rietveld refinement of xrd and nd: analysis of metastable qandilite

RIETVELD REFINEMENT WITH XRD AND ND: ANALYSIS OF METASTABLE QANDILITE-LIKE STRUCTURES

G. Kimmel1, J. W. Richardson2, R. Xu1, P. Ari-Gur1, E. Goncharov3, J. Zabicky3

1 Western Michigan University - USA, 2 Argonne National Laboratory – USA, 3Ben-Gurion University of the Negev - Israel, Email: [email protected]

ABSTRACT

Binary oxides of Mg and Ti were prepared by the sol-gel technique in various Mg/Ti atomic ratios, r. Metastable phases, produced between 873 and 973 K from xerogels with Mg/Ti atomic ratios in the range ~1.1 < r ≤ 2, were studied by XRD and ND. XRD analysis showed crystal structures similar to that of Mg2TiO4 (qandilite), an inverse spinel of cubic structure. The composition range where this occurred did not correspond to those of the conventional equilibrium phase diagrams. The data were processed by the Rietveld method. ND data led to improvements in the refinement of the atomic occupancies. Assuming some degree of tetragonal distortion was effective only in ND analysis. It was found by ND that the charge balance is kept only for a certain extent of mixed occupancy of the octahedral sites, leading to a molecular formula Mg2-2xTi1+xO4, were 0 < x < 0.25. Using this formula as a constraint in the Rietveld refinement with XRD improved the Rietveld refinement. An important contribution of XRD was in rejecting any attempt to introduce Ti in the tetrahedral sites.

INTRODUCTION

In the conventional phase diagram of the MgO-TiO2 system, Mg2TiO4 (qandilite) exists only above 1400 K as an inverse spinel cubic structure [1,2]. However, the amorphous xerogels of oxides with r = Mg/Ti atomic ratio in the range ~1.1 < r ≤ 2 yield a single phase with qandilite-like structures at ~873-973 K [2,3]. The X-ray diffractograms exhibit a severe line-broadening effect due to the nanometric size of the crystallites and the relatively large microstrain. Rietveld refinement with XRD led to the conclusion that the oxygen basis of the crystalline structure is identical to that of conventional qandilite [2,3]. In these previously reported works attention was paid to fitting the refinements to acceptable structural schemes, however, only in the present investigation attention is paid to correct chemical structures that include ionic charge balance.

Motivated by published studies [1,4] it was decided to conduct high-resolution neutron diffraction (ND) studies in combination with XRD. A transition to a Mg2TiO4 tetragonal structure was reported for qandilite samples treated for long periods below 940 K [1,4,5]. The possibility of formation of a tetragonal structure for the nonstoichiometric phases is examined in this work.

EXPERIMENTAL

Several qandilite-like phases (r from 1.5 up to 2.0) were synthesized by the sol-gel technique applied to solutions of Mg and Ti alkoxides in nonprotonic solvents, according to a procedure published elsewhere [6]. The xerogels were fired for 3 h at 873 K and rapidly cooled to room

