Low temperature sintering and dielectric properties of Ba<Subscript>3</Subscript>(VO<Subscript>4</Subscript>)<Subscript>2</Subscript> microwave ceramics using Co<Subscript>2</Subscript>O<Subscript>3</Subscript> additives

The Ba₃(VO₄)₂–x wt% Co₂O₃ (x?=?0.5–5) ceramics were prepared by the solid state reaction method in order to reduce the sintering temperature. The effects of the Co₂O₃ additions on the phase composition, microstructures, sintering characteristics and microwave dielectric properties of Ba₃(VO₄)₂ ceramics are investigated by an X-ray diffractometer, a scanning electron microscope and a network analyzer. As a result, the Q?×?f value of 54,000 GH, the ε _r of 14.6 and the τ_f value of +58.5 ppm/°C were obtained in the sample of the Ba₃(VO₄)₂–3 wt% Co₂O₃ ceramic sintered at the temperature of 925 °C, which is capable to co-fire with electrode metal of high conductivity such as Ag (961 °C). Moreover, the Q?×?f values of the sample with Co₂O₃ higher than that of 3 wt% additions decreased because of the formation of Ba₂V₂O₇ phase.     

Low temperature sintering and microwave dielectric properties of Li<Subscript>6</Subscript>Mg<Subscript>7</Subscript>Ti<Subscript>3</Subscript>O<Subscript>16</Subscript> ceramics with LiF additive for LTCC applications

Li₆Mg₇Ti₃O₁₆ ceramics were prepared by the conventional solid-state method with 1–5 wt% LiF as the sintering aid. Effects of LiF additions on the phase compositions, sintering characteristics, micro-structures and microwave dielectric properties of Li₆Mg₇Ti₃O₁₆ ceramics were investigated. The LiF addition could effectively lower the sintering temperature of Li₆Mg₇Ti₃O₁₆ ceramics from 1550 to 900 °C. For different LiF-doped compositions, the optimum dielectric permittivity (ε _r ) and quality factor (Q·f) values first increased and then decreased with the increase of LiF contents, whereas the temperature coefficient of resonant frequency (τ _f ) fluctuated between ??14.39 and ??8.21 ppm/°C. Typically, Li₆Mg₇Ti₃O₁₆ ceramics with 4 wt% LiF sintered at 900 °C exhibited excellent microwave dielectric properties of ε _r ?=?16.17, Q·f?=?80,921 GHz and τ _f ?=???8.21 ppm/°C, which are promising materials for the low temperature co-fired ceramics applications.     

Low temperature sintering and microwave dielectric properties of Zn<Subscript>3</Subscript>Mo<Subscript>2</Subscript>O<Subscript>9</Subscript> ceramic

Crystal structure and dielectric properties of Zn₃Mo₂O₉ ceramics prepared through a conventional solid-state reaction method were characterized. XRD and Raman analysis revealed that the Zn₃Mo₂O₉ crystallized in a monoclinic crystal structure and reminded stable up to1020 °C. Dense ceramics with high relative density (~ 92.3%) were obtained when sintered at 1000 °C and possessed good microwave dielectric properties with a relative permittivity (ε _r ) of 8.7, a quality factor (Q?×?f) of 23,400 GHz, and a negative temperature coefficient of resonance frequency (τ _f ) of around ??79 ppm/°C. With 5 wt% B₂O₃ addition, the sintering temperature of Zn₃Mo₂O₉ ceramic was successfully lowered to 900 °C and microwave dielectric properties with ε _r ?=?11.8, Q?×?f?=?20,000 GHz, and τ _f = ??79.5 ppm/°C were achieved.     

Tuning the temperature coefficient of resonant frequency for 8-layer twinned hexagonal perovskite Ba<Subscript>8</Subscript>ZnTa<Subscript>6</Subscript>O<Subscript>24</Subscript> ceramics

