and Y. After the x 1 argument was chosen, the linear regression model was established between Y and x 1 and x 3 , while x 2 was eliminated. For the mathematical model established by the enter method, the argument x 2 did not pass the test of normality. For the model established by the stepwise method, x 2 was eliminated and not introduced into the model. As mentioned above, as the impact of the temperature on the dielectric constant has already been included in the density, we did not need to consider the influence of both temperature and density simultaneously. it had also no self-correlation. Since there was only one argument, the DW was zero. From Table 5, the linear regression model can be obtained by the stepwise regression method as follow (Equation (4)): Y= −71.168 + 6.946x 1 + 25.799x 3 , Notes: B-Non standard regression coefficient; Beta-Standard regression coefficient; t-Statistics for test; p-value-Probability value under the corresponding F value; VIF-Variance inflation factor (when it is larger than 10, the re is serous multicollinearity); x 3 -Density; x 1 -Nd content.
The probability value p-value of both x 1 and x 3 were 0.000, which shows that the model had passed the significance test and the arguments x 1 and x 3 had statistical significance. The VIF values were 1.029 for x 1 and x 3 , and there was no multicollinearity between the arguments. Comparing the Beta value, it was found that the Beta value (0.906) of x 3 was much greater than that (0.229) of x 1 , which implies that x 3 was more important than x 1 . Figure 3 is the residual frequency distribution histogram and residual normal probability plot obtained by the stepwise method. As shown in Figure 3a, the residual frequency distribution histogram met the normal distribution. Figure 3b shows that almost all the points were on a line. By the Shapiro-Wilk test, the level of significance was more than 0.05. We thus concluded that the hypothesis about the error normality was reasonable and the model passed the test of the error normality.
Materials 2020, 13, x FOR PEER REVIEW 8 of 12 histogram met the normal distribution. Figure 3b shows that almost all the points were on a line. By the Shapiro-Wilk test, the level of significance was more than 0.05. We thus concluded that the hypothesis about the error normality was reasonable and the model passed the test of the error normality.

Comparison of Two Regression Models
For more than one argument, adjusted multiple correlation coefficient Ra 2 can be used to evaluate the quality of the model. The closer the value is to 1, the better the model is. In this study, Ra 2 were 0.943 and 0.940 for the enter regression model and the stepwise model, respectively. Therefore, both models were of high quality and could be used to explain the relationship between the dependent variable and the arguments. For the enter method, the argument x2 did not pass the significance test, which shows that temperature had little effect on the dielectric constant. As mentioned above, because there were close relationships between temperature and the density, it was not appropriate for them to be considered as impact factors at the same time. For Model 2 established by the stepwise method without x2, although it includes fewer arguments without all the information, it meets the purpose that the reasonable, simply and useful regression models should be established with the most suitable and the least arguments [27].

Guidance of the Model on Other Properties of Microwave Dielectric Ceramics
For microwave dielectric ceramics there are three important performances (1) dielectric constant εr, (2) quality factor Qf and (3) the temperature coefficient of resonant frequency τf. For the BaO-Ln2O3-TiO2 system, there was a generally recognized correlation between the three performances, that is the temperature coefficient τf was positively correlated and quality factor Qf was negatively correlated with the dielectric constant, which had been confirmed by many researches [19,21,30,31]. In this study, the same correlations among the three performances were also obtained, when referring to Figure 4. While y (i.e., molar content of Nd to Sm) from 0 to 0.4, εr and τf increased and Qf decreased with y. The suitable model was established for evaluating the effect of process parameters on dielectric constant by linear regression analysis. So, the effect of the same parameters on the quality factor and temperature coefficient could be estimated by the established model. According to the stepwise model, there was a positive correlation between Nd content and the dielectric constant, so there was also a positive correlation between Nd content and τf, but a negative correlation for Qf. Due to the small number of samples, there was no linear regression analysis on the quality factor and temperature coefficient of ceramics in this study. Especially for the temperature coefficient test, it would take a lot of time to get enough data. In the future work, we will further study Qf and τf by linear regression analysis.