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temperature; the obtained products were fine white powders. XRD was carried out in θ/2θ scans, by the Bragg-Brentano method, using a Philips APD system (fixed slits beam divergence 1º, reflected beam graphite monochromator). The beam source was a long-line focus copper anode powered by a PW1730 generator. The running conditions were: 45 kV, 40 mA, step-scan mode; 2θ range 10-155º, 2θ step 0.02º with 6 sec/step, with a proportional (Xe) detector. Similar samples were analyzed by ND. Data were collected on a 20 m time of flight. GPPD was used at the Intense Pulsed Neutron Source of Argonne National Laboratory, Illinois, USA. The target used for generation of the neutron beam was 238U. The samples were surrounded on both sides by 144 3He counters; the sample-to-detector distance was 1.5 m; the wavelength range was 0.02-0.57 nm. The structure refinement was made using the DBWS software [7] with XRD data and the GSAS software [8] with ND data. RESULTS AND DISCUSSION X-Ray diffraction. Fig. 1 shows diffractograms of qandilite-like phases with r = ~1.7. The xerogel fired at 873 K for 3 h shows broad lines belonging to a single-phase structure matching that of conventional Mg2TiO4 – cubic spinel structure. After 3 h at 1473 K a mixture of Mg2TiO4 and MgTiO3 appears, exhibiting sharp diffraction lines. The Rietveld refinement of the powder treated at 1473 K yielded well crystallized inverse spinel Mg(MgTi)O4 and ilmenite-type MgTiO3 structures. The ratio of Mg/Ti = r obtained from the quantitative analysis of the two phases, was 1.72, in good agreement with the r-value of the xerogel. In spite of the fact that the product treated at 873 K was a fine powder – with particle size smaller than 5 µm and spherical shape – which is ideal for powder diffraction, the Rietveld refinement yielded high values for the RB (R-Bragg) factor (0.09) and goodness of fit (GOF = 1.4). The deviation from stoichiometry should be associated with variation of the ion occupancies. However, the assumption that all ions site occupancies are refinable has lead to a solution that was chemically incorrect, namely, r = 0.5 instead of ~1.7, and the number of O2- ions per molecular formula was 5.4 instead of 4.0, with no charge balance; the RB factor was 0.05 and the GOF was 1.28. As the r value of the qandilite-like phases decreases the relative amount of Ti increases. Consequently the positive charge tends to increase and the oxygen amount should increase as well in order to keep the charge balance. Since it is impossible to consider oxygen interstitials it was assumed that the oxygen occupancy is saturated and both the tetrahedral and octahedral cation sites are occupied with Mg and Ti. This refinement failed because the Ti occupancy in the tetrahedral sites was negative. (RB = 0.05 and GOF = 1.28). Thus, other slight modifications were tried for the inverse spinel structure of qandilite: (i) The tetrahedral sites are fully occupied by Mg, whereas the deviations from stoichiometry

affect only the octahedral sites. The molecular formula for this case is Mg(Mg1-xTi1+x)O4. The results were r = 1.39 instead of ~1.7, RB = 0.076 and GOF =1.39.

(ii) As in (i) but refine also the Mg occupation in the tetrahedral sites. The molecular formula for this case is Mg1-x(Mg1-yTi1+y)O4. The results were r = 2.70 instead of ~1.7, RB = 0.05 and GOF = 1.28. The cationic charge was 7.24 instead of 8 (see Table 1 column XRD-c (a)).

(iii) Refine only the Mg occupation in the tetrahedral sites, keeping the octahedral sites with fixed and equal occupancies of Mg and Ti. The molecular formula is Mg1-x(MgTi)O4.


This gave the best agreement. The results were r = 1.88, reasonably close to ~1.7, RB = 0.046 and GOF = 1.22. The cationic charge was 7.76 instead of 8 (see Table 1 column XRD-c (b).

(iv) Refine as tetragonal. The molecular formula is Mg1-x(Mg0.5-yTi0.5+y)(Mg0.5-zTi0.5+z)O4. The RB and GOF values were lower than those of the cubic case, but r did not agree with the real composition and also the cationic charge was ~7 instead of 8 (see Tables 1 column XRD-t (c).) The tetrahedral sites were found to be occupied only by Mg with 15% vacancies. Occupancy of the octahedral sites was highly disordered and with more Mg than Ti.

Figure 1. Diffractogram of a qandilite-like phase (r = ~1.7) prepared by the sol-gel method and annealed at 873 and 1473 K for 3 h. The dotted vertical lines show the position of the diffraction lines of conventional Mg2TiO4.

