1 
Novel octahedral tilt system a+b+c+ in (1-x)Na0.5Bi0.5TiO3-xCdTiO3 solid solutions 
 
R. Ignatans, M. Dunce*, E. Birks, and A. Sternberg 
 
Institute of Solid State Physics, University of Latvia, Kengaraga 8, Riga, LV-1063, Latvia 
* Corresponding author. E-mail: marija.dunce@cfi.lu.lv. Tel.: +371 67260803. Fax: +371 67132778 
 
 
Abstract 
 
(1-x)Na0.5Bi0.5TiO3-xCdTiO3 solid solutions in the whole concentration range (0.0≤x≤1.0) were studied by means 
of x-ray diffraction, dielectric spectroscopy and polarization measurements. The study was mainly focused on 
crystalline structure of the compositions, depending on their place in the phase diagram. The solid solution 
system exhibits at least four different phases at room temperature, giving rise to paraelectric, ferroelectric and 
relaxor-ferroelectric behaviour. There were proposed appropriate space groups for each of these phases, using 
Rietveld refinement method for analysis of the x-ray diffraction patterns and taking into account polarization 
measurement results. Unexpected and unusual octahedral tilt systems – a+a+a+ and a+b+c+ – were found in 
certain CdTiO3 concentration ranges. The tilt system a+b+c+, which was detected in the ferroelectric phase, was 
evidenced for the first time, as it has been theoretically predicted, but never experimentally observed before in 
any material. It was shown that ferroelectricity in (1-x)Na0.5Bi0.5TiO3-xCdTiO3 solid solutions arises not only 
from the Ti+4 displacements, but also from the polar distortions in square planar and cubooctahedral cation A-
sites. Upon heating, at a phase transition from the ferroelectric to the paraelectric state, a+b+c+ tilt system 
transforms into a+a+a+. The studied compositions were compared with (1-x)Na0.5Bi0.5TiO3-xCaTiO3 solid 
solution system, as CdTiO3 and CaTiO3 are crystallographically very similar. It was revealed that both 
constituents behave very differently. CaTiO3 in (1-x)Na0.5Bi0.5TiO3-xCaTiO3, even in low concentrations, 
stabilizes solid solutions in its Pnma space group, unlike its counterpart CdTiO3 in the studied materials. 
 
KEYWORDS: structure, NBT, ferroelectrics, x-ray diffraction, tilt system, solid solution 
 
 
Introduction 
 
Na0.5Bi0.5TiO3 (NBT) and its solid solutions have been actively studied recently as one of the most promising 
lead-free ferroelectric materials to possibly substitute the lead containing ferroelectric materials widely used in 
industry [1-3]. Although the ground state structure of pure NBT at room temperature is still under discussion, 
the crystal structure of the poled specimens undoubtedly has the space group R3c and the a-a-a- oxygen 
octahedral tilt system according to Glazer notation [4-6]. At elevated temperatures NBT has two additional 
distinct phases. Between  280˚C and ~510˚C NBT has tetragonal symmetry with the space group P4bm and 
above this temperature region it becomes cubic with the space group m3Pm  [2,7]. Electron diffraction analysis 
of NBT in the temperature region above the depolarisation temperature (~200˚C) revealed superstructure 
maxima characteristic for the space group Pnma [8]. Recently coexistence of R3c and Pnma phases in non-poled 
NBT at room temperature was proposed [9]. 
CdTiO3 undergoes an irreversible first-order phase transition near 900˚C from an ilmenite to a perovskite 
phase which allows it to be synthesized with either of the structures [10]. The room temperature perovskite 
phase is non-polar and isostructural with CaTiO3 (space group Pnma, but sometimes nonstandard equivalent 
space groups Pbnm or Pcmn are used to describe the structure of CaTiO3) [11,12]. When a perovskite phase of 
CdTiO3 is cooled down to ~80 K, it undergoes a phase transition to a polar orthorhombic phase with the space 
group Pna21 [10]. 
(1-x)(Na0.5Bi0.5)TiO3-xCdTiO3 (abbreviated (1-x)NBT-xCdT) solid solution group has not been a popular 
research object probably due to toxicity of cadmium, which is comparable with toxicity of lead, and low 
expectations of good ferroelectric properties considering the non-polar nature of CdTiO3 at room temperature. 
Interestingly, the solid solutions with 0.4≤x≤0.6 are ferroelectric at room temperature. A distinct first order 
phase transition is revealed by discontinuity in temperature dependence of thermal expansion coefficient [13]. 
On the other hand, the solid solutions with 0.05≤x≤0.3 do not show any signs of a phase transition. They have 
lower dielectric response and exhibit relaxor properties [13]. The solid solutions with x≥0.7 are paraelectric at 
room temperature [13,14], just as CdTiO3. 
There have been few attempts to study crystal structure of (1-x)NBT-xCdT solid solutions. Authors in ref. 
[14] had used the obsolete method to obtain diffraction patterns. Since they had used only pseudocubic 
Manuscript Click here to download Manuscript Manuscript-revised.doc 
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approach, there was no information about possible space groups of these compounds. In our previous study [13], 
we had presented some information about structural properties of (1-x)NBT-xCdT solid solutions.  
The main purpose of this paper is to further clarify the crystal structure of (1-x)NBT-xCdT solid solutions. 
Our analysis is based on polarization measurements, x-ray diffraction, Rietveld analysis and group theoretical 
considerations. 
 
