Valence of Ti cations and its e ﬀ ect on magnetic properties of spinel ferrites Ti x M 1 (cid:1) x Fe 2 O 4 (M ¼ Co, Mn)

Powder samples of Ti x Co 1 (cid:1) x Fe 2 O 4 (0.0 # x # 0.4) and Ti x Mn 1 (cid:1) x Fe 2 O 4 (0.0 # x # 0.3) were synthesized using a conventional method for preparing ceramics. X-ray di ﬀ raction analysis con ﬁ rmed that the samples consisted of a single phase with a cubic (A)[B] 2 O 4 spinel structure. The average molecular magnetic moment ( m exp ) measured at 10 K decreased monotonically with increasing x for two series of samples. According to previous investigations, Ti 2+ and Ti 3+ cations are present in these ferrites, but there are no Ti 4+ cations; the magnetic moments of the Ti 2+ , Ti 3+ , and Mn 3+ cations are assumed to couple antiferromagnetically with those of the Mn 2+ , Co 2+ , Co 3+ , Fe 2+ , and Fe 3+ cations whenever they are at the (A) or [B] sublattice. The dependence of m exp of the two series of samples on the doping level x was ﬁ tted using a quantum-mechanical potential barrier, and the cation distributions in the two series of samples were obtained.

Many studies were carried out on the magnetic moment of and cation distribution in Ti-doped spinel ferrites. [12][13][14][15][16] In these investigations, all of the Ti cations were assumed to be tetravalent, but there have been disputes regarding the cation distribution. Dwivedi et al. prepared a series of samples, Co(Fe 1Àx Ti x ) 2 O 4 (x ¼ 0, 0.05, or 0.1), by conventional solidphase reactions; using X-ray photoelectron spectroscopy (XPS) they discovered that all Ti cations went into the octahedral sites. 12 Srinivasa Rao et al. prepared samples of the CoTi x Fe 2Àx O 4 (0.0 # x # 0.3) series; they thought that the Ti 4+ ions had the tendency to go to the [B] site, which affected the cation distribution in the samples. 13 Schmidbauer prepared samples of the Fe 1+x Cr 2À2x Ti x O 4 (0 # x # 1) series and concluded that there were Fe 2+ ions at the (A) and [B] sites, and all Cr and Ti cations occupied the B-sites. 14 Schmidbauer also prepared samples of two spinel ferrite series, Fe 2.4Àt Cr 0.6 Ti t O 4 (0 # t # 0.7) and Fe 2.1Àt Cr 0.9 Ti t O 4 (0 # t # 0.55), and assumed that all of the Ti 4+ ions entered the [B] sites. 15 However, when Kale et al. prepared Ti x Ni 1+x Fe 2À2x O 4 (0.0 # x # 0.7), they estimated the cation distribution at the (A) and [B] sites using X-ray diffraction and came to the conclusion that the fraction of Ti 4+ cations entering the (A) sites increased with increasing x, and it reached 0.5 when x ¼ 0.7. 16 In order to resolve these discrepancies regarding cation distributions in spinel ferrites, Xu et al. investigated the valence, distribution of cations and the magnetic structure of Ti-doped spinel ferrites [17][18][19] by using an O 2p itinerantelectron model. [20][21][22] They found an additional antiferromagnetic phase when Ti cations replaced a portion of the Ni or Fe cations in the spinel ferrites Ni 0.68 Fe 2.32 O 4 (ref. 17 and 18) and NiFe 2 O 4 , 19 and they offered the following explanation for the phenomenon: most of the Ti cations were Ti 2+ cations that occupied the [B] sites; the remaining Ti cations were Ti 3+ cations and there were no Ti 4+ cations; the magnetic moments of the Ti cations coupled antiferromagnetically with those of Fe and Ni cations whenever they were at the (A) or [B] sites.
The absence of Ti 4+ in an oxide has been conrmed by theoretical and experimental investigations. Cohen 23 and Cohen and Krakauer 24 used density functional theory to calculate the densities of states for valence electrons in the perovskite oxide BaTiO 3 . Their results indicated that the average valence of Ba is +2, which is the same as the traditionally accepted value, but the average valences of Ti and O are +2.89 and À1.63, respectively, which are different from the conventional results of +4 and À2, respectively. This calculation result was conrmed by the X-ray photoelectron spectra obtained by Wu et al., 25  spinel ferrites; they obtained estimated values between À1.6 and À1.8 for V alO of spinel ferrites, and they also dened the ionicity of an oxide as f i ¼ |V alO |/2, accompanied by calculated values of the iconicity of several cations in spinel ferrites. 27 Taking into account that there are O 1À ions in addition to O 2À ions, our group uses the O 2p itinerant-electron model 20 and the quantum mechanical potential barrier method 21,22 to investigate the cation distribution in several series of spinel ferrites. [28][29][30][31][32][33][34][35] In the study reported here, we prepared spinel ferrite samples of Ti x Co 1Àx Fe 2 O 4 (0.0 # x # 0.4) and Ti x Mn 1Àx Fe 2 O 4 (0.0 # x # 0.3) and measured the magnetic moment, m exp , of the samples at 10 K. The cation distribution in the samples was estimated by tting the measured values of m exp .