The same considerations were applied to other Mg/Ti ratios, r. Table 2 shows the results for r = 1.5. Columns XRD-c (a) XRD-c (b) and XRD –t (c) are related to cases (ii), (iii) and (iv), respectively. Neutron diffraction. Table 3 shows the results for the neutron diffraction without constraints, assuming that all the cationic sites contain Ti and Mg, and no oxygen vacancies (attempts to refine the oxygen occupancies failed). It was refined as cubic and as tetragonal. It seems that in both cases there is agreement between the occupancies and r. However, RB was lower for the tetragonal. Moreover, the tetragonal structure was similar to the low-temperature structure reported in the literature [4,5] but with more disorder. It is proposed for the nonstoichiometric qandilite-like phases that the molecular formula be written with four oxygen atoms, as in qandilite, paying attention to the charge balance. This can be achieved by the formula Mg2-2xTi1+xO4, with 0 ≤ x ≤ 0.25. Taking into account that the characteristic composition datum of the xerogel is the Mg/Ti atomic ratio, r, the correlations


shown in the equations r = (2 – 2x)/(1 + x) or x = (2 – r)/(2 + r) may be established between the composition of the qandilite-like phases and that of their original xerogels. Refinement with XRD and ND. Inspired by the ND results, it was decided to introduce charge balance as a constraint to the XRD refinement. Assuming the molecular formula Mg1-xMg1-xTi1+xO4, (Table 4) allowed refinement with the XRD data for a cubic system with charge balance. The results were r = 1.70, the same as the original, RB = 0.04 and GOF = 1.20. (Table 1 column XRD-c (d)). For the tetragonal system several charge-balanced molecular formulas were tried. The most successful one was: Mg1-xMg0.5+xTi0.5-xMg0.5-2xTi0.5+2xO4 (Table 5). The results were r = 1.69, the original, RB = 0.04 and GOF = 1.17. (Table 1 column XRD-t (e)). Usually the results for the refinement with XRD for a tetragonal system were slightly better than for the cubic (see also Table 2 columns XRD-c (d) and XRD-t (e)). Furthermore, in ND the cubic structure strongly deviates from the inverse spinel due to a high amount of Ti in the tetrahedral sites. This is not possible in XRD data refining because the Ti appears with negative occupancies. This discrepancy stems from the negative amplitude of neutron scattering of Ti, making the distinction between Ti and vacancies problematic in ND. Replacing the Ti in the tetrahedral sites by vacancies causes the ND refinement to yield the same results (see Tables 1 and 2, column ND-t (f)) as without constraints (see Table 3). A typical crystal structure for the nonstochiometric qandilite-like phase and balanced ion-charge, is shown in Table 6. The cation-anion bond calculated for XRD-t (e) are similar to the those reported by Millard et al. [5], however, that for the tetrahedral sites T-O is 0.200-0.205 nm, as compared to 0.199 in [5], for the octahedral the average bond lengths are 0.2050 nm for O1-O and 0.2015 nm for O2-O, as compared to 0.2081 and 0.1970 nm, respectively in [5]. The bond lengths for XRD-t (e) support the refined occupancies, suggesting that only Mg is on the T site, O1, O2 are mixed sites where O1 is richer in Mg and O2 is richer in Ti. In our case the smaller difference between O1-O and O2-O bond lengths is attributed to a higher degree of disorder. The vacancies of Mg in the T site may slightly increase the T-O bond length.

Table 1. Comparison of refinement results between XRD and ND for Mg/Ti = r ~1.7. c is cubic and t is tetragonal, cell volume in Å3. XRD-c (a)XRD-c (b) XRD-t (c) XRD-c (d) XRD-t (e) ND –t (f)a (Å) 8.4385 8.4385 5.9769 8.4385 5.9776 5.9730c (Å) 8.4073 8.4064 8.3960c/a' [a’=a·sqr(2)] 0.9946 0.9944 0.9940cell volume 300.34 300.37 299.54 cell volume = cubic 600.89 600.89 600.67 600.89 600.75 599.08 Occupancy: T -mg 0.8510 0.8813 0.8517 0.9195 0.9160 0.8688Occupancy: O1-Mg 0.6150 0.5 0.6617 0.4598 0.5840 0.5223Occupancy: O1-Ti 0.3850 0.5 0.3383 0.5402 0.4160 0.4777Occupancy: O2-Mg 0.6410 0.3320 0.4495Occupancy: O2-Ti 0.3590 0.6680 0.5504Mg per molecule 2.0810 1.8813 2.1544 1.8391 1.8320 1.8407Ti per molecule 0.7700 1.0000 0.6973 1.0804 1.0840 1.0281Mg/Ti ratio (r) 2.703 1.881 3.090 1.702 1.690 1.790Cations’ charge 7.24 7.76 7.10 8.00 8.00 7.79RB 0.050 0.046 0.026 0.042 0.041 0.05GOF for XRD 1.28 1.22 1.13 1.2 1.17