 
Materials and Methods 
 
(1-x)(Na0.5Bi0.5)TiO3-xCdTiO3 ((1-x)NBT-xCdT) ceramics were prepared by solid state reaction from chemical-
grade oxides as described in ref. [13]. Samples for dielectric and polarization measurements were prepared in 
form of 0.5 mm thick ceramic disks with Au electrodes fired at 500˚C. Samples for x-ray diffraction were 
prepared by crushing the ceramics in pestle and heating obtained powder at 700˚C for 30 minutes, then followed 
by cooling down to room temperature. 
X-ray diffraction patterns were obtained using PANalytical X’Pert PRO diffractometer equipped with 
multichannel solid-state detector PIXcel. Cu Kα1,2 radiation was used, the tube was set to 45 kV and 40 mA. 
Room temperature diffraction patterns were recorded with a step of 0.0131˚ from 10˚ to 150˚ degrees and 77.5 s 
counting time. Total acquisition time per diffraction pattern was ~1 hour. Anton Paar TTK 450 temperature 
camera and Anton Paar TCU 100 temperature control unit were used for non-ambient temperature experiments. 
Rietveld refinements [15] of obtained powder x-ray diffraction patterns were carried out with BGMN [16] 
and its graphical user interface Profex [17].  
Weak-field dielectric properties in temperature range from −120˚C to 350˚C and frequency region from 100 
Hz to 1 MHz were studied using an impedance analyser HP precision LCR meter 4284A with a measuring 
electric field of 0.4 V/cm. Rate of the temperature change was ~ 2˚C/min. Polarization hysteresis measurements 
were done using the Sawyer-Tower method in quasistatic limit. 
 
 
Results and discussion 
 
X-ray diffraction patterns (Fig. 1) indicate that solid solutions (1-x)NBT-xCdT with x≥0.7 are isostructural with 
pure CdTiO3, as it was expected. This is in agreement with completely paraelectric dielectric response (Fig. 
2(c)). Rietveld refinements were done in Pnma space group. Occupancies of Na+, Bi3+ and Cd2+ ions were fixed 
to corresponding chemical formula solid solutions. An example of fitted diffraction pattern can be seen in Fig. 1. 
Pnma space group is associated with mixed oxygen octahedral tilt system a-b+a- giving rise to both ½(ooe) and  
 
 
 