High-Q dielectric materials ilmenite MgTiO₃, columbite MgNb₂O₆ and cubic perovskite Ba₃NiTa₂O₉ with negative temperature coefficient of resonant frequency (τ _f ) were selected as candidates for compensating the τ _f of hexagonal perovskite Ba₈ZnTa₆O₂₄. X-ray diffraction data shows that Ba₈ZnTa₆O₂₄ coexists with Ba₃NiTa₂O₉ but is not compatible with MgTiO₃ and MgNb₂O₆ at high temperature. The τ _f for the mixed hexagonal/cubic perovskite Ba₈ZnTa₆O₂₄–Ba₃NiTa₂O₉ system is tunable via the temperature compensation effect and its quality factor may be improved via annealing the ceramics at high temperature to enhance the cation ordering in the cubic component. Permittivity ε _r  ~ 22–25, Q×f > 30,000 GHz and tunable τ _f within ±10 ppm/°C were achieved in the range of about 50–80 wt% Ba₃NiTa₂O₉ for the hexagonal/cubic perovskite composite Ba₈ZnTa₆O₂₄–Ba₃NiTa₂O₉ ceramics, which is suitable for the application as dielectric resonators and filters.     

Crystal structure,microstructure and microwave dielectric properties of novel MgAl<Subscript>2</Subscript>Ti<Subscript>3</Subscript>O<Subscript>10</Subscript> ceramic

In the present work, a novel MgAl₂Ti₃O₁₀ ceramic was obtained using a traditional solid-state reaction method. X-ray diffraction and energy dispersive spectrometer showed that the main MgAl₂Ti₃O₁₀ phase was formed after sintered at 1300–1450 °C. With rising the sintering temperature from 1300 to 1450 °C, the bulk density (ρ), relative permittivity (ε _r ) and Q?×?f value firstly increased, reached the maximum values (3.61 g/cm³, 14.9, and 26,450 GHz) and then decreased. The temperature coefficient of resonator frequency (τ _f ) showed a slight change at a negative range of ??94.6 to ??83.7 ppm/°C. When the sintering temperature was 1400 °C, MgAl₂Ti₃O₁₀ ceramics exhibited the best microwave dielectric properties with Q?×?f?=?26,450 GHz, ε _r ?=?14.9 and τ _f ?=???83.7 ppm/°C.     

Enhancing the microwave dielectric properties of Sr<Subscript>2</Subscript>Ca<Subscript>3</Subscript>Ta<Subscript>4</Subscript>TiO<Subscript>17</Subscript> ceramics by Zr Substitution

The compositions in Sr₂Ca₃Ta₄Ti_1?xZr_xO₁₇ (0?≤?x?≤?0.12) series were designed and fabricated by solid state sintering method. All the compositions formed single phases and crystallized in an orthorhombic crystal structure. Zr substitution led to the enhancing of the microwave dielectric properties by tuning the τ_f value through zero and increased the Q_uf_o value from 12,540 to 14,970 GHz with a slight decrease in ε_r. In the present study, a good combination of ε_r ~?51, Q_uf_o ~?145,43 GHz and τ_f ~ 3 ppm/°C were obtained for Sr₂Ca₃Ta₄Ti_0.90Zr_0.1O₁₇ ceramic sintered at 1575 °C for 4 h.     

Structure and electrical properties of MnO doped (Ba<Subscript>0.96</Subscript>Ca<Subscript>0.04</Subscript>)(Ti<Subscript>0.92</Subscript>Sn<Subscript>0.08</Subscript>)O<Subscript>3</Subscript> lead free ceramics

In this work, (Ba_0.96Ca_0.04)(Ti_0.92Sn_0.08)O₃–xmol MnO (BCTS–xMn) lead-free piezoelectric ceramics were fabricated by the conventional solid-state technique. The composition dependence (0 ≤ x ≤ 3.0 %) of the microstructure, phase structure, and electrical properties was systematically investigated. An O–T phase structure was obtained in all ceramics, and the sintering behavior of the BCTS ceramics was gradually improved by doping MnO content. In addition, the relationship between poling temperature and piezoelectric activity was discussed. The ceramics with x = 1.5 % sintering at temperature of 1330 °C demonstrated an optimum electrical behavior: d ₃₃ ~ 475 pC/N, k _p ~ 50 %, ε _r ~ 4060, tanδ ~ 0.4 %, P _r ~ 10.3 μC/cm², E _c ~ 1.35 kV/mm, T _C ~ 82 °C, strain ~0.114 % and \(d_{33}^{*}\) ~ 525 pm/V. As a result, we achieved a preferable electric performance in BaTiO₃-based ceramics with lower sintering temperature, suggesting that the BCTS–xMn material system is a promising candidate for lead-free piezoelectric ceramics.     