Table 2. Comparison of refinement results between XRD and ND for Mg/Ti = r ~1.5. c is cubic and t is tetragonal, cell volume in Å3. XRD-c (a) XRD-c (b) XRD-t (c) XRD-c (d) XRD-t (e) ND –t (f)a (Å) 8.4409 8.4409 5.9824 8.4414 5.9834 5.9721c (Å) 8.4020 8.4000 8.4079c/a' [a’=a·sqr(2)] 0.9931 0.9927 0.9955cell volume 300.70 300.7290 299.8760cell volume = cubic 601.40 601.40 601.40 601.51 601.4581 599.7520Occupancy: T -mg 0.8780 0.9060 0.8516 0.9222 0.9087 0.9973Occupancy: O1-Mg 0.6021 0.5000 0.4078 0.4611 0.5913 0.7388Occupancy: O1-Ti 0.3979 0.5000 0.5922 0.5389 0.4087 0.2612Occupancy: O2-Mg 0.8925 0.3174 0.0837Occupancy: O2-Ti 0.1075 0.6826 0.9163Mg per molecule 2.0822 1.906 2.1519 1.8444 1.8174 1.8198Ti per molecule 0.7958 1.000 0.6997 1.0778 1.0913 1.1775Mg/Ti ratio (r) 2.616 1.906 3.075 1.711 1.665 1.545Cations’ charge 7.35 7.82 7.10 8.00 8.00 8.35RB 0.054 0.050 0.039 0.049 0.048 0.092GOF for XRD 1.20 1.18 1.11 1.12 1.14

From the refinement with ND assuming cubic structures (Table 3), the T site included up to 25% Ti in most r ratios. This does not agree with the T-O bond lengths around 0.195 nm, which points to pure Mg occupancy. In the case of r = 1.5 there was no Ti but the refined r=1.786 was incorrect and the RB was high. These inconsistencies of the cubic structures favor the tetragonal ones.

Table 3. Neutron diffraction refined data without constraints. Cell volume in Å3.

Mg/Ti ratio (r) 2.0 1.9 1.8 1.5 Crystal system c t c t c t c t a (Å) 8.4257 5.9477 8.4257 5.9687 8.4294 5.9724 8.4330 5.9721c (Å) 8.4577 8.3964 8.3946 8.4330 8.4079c/a' [a’=a·sqr(2)] 1.0055 0.9947 0.9939 1.0000 0.9955cell volume 598.16 299.19 598.16 299.12 598.95 299.43 599.72 299.88cell volume = cubic 598.16 598.38 598.16 598.25 598.95 598.86 599.72 599.75 Occupancy: T -mg 0.7470 0.9375 0.7692 0.9076 0.8490 0.9368 1.0000 0.9983Occupancy T - Ti 0.2530 0.0625 0.2308 0.0924 0.1510 0.0632 0.0017Occupancy: O1-Mg 0.6134 0.8104 0.6073 0.7744 0.5550 0.6951 0.4616 0.7388Occupancy: O1-Ti 0.3866 0.1896 0.3927 0.2256 0.4450 0.3049 0.5384 0.2612Occupancy: O2-Mg 0.2548 0.3047 0.2803 0.0837Occupancy: O2-Ti 0.7452 0.6953 0.7197 0.9163Mg per molecule 1.9738 2.0027 1.9838 1.9867 1.9590 1.9122 1.9232 1.8208Ti per molecule 1.0262 0.9973 1.0162 1.0133 1.0410 1.0246 1.0768 1.1792Mg/Ti ratio (r) 1.923 2.008 1.952 1.961 1.882 1.866 1.786 1.544Cations’ charge 8.05 7.99 8.03 8.03 8.08 7.92 8.15 8.36RB 0.135 0.056 0.117 0.054 0.107 0.05 0.132 0.092