 
Fig. 1 X-ray diffraction pattern and Rietveld fit with Pnma space group of composition 0.2NBT-0.8CdT as a 
representative of x≥0.7 solid solutions (the inset represents a magnified fragment of the pattern). Asterisk marks 
a small TiO2 impurity. Two superstructure reflections are marked with arrows – ½(310)PC corresponding to the 
in-phase tilting and ½(311)PC corresponding to the out-of-phase tilting of the oxygen octahedra. Maxima are 
indexed according to pseudo-cubic perovskite cell 
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½(ooo) superstructure reflections (where o is odd and e is even Miller index) which are associated with in-phase 
and out-of-phase octahedral tilting [6]. Additionally there are several superstructure reflections which do not 
correspond to either of tilting patterns (so called concert reflections). These reflections appear when antiparallel 
cation shifts are present in the crystalline structure or both of the oxygen octahedral tilting patterns are present. 
In the case of CdTiO3 and CaTiO3 both conditions are fulfilled [18]. 
(1-x)NBT-xCdT solid solutions in the concentration range 0.4≤x≤0.6 are ferroelectric, as it can be seen from 
polarization hysteresis loops in Fig. 2(d). Also dielectric spectroscopy measurements (Fig. 2(b)) indicate that 
ferroelectric phase transition occurs at elevated temperatures. The obtained diffraction patterns of these solid 
solutions are considerably different from those of the compositions with x≥0.7, as it is shown in Fig. 3 for the 
composition with x=0.6 composition. Superstructure reflections of ½(ooo) type and concert reflections cannot be  
 
 
 
 
Fig. 2 Temperature-frequency dependences of dielectric permittivity (solid lines) and loss tangent (dashed lines) 
for (1-x)NBT-xCdT compositions with x=0.3 (a), x=0.6 (b) and x=0.8 (c), as well as polarization hysteresis loops 
of (1-x)NBT-xCdT solid solutions with different CdTiO3 concentrations x 
 
 
 
 
Fig. 3 X-ray diffraction pattern  and Rietveld fit with Imm2 space group of 0.4NBT-0.6CdT composition as a 
representative of 0.4≤x≤0.6 solid solutions (the inset represents a magnified fragment of the pattern). Asterisk 
indicates the peak from small TiO2 impurity, arrows indicate a few superstructure maxima of ½(ooe) type. 
Maxima are indexed according to pseudo-cubic perovskite cell 
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seen in the diffraction patterns, only strong ½(ooe) superstructure reflections are visible, which is a clear sign of 
an in-phase oxygen octahedral tilt system. Additionally, 111PC is not split, while 200PC reflection is split into 
three maxima, two of which are not well-separated (see Figs. 4(a) and 4(b)). This is consistent with the 
orthorhombic symmetry. Such observations point to the a+b+c+ octahedral tilting pattern according to Glazer’s 
scheme and the Immm space group [19]. This space group cannot be correct, as the studied solid solutions in the 
CdTiO3 concentration range 0.4≤x≤0.6 are ferroelectric (essentially the same diffraction pattern was obtained for 
both poled and unpoled samples), but Immm space group is centrosymmetric. Rietveld calculations for this space 
group led to a good description of the main perovskite maxima and correct positions of superstructure 
reflections, while the calculated intensity of superstructure reflections was approximately half as large as the 
measured intensity. All the above mentioned indicates that the proper space group must be very similar to 
Immm. The group-theoretical analysis performed by Stokes et al [20] extracted a polar space group Imm2 which 
is a subgroup of Immm. Essentially, in the space group Imm2 additional atomic displacements along Z axis are 
allowed, unlike in the non-polar supergroup. Initial atomic coordinates for Rietveld refinement were calculated 
by ISODISTORT tool [21]. Rietveld refinement in this space group gives a very good description of 
experimental diffraction pattern, as it can be seen in Fig. 3.
 
 
 
 
Fig. 4 X-ray diffraction maxima 200PC and 211PC of 0.4NBT-0.6CdT composition. The graphs on the left – (a) 
and (c) – correspond to calculation results obtained using Rietveld model with anisotropic micro-strains, the 
graphs on the right – to calculation results obtained using isotropic micro-strain model – (b) and (d); Black solid 
line corresponds to experimental data, red dashed line – to calculated data, blue solid line – to difference 
between experimental and calculated data, tick marks indicate positions of Bragg reflections responsible for 
splitting of the pseudo-cubic maxima 
 