Effects of LiF addition on the sintering behavior and microwave dielectric properties of Li<Subscript>2</Subscript>Mg<Subscript>3</Subscript>SnO<Subscript>6</Subscript> ceramics

Li₂Mg₃SnO₆ (abbreviation for LMS) ceramics doped with 1–4 wt% lithium fluoride (LiF) were prepared by the conventional solid-state reaction method. The effects of LiF addition on the phase compositions, sintering behaviors and microwave dielectric properties of LMS ceramics were investigated. A small amount of LiF addition could effectively decrease the sintering temperatures due to the liquid phase in the sintering process and induced no apparent degradation of the microwave dielectric properties. The optimized quality factor values for each composition firstly increased and then decreased with the increase of the LiF content. Whereas, the optimized dielectric permittivity increased with increasing of the LiF content. Distinguished microwave dielectric properties with a dielectric constant (ε _r) of 11.13, a quality factor (Q·f) of 104,750 GHz, and a temperature coefficient of resonant frequency (τ _f ) of ?10.83 ppm/°C were obtained for LMS ceramics sintered at 950?°C doped with 3 wt% LiF, which showed that the materials were suitable for the low temperature co-fired ceramics applications (LTCC).     

Effect of Ni<Superscript>2+</Superscript> substitution on crystal structure and microwave dielectric properties for MgZrNb<Subscript>2</Subscript>O<Subscript>8</Subscript> ceramics

Monoclinic structured Mg_1?xNi_xZrNb₂O₈ (0?≤?x?≤?0.12) ceramics were synthesized for the first time through traditional solid-state reaction process and pure phase were obtained in all range. Rietveld refinement was used to analyze the crystal structure. With the increase of Ni²⁺ substitution amount, ε _r decreased, Q?×?f rose first then fell, τ _f shifted for the positive direction. Bond ionicity, lattice energy and bond energy were separately calculated to investigate the correlations with microwave dielectric properties. Typically, ceramics samples with the composition of Mg_0.92Ni_0.08ZrNb₂O₈ sintered at 1280 °C for 4 h exhibited the optimum microwave dielectric properties: ε _r ?=?24.58, Q?×?f?=?74534.1 GHz, τ _f ?=???49.11 ppm/°C, which could be a promising material for application.     

Effects of sintering temperature on the properties of Mn/Y codoped Ba<Subscript>0.67</Subscript>Sr<Subscript>0.33</Subscript>TiO<Subscript>3</Subscript> ceramics for tunable application

(Ba_0.67Sr_0.33)_1?3x/2Y _x Ti_1?y/2Mn _y O₃ [BST(Mn + Y), x = 0.006, y = 0.005] ceramics were fabricated by using citrate–nitrate combustion derived powder. Microstructure and dielectric properties of the BST(Mn + Y) ceramic samples were investigated within the sintering temperature ranged from 1220 to 1300 °C. Sintering temperature has a great influence on the microstructure and electrical properties of the ceramic samples. The dielectric properties, ferroelectric properties, and tunability are enhanced by optimizing sintering temperature. The relatively high tunability of 40 % (1.5 kV/mm DC field, 10 kHz) was obtained, and relatively low dielectric loss, <0.0052 (at 10 kHz, 20 °C) was acquired for BST(Mn + Y) samples sintered at 1275 °C for 3 h. Both the low dielectric loss and enhanced tunable properties of BST(Mn + Y) are useful for tunable devices application.     

Very high thermal stability with excellent dielectric,and non-ohmics properties of Mg-doped CaCu<Subscript>3</Subscript>Ti<Subscript>4.2</Subscript>O<Subscript>12</Subscript> ceramics

In this work, the nominal CaCu_3?xMg_xTi_4.2O₁₂ (0.00, 0.05 and 0.10) ceramics were prepared by sintering pellets of their precursor powders obtained by a polymer pyrolysis solution method at 1100 °C for different sintering time of 8 and 12 h. Very low loss tangent (tanδ)?<?0.009–0.014 and giant dielectric constant (ε′) ～?1.1?×?10⁴–1.8?×?10⁴ with excellent temperature coefficient (Δε′) less than ±?15% in a temperature range of ??60 to 210 °C were achieved. These excellent performances suggested a potent application of the ceramics for high temperature X8R and X9R capacitors. It was found that tanδ values decreased with increasing Mg²⁺ dopants due to the increase of grain boundary resistance (R_gb) caused by the very high density of grain, resulting from the substitution of small ionic radius Mg²⁺ dopants in the structure. In addition, CaCu_3?xMg_xTi_4.2O₁₂ ceramics displayed non-linear characteristics with the significant enhancements of a non-linear coefficient (α) and a breakdown field (E_b) due to Mg²⁺doping. The high values of ε′ (14012), α (13.64) and E_b (5977.02 V/cm) with very low tanδ value (0.009) were obtained in a CaCu_2.90Mg_0.10Ti_4.2O₁₂ ceramic sintered at 1100 °C for 8 h.     