Table 4. Occupancies assuming charge balance in the cubic system. Molecular formula: Mg1-xMg1-xTi1+xO4.

T-Mg 1 – x O-Mg 1 – x O-Ti 1 + x

Table 5. Occupancies assuming charge balance in the tetragonal system. Molecular formula: Mg1-xMg0.5+xTi0.5-xMg0.5-2xTi0.5+2xO4.

T-Mg 1 – x O1-Mg 0.5 + x O1-Ti 0.5 – x O2-Mg 0.5 – 2x O2-Ti 0.5 + 2x

Table 6. Crystal structure Mg/Ti = r~1.7, and balanced ion charge,

RB = 0.04; GOF = 1.2; refined r = 1.69, obtained from XRD. Spacegroup: P4122. System: Tetragonal. Cel parameters a,c [Å]: 5.9776, 8.4064. name Wyck x y z site occupancy Tetr 4c 0.2514 0.2514 0.3750 0.8928 Mg2+ Oct1 4a 0.0000 0.2362 0.0000 0.6072 Mg2+ and 0.3928 Ti4+ Oct2 4b 0.5000 0.2451 0.0000 0.2855 Mg2+ and 0.7145 Ti4+ Oxy1 8d -0.0160 0.7471 0.2544 1 O2- Oxy2 8d 0.5249 0.2592 0.2297 1 O2-

SUMMARY AND CONCLUSIONS It is possible to represent stoichiometric and nonstoichiometric qandilite-like structures by a single molecular formula: Mg2-2xTi1+xO4, enforcing charge balance. The best agreement between calculated and observed data was obtained for tetragonal structures similar to the model reported in the literature for stoichiometric structures of Mg2TiO4 [4,5] with two modifications: In the case of nonstoichiometric qandilite-like phases Mg occupation of the tetragonal sites takes place with vacancies, and the octahedral sites have more disorder than in the stoichiometric model. The ND needed fewer constraints than the XRD refinements. Acknowledgement. The neutron diffraction work benefited from the use of the Intense Pulsed Neutron Source at Argonne National Laboratory. This facility is funded by the U.S. Department of Energy under Contract W-31-1009-ENG-38. REFERENCES [1] Wechsler, B.A; Navrotsky, A., J. Solid State Chem., 1984, 55, 165-180. [2] Kimmel, G.; Zabicky, J., Adv. X-Ray Anal. , 1998, 42, 238-244. [3] Kimmel, G.; Zabicky J., Mater. Sci. Forum, 1998, 278-281, 624-629. [4] Wechsler, B.A.; von Dreele, R.B., Acta Cryst. B, 1989, 45, 542-549. [5] Millard, R.L.; Peterson, R.C.; Hunter, B.K., Am. Mineral, 1995, 80,885-896. [6] Zabicky, J.; Zevin, L.; Simon, E.; Shneivais, A.; Sason, U.; Abramovich, L.; Ondracek, G.;

Schüller, M.; Fredel, M, Nanostruct. Mat., 1993, 3, 77-82. [7] Young, R.A.; Sakthivel, A.; Moss, T.S.; Paiva-Santos, C.O., J. Appl. Cryst., 1995, 28, 366-

367. [8] von Dreele, R.B.; Jorgensen, J.D.; Windsor, C.G., J. Appl. Cryst., 1982, 15, 581-589.


rietveld refinement of xrd and nd: analysis of metastable qandilite

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