Comparison of Two Regression Models
For more than one argument, adjusted multiple correlation coefficient R a 2 can be used to evaluate the quality of the model. The closer the value is to 1, the better the model is. In this study, R a 2 were 0.943 and 0.940 for the enter regression model and the stepwise model, respectively. The refore, both models were of high quality and could be used to explain the relationship between the dependent variable and the arguments. For the enter method, the argument x 2 did not pass the significance test, which shows that temperature had little effect on the dielectric constant. As mentioned above, because there were close relationships between temperature and the density, it was not appropriate for them to be considered as impact factors at the same time. For Model 2 established by the stepwise method without x 2 , although it includes fewer arguments without all the information, it meets the purpose that the reasonable, simply and useful regression models should be established with the most suitable and the least arguments [27].

Guidance of the Model on Other Properties of Microwave Dielectric Ceramics
For microwave dielectric ceramics there are three important performances (1) dielectric constant ε r , (2) quality factor Qf and (3) the temperature coefficient of resonant frequency τ f . For the BaO-Ln 2 O 3 -TiO 2 system, the re was a generally recognized correlation between the three performances, that is the temperature coefficient τ f was positively correlated and quality factor Qf was negatively correlated with the dielectric constant, which had been confirmed by many researches [19,21,30,31]. In this study, the same correlations among the three performances were also obtained, when referring to Figure 4. While y (i.e., molar content of Nd to Sm) from 0 to 0.4, ε r and τ f increased and Qf decreased with y. The suitable model was established for evaluating the effect of process parameters on dielectric constant by linear regression analysis. So, the effect of the same parameters on the quality factor and temperature coefficient could be estimated by the established model. According to the stepwise model, the re was a positive correlation between Nd content and the dielectric constant, so there was also a positive correlation between Nd content and τ f , but a negative correlation for Qf. Due to the small number of samples, the re was no linear regression analysis on the quality factor and temperature coefficient of ceramics in this study. Especially for the temperature coefficient test, it would take a lot of time to get enough data. In the future work, we will further study Qf and τ f by linear regression analysis.   Figure 4 is the microwave dielectric properties of ceramics sintered at 1400 °C with different Nd content. At 1400 °C, the ceramics showed better properties than those sintered at other temperature. The scanning electron microscopes (SEMs) are illustrated in Figure 5. From Figure 5a, the ceramics sintered at 1350 °C appeared as incomplete grain growth, while the ceramics at 1450 °C (see Figure 5c) show overheated and some grains were melted together, which had the tendency of secondary grain growth who led to the enlargement of internal pores. However, the ceramics sintered at 1400 °C present grains with complete growth and uniform size. So, only the properties of ceramics sintered at 1400 °C were chosen to be discussed. As shown in Figure 4a, with the increase of Nd content, the dielectric constants of ceramics increased gradually, which was due to the contributions of cell volume and ionic polarizability of Nd dopant. Ba6−3x(Sm1−yNdy)8+2xTi18O54 (x =2/3) solid solution is the like-perovskite tungsten bronze structure with the ABO3 type. The  Figure 4 is the microwave dielectric properties of ceramics sintered at 1400 • C with different Nd content. At 1400 • C, the ceramics showed better properties than those sintered at other temperature. The scanning electron microscopes (SEMs) are illustrated in Figure 5. From Figure 5a, the ceramics sintered at 1350 • C appeared as incomplete grain growth, while the ceramics at 1450 • C (see Figure 5c) show overheated and some grains were melted together, which had the tendency of secondary grain growth who led to the enlargement of internal pores. However, the ceramics sintered at 1400 • C present grains with complete growth and uniform size. So, only the properties of ceramics sintered at 1400 • C were chosen to be discussed. As shown in Figure 4a, with the increase of Nd content, the dielectric constants of ceramics increased gradually, which was due to the contributions of cell volume and ionic polarizability of Nd dopant. Ba 6−3x (Sm 1−y Nd y ) 8+2x Ti 18 O 54 (x = 2/3) solid solution is the like-perovskite tungsten bronze structure with the ABO 3 type. The superlattice exists in the direction of c axis, which is due to the tilting of the titanium-oxygen octahedron [32]. The size and filling degree of ions at the A site will affect the tilt degree of Ti-O octahedron, and then change the cell volume. Nd dopant will occupy the A site, so the substitution of Nd for Sm affects the tilt degree of octahedron and cell volume. According to Clausius-Mossotti Equation (5), the dielectric constant of ceramics is determined by the cell volume V m and the total ionic polarizability α D , namely [26]: Materials 2020, 13, x FOR PEER REVIEW 10 of 12 superlattice exists in the direction of c axis, which is due to the tilting of the titanium-oxygen octahedron [32]. The size and filling degree of ions at the A site will affect the tilt degree of Ti-O octahedron, and then change the cell volume. Nd dopant will occupy the A site, so the substitution of Nd for Sm affects the tilt degree of octahedron and cell volume. According to Clausius-Mossotti Equation (5), the dielectric constant of ceramics is determined by the cell volume Vm and the total ionic polarizability αD, namely [26]: The ionic polarizabilities of Nd 3+ and Sm 3+ were 5.01 Å 3 and 4.74 Å 3 respectively. So, the substitution of Nd 3+ at A site influences not only the cell volume Vm, but also the total ionic polarizability αD, which resulted in the change of dielectric constant εr. The dopant element Nd with a large ionic radius and large ionic polarizability relative to Sm will increase the unit cell volume and then lead to the enlargement of the octahedral B site occupied by Ti 4+ in the TiO6 octahedron, which accounted for the increases of the ionic electronic polarizability and then the dielectric constant. This was also confirmed by Wang et al. [15] and Valant et al. [33]. The dielectric constant is the synergistic effect of the unit cell volume and polarizability. Similarly, the lanthanum (La) element dopant with a large ionic radius of 1.36 Å and ionic polarizability of 6.03 Å 3 will also increase the dielectric constant of ceramics, but decrease the quality factor. The influence of the La dopant will be reported elsewhere. This study will benefit the prediction of the dielectric constant of ceramics with different dopants and guide the investigation of other ceramic systems.