Sample preparation
Spinel ferrites Ti x Co 1Àx Fe 2 O 4 (0.0 # x # 0.4; hereaer referred to as the Co-series) and Ti x Mn 1Àx Fe 2 O 4 (0.0 # x # 0.3; here-aer referred to as the Mn-series) were prepared using the method of solid-phase reaction. 17 The analytical reagent (AR)grade chemicals CoO, MnO 2 , Fe 2 O 3 , and TiO 2 were used as the starting materials. First, stoichiometric amounts of each chemical were mixed together, ground for 8 h in an agate mortar, and then calcined at 1173 K for 5 h. The calcined materials were then ground again for 1 h. The ground powder was calcined at 1473 K for an additional 5 h, and then further ground for 1 h. Next, the twice calcined and thrice ground powder was pressed into pellets at a pressure of 10 4 kg cm À2 and then sintered at 1673 K for 10 h in a tube furnace under an argon ow. The sintered pellets were then ground for 30 min in an agate mortar, and the resulting powder was used for the measurements.

Sample characterization
The crystal structure of the samples was determined by analyzing their X-ray diffraction (XRD) patterns, which were measured with an X-ray diffractometer (X'pert Pro, PANalytical, The Netherlands) with Cu K a (l ¼ 1.5406Å) radiation at room temperature. The data were collected in the 2q range of 15-120 with a step size of 0.0167 . The working current and voltage were 40 mA and 40 kV, respectively. The magnetic hysteresis loops of the samples were measured using a physical properties measurement system (PPMS, Quantum Design Corporation, USA) at 10 and 300 K. 3) samples, which indicate that they consisted of a single-phase with a cubic spinel structure of space group Fd 3m. The XRD data were tted using the X'Pert HighScore Plus soware (PANalytical, The Netherlands) and the Rietveld powder-diffraction prole-tting technique. 36 The ions O (32e), A (8b) and B (16c) were located at the positions (u, u, u), (0.375, 0.375, 0.375), and (0, 0, 0), respectively. We obtained the crystal structure data, including the crystal lattice constant, a, the oxygen position parameters, u, the distances from the O anions to the cations at the (A) and [

Analysis of X-ray diffraction patterns
the X'Pert HighScore Plus soware, and they were found to be greater than 100 nm. Therefore, surface effects of the crystallites are expected to be very weak in all samples. Fig. 2 shows the dependence of the lattice parameter a on the Ti-doping level, x, in the two series of samples. It can be seen that with increasing x, a increased for the Co-series and decreased for the Mn-series. The different trends in the lattice constant were related to the cation radii, magnetic ordering, and cohesive energies of the samples. Fig. 3 and 4 show the magnetic hysteresis loops of the two series of samples measured at 10 and 300 K. From these gures, we obtained the specic saturation magnetization (s S ) measured at 10 and 300 K and the magnetic moment (m exp ) per formula unit of each sample at 10 K, as listed in Table 2. It can be seen that the values of s S for the two series of samples gradually decreased with increasing x at both 10 and 300 K.