Influence of Nd doping on microwave dielectric properties of SrTiO<Subscript>3</Subscript> ceramics

Sr_1?x Nd _x TiO₃ (x?=?0.08–0.14) ceramics were prepared by conventional solid-state methods. The analysis of crystal structure suggested Sr_1?x Nd _x TiO₃ ceramics appeared to form tetragonal perovskite structure. The relationship between charge compensation mechanism, microstructure feature and microwave dielectric properties were investigated. Trivalent Nd³⁺ substituting Sr²⁺ could effectively decrease oxygen vacancies. This reduction and relative density were critical to improve Q?×?f values of Sr_1?x Nd _x TiO₃ ceramics. For ε _r values, incorporation of Nd could restrain the rattling of Ti⁴⁺ cations and led to the reduction of dielectric constant. The τ _f values were strongly influenced by tilting of oxygen octahedral. The τ _f values decreased from 883 to 650 ppm/°C with x increasing from 0.08 to 0.14. A better microwave dielectric property was achieved for composition Sr_0.92Nd_0.08TiO₃ at 1460 °C: ε _r ?=?160, Q?×?f?=?6602 GHz, τ _f ?=?883 ppm/°C.     

Effect of Al<Subscript>2</Subscript>O<Subscript>3</Subscript> addition on the microstructure and electrical properties of LaMnO<Subscript>3</Subscript>-based NTC thermistors

A study to develop a new system of negative temperature coefficient thermistors for wide temperature range, A series of Mn-based perovskite-structured ceramics of composition (LaMn_1?x Al _x O₃)_0.9(Al₂O₃)_0.1 has been synthesized by conventional solid state reaction at 1350?°C. The X-ray diffraction patterns showed that for all the samples, the substitution of manganese by aluminum up to x?=?0.1 preserved the rhombohedral perovskite LaMnO₃-like phase. For x?=?0.2, apart from the LaMnO₃-like structure, a second perovskite phase based on the cubic LaAlO₃ structure was formed. For x?=?0.3 and 0.4, the phase present was LaAlO₃ -type structure. The grain sizes of the sintered body detected by scanning electron microscope were decreased with increasing Al₂O₃ content. The resistivity increases with increasing the Al content. The obtained values of ρ _25?°C and B _25/50 and E _a are in the range of 10–13103 Ω cm, 1813–2794 K, 0.156–0.241 eV, respectively. The resistance variation (ΔR/R) was <0.241% and the minimum value (0.0483%) was obtained for aging at 125?°C at 500 h. The aim of this work was explored new composite ceramics materials, which could be used as potential candidates for wide temperature range from ?100 to 500?°C thermistors applications.     

Strong up-conversion luminescence and electrical properties of SrBi<Subscript>4</Subscript>Ti<Subscript>4</Subscript>O<Subscript>15</Subscript> multifunctional ceramics by Er<Superscript>3+</Superscript> doping

Novel green-emitting piezoelectric ceramics of SrBi_4?x Er _x Ti₄O₁₅ (SBT-xEr) were prepared. Strong up-conversion with bright green (524 and 548 nm) and a relatively weak red (660 nm) emission bands were obtained under 980 nm excitation at room temperature, which is attributed to the intra 4f–4f electronic transition of (²H_11/2, ⁴S_3/2)–⁴I_15/2 and the transition from ⁴F_9/2 to ⁴I_15/2 of Er³⁺ ions, respectively. Simultaneously, Er³⁺ doping promotes the electrical properties. At 0.8 mol%Er, the optimal electric properties with high Curie temperature of T _c?~527?°C, large remanent polarization of 2P _r?~14.92 μC/cm² and piezoelectric constant of d ₃₃?~17 pC/N was achieved. As a multifunctional material, Er³⁺ doped SBT showed a great potential to be used in 3D-display, bio-imaging, solid state laser and optical temperature sensor.     