Conclusions
Microwave dielectric ceramics with high dielectric constants were prepared. The two kinds of linear regression models were established and compared. The model built by the stepwise method had more practical significance than that by the enter method. The better model was Y = −71.168 + 6.946x1 + 25.799x3, where Y, x1 and x3 represent the dielectric constant, Nd content and the density, The ionic polarizabilities of Nd 3+ and Sm 3+ were 5.01 Å 3 and 4.74 Å 3 respectively. So, the substitution of Nd 3+ at A site influences not only the cell volume V m , but also the total ionic polarizability α D , which resulted in the change of dielectric constant ε r . The dopant element Nd with a large ionic radius and large ionic polarizability relative to Sm will increase the unit cell volume and then lead to the enlargement of the octahedral B site occupied by Ti 4+ in the TiO 6 octahedron, which accounted for the increases of the ionic electronic polarizability and then the dielectric constant. This was also confirmed by Wang et al. [15] and Valant et al. [33]. The dielectric constant is the synergistic effect of the unit cell volume and polarizability. Similarly, the lanthanum (La) element dopant with a large ionic radius of 1.36 Å and ionic polarizability of 6.03 Å 3 will also increase the dielectric constant of ceramics, but decrease the quality factor. The influence of the La dopant will be reported elsewhere. This study will benefit the prediction of the dielectric constant of ceramics with different dopants and guide the investigation of other ceramic systems.

Conclusions
Microwave dielectric ceramics with high dielectric constants were prepared. The two kinds of linear regression models were established and compared. The model built by the stepwise method had more practical significance than that by the enter method. The better model was Y = −71.168 + 6.946x 1 + 25.799x 3 , where Y, x 1 and x 3 represent the dielectric constant, Nd content and the density, respectively, which disclosed the relationship between the dielectric constants and Nd content and the density of Ba 6−3x (Sm 1−y Nd y ) 8+2x Ti 18 O 54 (x = 2/3) microwave dielectric ceramics. Based on the model, it is clear that the density (x 3 ) was more important to the dielectric constant (Y) than Nd content (x 1 ), owing to the standardized coefficient Beta value that was 0.906 for x 3 but 0.229 for x 1 , while a large Beta value indicates that it was more significant. This model will help to instruct how to optimize the preparation technology in order to obtain high dielectric constant microwave ceramics.