Analysis of magnetic properties of the samples
4 Estimation of cation distributions by fitting the samples' magnetic moments at 10 K Following the procedure reported by Xu et al., [18][19][20] we used the O 2p itinerant-electron model 20 and the quantum mechanical potential barrier method 21,22 to t the magnetic moments measured at 10 K as a function of x and estimate the cation distribution in all samples. During the tting process, the following factors were taken into account: Factor 1: since there were O 1À ions in addition to O 2À ions, the ionicity of the cations in the samples was distinctly lower than 1.0, as shown in Table 3; the values listed in Table 3  sublattice). (iii) In a given sublattice that is constrained by Hund's rules and by the fact that an itinerant electron has constant spin direction, the direction of the magnetic moments of cations with the 3d electron number of n d # 4 will couple antiferromagnetically to those of the cations with n d $ 5 at either the (A) sites or the [B] sites of a spinel ferrite. Therefore, the directions of the magnetic moments of Ti 3+ (3d 1 ), Ti 2+ (3d 2 ), and Mn 3+ (3d 4 ) were antiparallel to those of the magnetic  Table 3. Factor 3: we assumed that there is a square potential barrier between a pair of anion and cation. 21 The height and the width of the potential barrier are related to the cation ionization energy and the distance between the cation-anion pair. The content ratio (R) of the different cations is therefore related to the probability of the last ionized electrons transmitted through the potential barriers, and the following equation can be obtained: where nanometers (nm) and electron-volts (eV) are used as the units of length and energy; P C (or P D ) stands for the probability of the last ionized electron of the C (or D) cation jumping to the anions through the potential barrier with the height V C (or V D ) and the width r C (or r D ). V C and V D are the ionization energies of the last ionized electron of the cations C and D, and r C and r D are the distances from the cations C and D to the anions. The parameter c is a barrier shape-correcting constant related to the different extents to which the shapes of the two potential barriers deviate from a square barrier.
Factor 4: we considered the Pauli repulsion energy of the electron cloud between adjacent cations and anions. This can be taken into account using the effective ionic radius: 37 smaller ions tend to enter the sites with smaller available space in the lattice. It is worth noting that the volumes of the (A) sites are smaller than those of the [B] sites in spinel ferrites.
Factor 5: during the thermal treatment of the samples, the tendency to balance the electrical charge density forced some of the divalent cations (with large effective ionic radii) to enter the (A) sites (with smaller available space) from the [B] sites (with large available space), jumping over an equivalent potential barrier, V BA , because cations at the (A) sites have four adjacent oxygen ions while cations at the [B] sites have six adjacent oxygen ions. V BA is related to the ionization energy, ionic radius, and the thermal-treatment temperature. We assumed V BA of the ferrite samples can be expressed by the following equations: 34  Table 3.  Table 2 Specific saturation magnetization measured at 10 K (s S-10 K ) and 300 K (s S-300 K ) for the two series of samples; m exp is the experimental magnetic moment per formula unit of a sample, which was calculated using s S-10K x s S-10 K (A m 2 kg À1 ) s S-300 K (A m 2 kg À1 ) m exp (m B per formula) It can be seen from eqn (4) that y 1 + y 2 + y 3 + y 4 + y 5 + y 6 ¼ 1, where N 3 is the number of trivalent cations per formula unit. The parameters f Ti , f Fe , and f M ¼ f Co (or f Mn ) represent the ionicities of the Ti, Fe, and Co (or Mn) ions, 27 whose values are shown in Table 3. Eqn (7) suggests that when the ionicity of all cations are 1.00, the sum of the valence of all cations is 8.00, while N 3 ¼ 2.00. In fact, the ionicity of each cation is lower than 1.00 (see Table 3), resulting in N 3 < 2.00. From eqn (4), we have R B1 x 1 À y 1 À y 4 3 À x 1 À x 2 À y 3 À y 6 ¼ z 1 z 3 ; R B2 x 2 À y 2 À y 5 3 À x 1 À x 2 À y 3 À y 6 ¼ z 2 z 3 ; where R A1 , R A2 , R A4 , R A5 , and R A6 represent the probability ratios of the Ti 3+ , Co 3+ (Mn 3+ ), Ti 2+ , Co 2+ (Mn 2+ ), and Fe 2+ ions, respectively, with respect to the Fe 3+ ions at the (A) sites, while R B1 and R B2 represent the probability ratios of the Ti 3+ and Co 3+ (Mn 3+ ) ions with respect to the Fe 3+ ions at the [B] sites. From eqn (5) and (8), we can obtain From eqn (6) and (9), we have According to the above-mentioned quantum mechanical potential barrier method for estimating the cation distributions in spinel ferrites, 21,22 which is similar to eqn (1), the content ratios R A1 , R A2 , R A4 , R A5 , and R A6 at the (A) sites and R B1 and R B2 at the [B] sites can be rewritten as   Table 3). Therefore, we can calculate the average magnetic moment per formula unit of a sample from eqn (4): ; (19) where  (8) contains ve equations and eqn (9) contains two equations. Therefore, we needed to obtain the values of at least two independent parameters, such as c v and V BA (Ti 2+ ), in order to t the observed values of m exp of a sample at 10 K.
Using the above parameters and equations, we tted the dependence of m exp on x for the two series of samples. The points and curves in Fig. 5 represent the observed and   calculated magnetic moments, m exp and m C , of the samples. It can be seen that the tted curves are very close to the experimental results. In the tting process, we obtained the cation distribution and other data, as listed in Tables 4 and 5. The cation distribution is shown as a function of x for the two series of samples in Fig. 6 and 7.  From Tables 4, 5, Fig. 6 and 7, we found that the tting parameters and the cation distribution in the samples had certain characteristics, as discussed in the following subsections.

Fitting parameters: c v and V BA
During the tting process, we determined that the potential barrier shape-correcting constant c v was equal to 1.  (2) and (3), and they appear reasonable in the context of a physics problem.

