Materials Science and Engineering B 111 (2004) 150–155

Neodymium ion dopant effects on the phase transformationin sol–gel derived titania nanostructures

Andrew Burnsa,b, G. Hayesc, W. Li a, J. Hirvonend, J. Derek Demareed, S. Ismat Shaha,e,∗a Department of Material Science and Engineering, University of Delaware, 208 Dupont Hall, Newark, DE 19716, USA

b Department of Chemistry and Biochemistry, University of Delaware, Newark, DE 19716, USAc Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA

d US Army Research Laboratory, Aberdeen Proving Ground, MD, USAe Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA

Received 22 August 2003; accepted 5 April 2004

Abstract

Doped and undoped nanostructured titanium dioxide was synthesized via a sol–gel method under varying conditions to determine the effectsof neodymium ion doping on the titania lattice. Specifically, the effects of doping on the anatase to rutile phase transformation were studied.Samples were analyzed by a variety of techniques, including X-ray diffraction, Rutherford back-scattering spectroscopy, scanning electronmicroscopy, and energy dispersive X-ray spectrometry to investigate the thermodynamic and structural effects of dopant addition. Analysisof the anatase to rutile phase transformation in TiO2 showed a maximum increase in activation energy for 0.1 mol% Nd3+ doped TiO2 withno further response at higher dopant concentrations. This was supported by structural analyses, which showed distortion only along thec-axisof anatase TiO2 with a maximal distortion at 0.1 mol% Nd3+ and no further response at higher concentration. This is due to a combination ofinterstitial and substitutional accommodation of the dopant ions.© 2004 Elsevier B.V. All rights reserved.

Keywords: Titanium dioxide; Neodymium; Doping effects; Phase transitions; Sol–gel

1. Introduction

Nanostructured semiconductors have been studied exten-sively for their desirable photocatalytic[1–4], optical [5],and electronic[6,7] properties, as well as their applicationin photovoltaics[3,8,9]. For photocatalysis, one of the mostpromising examples of this class of compounds is titaniumdioxide. The greater interest in TiO2 rests in the fact thatit is non-toxic, inexpensive, highly photoactive, and easilysynthesized and handled.

The optical and electronic properties of nanostructuredTiO2 can be tailored by a variety of measures, includingthermal treatments[10–13], supported film growth[14],and metal-ion doping[15,16]. In particular, doping withlanthanide metal ions, such as neodymium (Nd3+), hasbeen shown to increase photocatalytic efficiency for se-

∗ Corresponding author. Tel.:+1-302-831-1618;fax: +1-302-831-4545.

E-mail address: ismat@udel.edu (S.I. Shah).

lected reactions[2,15]. Therefore, it is critical to assess theeffects of lanthanide-ion doping on the structure of titaniawhich will allow greater control over the desired properties.Specifically, control over the anatase to rutile conversiontemperature for a chosen dopant concentration, would allowparticular ratios of phases to be produced for mixed-phaseapplications[17].

The photocatalytic activity of doped TiO2 is also greatlyaffected by the location and the nature of the dopant. Forexample, neodymium is known to increase the degradationof chlorophenols, whereas iron is completely ineffectivein these degradations[15]. Dopant effectiveness is partlyrelated to the ionic radius and oxygen affinity of the dopantion. The comparatively large Nd3+ ion produces a localizedcharge perturbation when present substitutionally in TiO2.Whereas, an Fe3+ ion has a comparable ionic radius to Ti4+and is not useful for the enhancement of photocatalyticactivity for 2-chlorophenol degradation. It is, therefore, im-portant to understand the location of the dopant in the TiO2lattice. An analysis of the effects of dopant addition on the

0921-5107/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.mseb.2004.04.008

A. Burns et al. / Materials Science and Engineering B 111 (2004) 150–155 151

lattice parameters of titania will allow insight into the wayin which the dopant ions are incorporated into the lattice.

Titanium dioxide naturally occurs primarily in three crys-talline phases: brookite (orthorhombic), anatase (tetragonal),and rutile (tetragonal). Upon annealing, metastable anatase,and brookite undergo first-order phase transformations to-wards the stable rutile phase. There are several pathways thisphase transformation can take including: anatase to rutile,anatase to brookite to rutile, brookite to rutile, and brookiteto anatase to rutile. The transition sequence is dependent onthe experimental conditions and the properties of the initialsample including particle size, initial phase, annealing tem-perature, etc.[11,18].

The kinetics of phase transformation in undoped titania isextensively reviewed by Zhang and Banfield[19], who com-pared results and methodologies for the past several decadesof research. For bulk TiO2, the activation energy falls be-tween roughly 350 and 700 kJ/mol, depending on grainsize and synthetic method. Zhang and Banfield reportedconversion temperatures in the range of 900–1100◦C. Fornanocrystalline titania, recent research has shown activationenergies in the vicinity of 150–200 kJ/mol and phase trans-formation temperatures in the range of 700–900◦C [20].Initial particle size plays a crucial role in the phase trans-formation due to the effects of its related surface energyor surface enthalpy. Banfield and co-workers[13,18,21]investigated the transformation of anatase and/or brookiteto rutile as a function of initial particle size and found thatthe relative stability of anatase increases with decreasingparticle size. However, these studies are based on pure TiO2.