Frequency and temperature dependence of electrical properties of barium and gadolinium substituted SrBi<Subscript>2</Subscript>Nb<Subscript>2</Subscript>O<Subscript>9</Subscript> ceramics

Barium strontium gadolinium bismuth niobate (Ba_0.1Sr_0.81Gd_0.06Bi₂Nb₂O₉, BSGBN) ceramics were prepared by using the conventional solid-state reaction method. The dielectric permittivity, modulus and impedance spectroscopy studies on BSGBN were investigated in the frequency range, 45 Hz–5 MHz and in the temperature range from room temperature (RT) to 570 °C. The dielectric anomaly with a broad peak was observed at 470 °C. Simultaneous substitution of Ba²⁺ and Gd³⁺ increases the transition temperature of SrBi₂Nb₂O₉ (SBN) from 392 to 470 °C. XRD studies in BSGBN revealed an orthorhombic structure with lattice parameters a = 5.4959 Å, b/a = 1.000, c = 25.0954 Å. Impedance and modulus plots were used as tools to analyse the sample behaviour as a function of frequency. Cole-Cole plots showed a non-Debye relaxation. Also, dc and ac conductivity measurements were performed on BSGBN. The electric impedance which describes the dielectric relaxation behaviour is fitted to the Kohlrausch exponential function. Near the phase transition temperature, a stretched exponential parameter β indicating the degree of distribution of the relaxation time has a small value.     

Characterization of Zn doped MgTiO<Subscript>3</Subscript> ceramics: an approach for RF capacitor applications

In the present study the effect of Zn substitution on densification, microstructure, microwave and broad band dielectric properties of MgTiO₃ ceramics were investigated. The (Mg_1?x Zn _x )TiO₃ (x?=?0.01–0.07) ceramics have been prepared by the conventional solid-state reaction method. The sintering conditions were optimized to obtain the best dielectric properties with maximum relative densities. The microwave dielectric properties are heavily influenced by the amount of x concentration. The optimum dielectric properties of ε _r ~ 17.34, Q?×?f _o ~ 274 THz, τ _f ~ -40.3 ppm/^oC is obtained for (Mg_0.95Zn_0.05)TiO₃ ceramics sintered at 1275?°C. The broad band dielectric properties of (Mg_0.95Zn_0.05)TiO₃ ceramics were measured in the frequency range of 1–100 MHz, and temperature range of 133–483 K. Interestingly, the broad band dielectric properties show relaxation behaviour with frequency. The higher temperature dielectric spectrum of (Mg_0.95Zn_0.05)TiO₃ (MZT) ceramics displayed a distinct dispersion, which is shifting towards a lower frequency side. The observed dielectric relaxation behavior is analyzed using Cole–Cole plot. Furthermore, voltage dependent capacitance behavior at different frequencies is studied for the MZT sample, and it’s interesting to note that the capacitance is stable with the variation in voltage. The electrical conductivity study is carried out as a function of frequency and temperature for MZT sample and the activation energy is calculated by using Arrhenius equation, which is found to be 0.07 eV at 10 MHz. The obtained dielectric response of MZT ceramics are suitable for dielectric resonator and type-1 RF capacitor applications.     

Scaling behavior of dynamic hysteresis in Bi<Subscript>3.15</Subscript>Nd<Subscript>0.85</Subscript>Ti<Subscript>3</Subscript>O<Subscript>12</Subscript> ceramics

The scaling behavior of dynamic hysteresis was investigated in Bi_3.15Nd_0.85Ti₃O₁₂ bulk ceramics at a frequency of 1–1000 Hz and an external electric field amplitude of 79–221 kV/cm. The scaling behavior at low amplitude (E ₀ ≤ 114 kV/cm) takes the form of \(\langle A \rangle \propto f^{ - 0.013} E_{0}^{0.7}\) for low frequency (f ≤ 200 Hz) and \(\langle A \rangle \propto f^{ - 0.013} E_{0}^{0.22}\) for high frequency (f > 200 Hz), where \(\langle A \rangle\) is the area of hysteresis loop and f and E ₀ are frequency and amplitude of external electric field, respectively. At high amplitude (E ₀ > 114 kV/cm), we obtain \(\langle A \rangle \propto f^{0.011} E_{0}^{1.163}\) at low frequency and \(\langle A \rangle \propto f^{ - 0.015} E_{0}^{0.7}\) at high frequency. At low E ₀, the contribution to the scaling relation mainly results from reversible domain switching, while at high E ₀ reversible and irreversible domain switching concurrently contribute to the scaling relation.     