Valence and distribution of Ti cations
(i) The ratio of Ti 2+ ions at the (A) and [B] sites to the Ti-doping level, x, is more than 72% in the Co-series samples ( Fig. 6(b)) and about 89% in the Mn-series samples ( Fig. 7(b)). This result is similar to that measured using XPS and reported by Dupin et al.; they found that the average O ionic valence is À1.15 for TiO 2 , which suggests that 70% of Ti cations in TiO 2 are Ti 2+ ions. 26 Therefore, the conventional view 12-16 that all Ti cations in an oxide are Ti 4+ ions needs to be modied.
(ii) The ratio of Ti 2+ cations that entered the [B] sites to x increased from 58% (x ¼ 0.1) to 63% (x ¼ 0.4) in the Co-series samples ( Fig. 6(b)), and this ratio remained at 79% from x ¼ 0.1 to x ¼ 0.3 in the Mn-series samples ( Fig. 7(b)). This result is similar to that reported by Xu et al., who found the ratio of Ti 2+ cations that entered the [B] sites to x was 81% in Ti-doped ferrite Ni 0.68 Fe 2.32 O 4 . 18 (iii) The ratio of Ti cations, including Ti 2+ and Ti 3+ , that entered the [B] sites to x increased from 67% (x ¼ 0.1) to 72% (x ¼ 0.4) in the Co-series samples (Fig. 6(d)), and this ratio was 85% in the Mn-series samples (Fig. 7(d)). This result appears to be a balance between the contrasting results reported by several authors: 12 (Fig. 6(d)). This ratio is very close to that reported by Shang et al. 30 for Co 1Àx Cr x Fe 2 O 4 . This result is also close to that reported by Wakabayashi et al. for a CoFe 2 O 4 lm with thickness of 11 nm, which was based on so X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) combined with cluster model calculations. 38 (ii) The ratio of Co 2+ cations that entered the [B] sites to the total Co cation content increased from 67% (x ¼ 0.0) to 73% (x ¼ 0.4). This result is similar to that reported by Shang et al., who found that the ratio of Co 2+ cations that entered the [B] sites ranged from 64% (x ¼ 0.0) to 59% (x ¼ 0.8) in Co 1Àx Cr x Fe 2 O 4 . 30

Distribution of Mn cations in Mn-series
The ratio of Mn 2+ cations that entered the [B] sites to the total Mn cation content was 61%. The ratio of Mn ions, including Mn 2+ and Mn 3+ cations, that entered the [B] sites to the total Mn cation content was 73%. This result is similar to that reported by Xu et al. 20

Entry of few Co and Mn cations into the (A) sites
It can be seen from Fig. 6(c) and 7(c) that a few of the Co (Mn) cations entered the (A) sites of the Co (Mn) series samples. This is in accordance with the observed results from XRD mentioned in Section 3.1: the ratio of observed to ideal values of A-O distance for MnFe 2 O 4 , 1.09, is higher than that for CoFe 2 O 4 , 1.04, because the effective radius of Mn is greater than that of Co (see Table 3). This suggested that a few of the Mn (Co) cations entered the (A) sites of MnFe 2 O 4 (CoFe 2 O 4 ).

Conclusions
The single-phase spinel ferrites Ti x Co 1Àx Fe 2 O 4 (0.0 # x # 0.4) and Ti x Mn 1Àx Fe 2 O 4 (0.0 # x # 0.3) were prepared using the conventional method for preparing ceramics. The samples were found to consist of a single phase with a cubic spinel structure. The lattice constant increased in the Co-series samples and decreased in the Mn-series samples with increases in the dopant level, x. The values of m exp of the two series of samples, measured at 10 K, decreased approximate linearly with increasing x.
The dependence of m exp on x for the two series of samples was tted using a quantum-mechanical potential barrier method. The tted magnetic moments were very close to the experimental results. In the tting process, the cation distributions of the two series of samples were obtained.
The cation distributions and the magnetic structure obtained in this study are distinctly different from those reported by other groups: (i) there were Ti 2+ and Ti 3+ ions, but no Ti 4+ ions, in our samples. (ii) The ratio of Ti 2+ cations that entered the [B] sites to the Ti-doping level, x, increased from 58% (x ¼ 0.1) to 63% (x ¼ 0.4) in the Co-series samples, and this ratio was 79% from x ¼ 0.1 to x ¼ 0.3 in the Mn-series samples. (iii) The magnetic moments of Ti 2+ , Ti 3+ , and Mn 3+ ions (with 3d electron number of n d # 4) coupled antiferromagnetically with other cations (n d $ 5) whenever they were at the (A) or [B] sublattice.