 
According to ref. [18,22] and our knowledge, a+b+c+ octahedral tilt system has never been experimentally 
detected before. In the Imm2 space group there are four non-equal A-cation sites. Three of them have distorted 
square planar environment and one has distorted cubooctahedral coordination (atomic position located at the 
origin). A proper description of the superstructure reflections is only possible with displacements of the A site 
cations relative to the surrounding oxygen anions along Z axis. Rietveld refinements also show Ti4+ off-centring 
in the oxygen octahedron, which indicates that ferroelectricity arises from both – the A site and the B site ion 
displacements. 
Structural parameters of 0.4NBT-0.6CdT solid solution are summarized in Table 1 and the corresponding 
model is shown in Fig. 5 [23]. The isotropic Debye-Waller factors (Biso) of the A-site cations were refined as one 
parameter between all four sites, as it represents more static disorder rather than thermal one. Similarly the 
isotropic Debye-Waller factors of the oxygen atoms were constrained to one refinement parameter, as it 
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drastically reduces the number of variables. The low scattering amplitude of oxygen atoms justifies this 
approach. 
 
 
Table 1 Structural parameters of 0.4NBT-0.6CdT and 0.7NBT-0.3CdT solid solutions at room temperature 
 
0.4NBT-0.6CdT 
Atoms Wyckoff pos. X Y Z Biso (Å2) 
Na/Bi/Cd a 0 0 0a 
1.582 (13) 
Na/Bi/Cd a 0 0 0.53633 (85) 
Na/Bi/Cd b 0 0.5 0.02520 (53) 
Na/Bi/Cd b 0 0.5 0.49148 (83) 
Ti e 0.25944 (53) 0.24280 (33) 0.25159 (59) 0.112 (29) 
O e 0.2085 (12) 0.2887 (12) -0.01245 (74) 
1.761 (66) 
O c 0.2806 (12) 0 0.19602 (83) 
O c 0.2892 (10) 0 0.7567 (25) 
O d 0 0.77521(99) 0.3257 (11) 
O d 0 0.7832 (10) 0.7494 (21) 
Space group Imm2; a=7.64794 (10) Å, b= 7.665550 (99) Å, 7.712358 (77) Å 
Rwp=1.31% χ2=3.40 
0.7NBT-0.3CdT 
Atoms Wyckoff pos. X Y Z Biso (Å2) 
Na/Bi/Cd a 0 0 0 
3.126 (19) 
Na/Bi/Cd b 0 0.5 0.5 
Ti c 0.25 0.25 0.25 0.628 (19) 
O e 0 0.20724 (51) 0.26955 (68) 1.834 (68) 
Space group 3Im ; a=7.711396 (24) 
Rwp=1.52% χ2=1.81 
a Z coordinate is fixed to deal with the floating origin in the polar space group. 
 
 
 
 
Fig. 5 Structural model of 0.4NBT-0.6CdT solid solution (a) and four nonequal A-site cation positions: (b) 
Wyckoff position a with Z=0; (c) Wyckoff position a with Z=0.53633; (d) Wyckoff position b with Z=0.02520; 
(e) Wyckoff position b with 0.49148. In each A-site position, cations are shifted along c axis in relation to 
surrounding oxygen anions, creating a dipole moment 
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Ferroelectricity in 0.4≤x≤0.6 solid solutions arises due to  Bi3+ ions which condensate unstable polar phonon 
mode found in the pure CdTiO3 at the Γ point [24]. Interaction of Bi 6s with O 2p results in set of bonding and 
antibonding states. Antibonding states can be stabilized by mixing them with empty Bi 6p states, but the site 
symmetry must be such that sp mixing is allowed [25]. In this case, the second order Jahn-Teller distortion leads 
to a polar structure due to already present polar instability from the CdTiO3 part of the solid solution [25].  
Although the A-cation ordering is possible in the Imm2 space group due to the four non-equal A-cation sites, 
meaningful results or improved description of diffraction patterns were not obtained assuming several different 
A-site ordering schemes neither in x=0.6 nor in other solid solutions in the studied system. Therefore Rietveld 
refinement was done as before by fixing atomic occupancies of Na+, Bi3+ and Cd2+ ions in every A-site 
according to chemical formula, which always gives a good fit and meaningful results. 
Anisotropic microstrain model was employed for (1-x)NBT-xCdT solid solutions in the concentration region 
0.4≤x≤0.6, which gave a better fit to the diffraction patterns (Figs. 4(a) and 4(c)) as compared with traditional 
isotropic microstrain model (Figs. 4(b) and 4(d)). It must be mentioned that, for the solid solution with x=0.7, 
which undoubtedly has space group Pnma, fitting of diffraction pattern is also improved by anisotropic 
microstrain model, while for the composition with x=0.8 it is not. Since the composition with x=0.7 is located 
close to the concentration range where ferroelectric phase exists, the observed anisotropic microstrains can be 
explained by influence of  residual ferroelectric state. The presence of weak maximum in the temperature 
dependence of dielectric permittivity [13] supports such an assumption. 
When the solid solutions in the concentration range 0.4≤x≤0.6 are heated, a phase transition from the 
orthorhombic Imm2 to the cubic 3Im  space group occurs (accordingly from a+b+c+ to a+a+a+ oxygen 
octahedral tilt system). Particular symmetry of the high-temperature phase can be identified by loss of splitting 
of the 200PC maximum (Fig. 6) and still visible superstructure reflections of ½(ooe) type corresponding to the in-
phase tilt system. The three non-equivalent distorted square planar sites in Imm2 (see Fig. 5) transform into one 
 