Dielectric relaxation phenomena in the compound: LiCo<Subscript>3/5</Subscript>Mn<Subscript>1/5</Subscript>Cu<Subscript>1/5</Subscript>VO<Subscript>4</Subscript>

Solution-based chemical method has been used to produce LiCo_3/5Mn_1/5Cu_1/5VO₄ ceramics. The formation of the compound is checked by X-ray diffraction analysis and it reveals an orthorhombic unit cell structure with lattice parameters of a = 9.8262 Å, b = 3.0706 Å, c = 14.0789 Å. Field emission scanning electron micrograph indicates a polycrystalline texture of the material with grains of unequal sizes (~0.2 to 3 μm). Complex impedance spectroscopy technique is used to study the dielectric properties. Temperature dependence of dielectric constant (ε _r) at various frequencies exhibits the dielectric anomalies in ε _r at T _c (transition temperature) = 245, 255, 260 and 265 °C with (ε_r)_max. ~458, 311, 214 and 139 for 50, 100, 200 and 500 kHz, respectively. Frequency dependence of tangent loss at various temperatures shows the presence of dielectric relaxation in the material.     

Microwave dielectric properties of Bi(Sc<Subscript>1/3</Subscript>Mo<Subscript>2/3</Subscript>)O<Subscript>4</Subscript> ceramics for LTCC applications

A novel microwave dielectric ceramics Bi(Sc_1/3Mo_2/3)O₄ with low firing temperature were prepared via the solid reaction method. The specimens have been characterized using scanning electron microscopy, X-ray diffraction, Raman spectroscopy and DC conductivity. The Bi(Sc_1/3Mo_2/3)O₄ ceramics showed B-site ordered Scheelite-type structure with space group C2/c. Raman analysis indicated that prominent bands were attributed to the normal modes of vibration of MoO₄ ^2? tetrahedra. The dielectric loss of Bi(Sc_1/3Mo_2/3)O₄ ceramics can be depended strongly the bulk conductivity by DC measurement. The superior microwave dielectric properties are achieved in the Bi(Sc_1/3Mo_2/3)O₄ ceramic sintered at 875 °C/4 h, with dielectric constant?~?25, Q?×?f ~?51,716 GHz at 6.4522 GHz and temperature coefficient of resonance frequency ~???70.4 ppm/°C. It is a promising microwave dielectric material for low-temperature co-fired ceramics technology.     

Ultralow thermal conductivity of cerium-doped Nd<Subscript>2</Subscript>Zr<Subscript>2</Subscript>O<Subscript>7</Subscript> over a wide doping range

In this paper, we report an ultralow thermal conductivity and a high-temperature phase stability of the (Nd_1?x Ce _x )₂Zr₂O_7+x system over the temperature range from room temperature to 1600 °C and over a wide composition range (0.2 ≤ x ≤ 0.8), and the (Nd_1?x Ce _x )₂Zr₂O_7+x system is therefore considered a strong candidate material for the fabrication of next-generation high-temperature thermal barrier coatings. The observed thermal conductivities (0.65–1.0 W/mK) are about 60–40% lower than those of undoped Nd₂Zr₂O₇ over the same temperature range (100–700 °C) and indicate a glass-like behavior. For comparison, the variation in the thermal conductivity with the temperature of the (Gd_1?x Ce _x )₂Zr₂O_7+x system with similar point defects was also measured, and the observed behavior was almost the same as that of undoped Gd₂Zr₂O₇ and was mostly determined by phonon–phonon scattering (λ ∝ 1/T). The effect of point defect scattering and strong phonon scattering sources (rattlers) on the thermal conductivity is also discussed in this paper. The results of this study suggest that the ultralow thermal conductivity of (Nd_1?x Ce _x )₂Zr₂O_7+x can be attributed to the presence of rattlers because of the large difference between the ionic radii of the Nd³⁺ and Ce⁴⁺ ions.