 
 
 
Fig. 6 Contour plot of temperature dependence of 200PC maximum for (1-x)NBT-xCdT composition with x=0.6. 
Horizontal line at 190˚C indicates phase transition temperature 
 
 
non-distorted square planar site during the phase transition. The polar distorted cubooctahedrally coordinated A-
site in Imm2 transforms into a nonpolar, but still slightly distorted cubooctahedrally coordinated site 
characteristic for 3Im space group [22]. Observed diffraction pattern of the high-temperature phase in the 
concentration region 0.4≤x≤0.6 is analogous to the one in Fig. 7, and refined crystallographic model is 
equivalent to the one in Fig. 8. The phase transition temperature observed by XRD (Fig. 6) is in a good 
agreement with the dielectric permittivity measurement data – Fig. 2(b). 
 
 
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Fig. 7 X-ray diffraction pattern and Rietveld fit with 3Im space group for (1-x)NBT-xCdT composition with  
x=0.3 (the inset represents a magnified fragment of the pattern). Arrows indicate some of the visible 
superstructure reflections 
 
 
 
Fig. 8 Structural model of 0.7NBT-0.3CdT solid solution. On the left (a), slightly distorted cubooctahedral 
coordination sites and, on the right (b), square planar coordination sites are shown 
 
 
(1-x)NBT-xCdT solid solutions in the concentration range 0.05≤x≤0.3 are also cubic with the space 
group 3Im , just as the high-temperature phase in the concentration range 0.4≤x≤0.6. Dielectric spectroscopy in 
0.05≤x≤0.3 concentration range shows typical relaxor-ferroelectric dielectric response. Usually perovskite 
relaxor ferroelectrics have averaged mPm3  symmetry [26], which does not have octahedral tilts. But in the solid 
solutions with 0.05<x≤0.3 we have observed classical relaxor ferroelectric behaviour and a space group with 
octahedral tilting. Rietveld refinements gave very good fits to both – main perovskite and superstructure maxima 
(Fig. 7). The crystallographic model created from refined structural parameters (Table 1) can be seen in Fig. 8. 
It is evident that Cd2+ is responsible for the quite extraordinary square planar coordination in NBT containing 
solid solutions. Formation of structures with a low coordination number is characteristic feature of closed-shell 
d10 ions and  results from d-s and d-s-p hybrid orbitals [27]. 
Although CaTiO3 is isostructural with CdTiO3 at room temperature, its solid solutions with NBT behave in a 
different manner. These solid solutions even with low CaTiO3 concentrations (x>0.05) stabilize in the Pnma 
space group [9]. So (1-x)NBT-xCaTiO3 system does not exhibit any crystal structures not related to one or 
another component of the solid solution, contrary to (1-x)NBT-xCdT system. This again emphasizes the fact that 
the Pnma space group is a completely stable form of CaTiO3, but not of CdTiO3 [24].  
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The unit cell parameters of 3Im , Imm2 and Pnma phases were recalculated to pseudocubic perovskite unit 
cell parameters aPC, bPC and cPC (Fig. 9) for a simpler comparison. In the 3Im  space group and the Imm2 space 
group case recalculation is very easy as their unit cell parameters are equal to doubled pseudocubic unit cell 
parameters. In case of Pnma phase, basing on simple geometric considerations, one can derive pseudocubic unit 
cell parameters aPC, bPC, cPC and angle βPC from actual orthorhombic lattice parameters (aPnma, bPnma, cPnma): 
 
 2 2
1
2
PC PC Pnma Pnmaa c a c    , (1) 
 
1
2
PC Pnmab b  , (2) 
 
2 2
1
2 2
cos Pnma PnmaPC
Pnma Pnma
a c
a c
 
 
  
 
. (3) 
 
 
 
 
Fig. 9 Pseudocubic unit cell parameters of (1-x)NBT-xCdT solid solution system. Solid solutions with x≤0.05 
resemble pure NBT and the corresponding unit cell parameters are not shown 
 
 
At very low CdTiO3 concentrations (x≤0.05), x-ray diffraction patterns of (1-x)NBT-xCdT highly resemble 
pure NBT, as it is expected. 
 
 
Conclusions 
 
Oxygen octahedral tilt system a+b+c+, which has never been detected earlier, is observed in ferroelectric (1-
x)NBT- xCdT solid solutions in 0.4≤x≤0.6 concentration range. Space group Imm2 corresponding to a+b+c+ tilt 
system is determined by analysing main perovskite and superstructure maxima with Glazer’s method, taking in-
to account ferroelectric behaviour of these solid solutions observed in polarization hysteresis loops. Structural 
phase transition from Imm2 to 3Im space group (accordingly from a+b+c+ to a+a+a+ oxygen octahedral tilt 
system) is observed for solid solutions in the 0.4≤x≤0.6 concentration range. Bi3+ induced condensation of the 
unstable phonon mode at the Γ point is responsible for the stabilization of the ferroelectric phase in this 
concentration region. 
Relaxor ferroelectric solid solutions in the concentration range 0.05≤x≤0.3 are found to be in the 3Im  space 
group. X-ray diffraction patterns at the elevated temperatures (up to 300˚C) of these solid solutions do not show 
any structural phase transition. 
Origins of the square planar coordination in the 3Im  and Imm2 space groups are connected with presence of 
the closed shell Cd2+ ions in the solid solution. 
 
 
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Acknowledgements 
 
This work has been supported by National Research Program in the framework of project “Multifunctional 
Materials and composites, photonics and nanotechnology (IMIS2)”. 
 
 
References 
 
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Institute of Solid State Physics, University of Latvia as the Center of Excellence has received funding from the European Union’s
Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-TeamingPhase2 under grant agreement No. 739508,
project CAMART²
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Table 1 Structural parameters of 0.4NBT-0.6CdT and 0.7NBT-0.3CdT solid solutions at room 
temperature 
 
0.4NBT-0.6CdT 
Atoms Wyckoff pos. X Y Z Biso (Å
2) 
Na/Bi/Cd a 0 0 0a 
1.582 (13) 
Na/Bi/Cd a 0 0 0.53633 (85) 
Na/Bi/Cd b 0 0.5 0.02520 (53) 
Na/Bi/Cd b 0 0.5 0.49148 (83) 
Ti e 0.25944 (53) 0.24280 (33) 0.25159 (59) 0.112 (29) 
O e 0.2085 (12) 0.2887 (12) -0.01245 (74) 
1.761 (66) 
O c 0.2806 (12) 0 0.19602 (83) 
O c 0.2892 (10) 0 0.7567 (25) 
O d 0 0.77521(99) 0.3257 (11) 
O d 0 0.7832 (10) 0.7494 (21) 
Space group Imm2; a=7.64794 (10) Å, b= 7.665550 (99) Å, 7.712358 (77) Å 
Rwp=1.31% χ2=3.40 
0.7NBT-0.3CdT 
Atoms Wyckoff pos. X Y Z Biso (Å
2) 
Na/Bi/Cd a 0 0 0 
3.126 (19) 
Na/Bi/Cd b 0 0.5 0.5 
Ti c 0.25 0.25 0.25 0.628 (19) 
O e 0 0.20724 (51) 0.26955 (68) 1.834 (68) 
Space group 3Im ; a=7.711396 (24) 
Rwp=1.52% χ2=1.81 
a Z coordinate is fixed to deal with the floating origin in the polar space group. 
Table 1 Click here to download Table Table1.doc