For doped TiO2, phase transformation analysis is com-plicated by dopant-related effects not only on particle sizebut also on the lattice thermodynamics through altered lat-tice parameters. Previous studies on the phase transforma-tion characteristics of doped TiO2 have shown stabilizationof both anatase and rutile phases for different experimen-tal conditions. For example, the work of Arroyo et al.[22]on manganese (Mn2+) doped titania showed stabilization ofthe anatase phase for low dopant levels (∼0.5 mol%), priorto segregation of the dopant to the surface at higher con-centrations (∼3 mol%) and stabilization of the rutile phase.The work of Arbiol et al. [23] on niobium-doped sam-ples showed a similar trend. For low dopant concentration(∼3 mol%) the anatase phase was stabilized, as evinced byan elevated annealing temperature needed to introduce ru-tile peaks in XRD and Raman spectroscopy[22]. Similarto the results of Arbiol et al.[23], at higher dopant con-centrations (>4 mol%), formation and segregation of NbOwas found to be the dominant feature of the system, sta-bilizing the rutile phase. It must be emphasized that bothof these previously reported dopant studies dealt with dou-bly oxidized ions, whereas neodymium is a triply oxidizedion, which changes its interaction in the lattice of Ti4+ andO2−. The work of Zhang and Reller[24] on TiO2-doping byFe4+ and Ni4+ also showed evidence of segregated phases,such as Ti2Fe2O5. In this paper, we present the results of

the investigation of Nd3+ doping on the structural proper-ties of TiO2 and its effect on the anatase to rutile phasetransformation.

2. Experimental

Titanium dioxide nanostructures have been fabricated bymany methods including metalorganic chemical vapor de-position (MOCVD)[25], coprecipitation[26], TiCl4 oxida-tion [27], and sol–gel processing[28,29]. In this study, weused a sol–gel method[30] because it is capable of pro-ducing highly homogeneous metal oxide nanoparticles aswell as its facility for dopant addition. The doped titaniananoparticles were synthesized from titanium tetrachloride(Fluka 98%), and neodymium (III) acetylacetonate hydrate(Aldrich). The dopant stoichiometry was controlled by dis-solving the Nd3+ precursor in ethanol (Pharmco 200 proof)prior to the drop-wise addition of TiCl4. The reaction wasperformed at room temperature while stirring in a fume hoodto avoid exposure to the Cl2 and HCl gases that evolve dur-ing the reaction. The resulting yellow solution was allowedto rest until all gas evolution had ceased and the solutionreturned to room temperature.

All samples were dip-coated onto quartz and boron-doped(p-type) silicon substrates at a withdrawal rate of 3 cm/min.The coated substrates were allowed to dry in a dessicator,followed by calcination for 30 min in a box furnace operat-ing between 700 and 900◦C in an ambient atmosphere. ForXRD analysis, several layers of TiO2 were deposited, witha calcination step following each dipping cycle.

Structural characterizations of the doped and un-dopedTiO2 samples were done by X-ray diffraction (XRD).θ–2θscans were recorded at several resolutions using Cu K� ra-diation in a Rigaku D-Max B diffractometer equipped witha graphite crystal monochromator. Analysis of the result-ing data was completed using XFIT[31] software to de-termine peak position, width, and intensity. Full-width athalf-maxima (FWHM) data was analyzed by Scherrer’s for-mula [32] to determine average particle sizes. The instru-mental broadening for the Rigaku D-Max B diffractometerwas 0.011◦, which was subtracted from the peak width inorder to calculate the particle size. Since peak width used inthe Scherrer’s formula also includes the line broadening dueto strain in the particles, measurements of particle size andsize distribution were also carried out by scanning electronmicroscopy (SEM) on a JEOL SEM, operating at 15 kV. TheXRD and SEM results show very similar particle sizes.

The dopant concentration was verified by Ruther-ford back-scattering spectroscopy (RBS), as well as en-ergy dispersive X-ray spectroscopy (EDX) performed inarea-averaging mode on samples prepared on quartz sub-strates. Rutherford back-scattering spectroscopy was usedto determine the Nd3+ concentration in the TiO2 lattice. A2 MeV He+ source was used. The work was carried out atthe Army Research Laboratories in Aberdeen, Maryland.

152 A. Burns et al. / Materials Science and Engineering B 111 (2004) 150–155

To determine the effects of the rutile/anatase mixture, amethod for determining the relative mass concentrations wasneeded. The mass fraction of rutile (xr) in the crystal latticecan be calculated based on the relationship between the in-tegrated intensities of anatase (1 0 1) and rutile (1 1 0) peaksby the following formula developed by Spurr and Myers[33]:

xrutile = 1

1 + K(Ia/Ir)(1)

where Ia and Ir are the integrated peak intensities of theanatase and rutile peaks, respectively. The empirical constantK was determined via an XRD analysis of powders of knownproportions of pure anatase and pure rutile TiO2, and is equalto 0.79.

In order to measure the effects of Nd3+ addition to theTiO2 lattice, the lattice constants of samples containing vary-ing amounts of dopant were measured. For this analysis, onlysamples containing the anatase phase were used, to avoid theeffects of rutile crystals segregated out of the anatase sys-tem. Samples containing 0.0, 0.1, 0.2, and 0.35 mol% Nd3+were annealed at 700◦C for 30 min before analysis by XRD.For the anatase crystal system the lattice constants ‘a’ and‘c’ were determined from two appropriate reflections (h k l)using the following formula[32]:

d = 1√(h/a) + (k/a) + (l/c)

(2)

The value ofd, for an XRD peak can be determined fromthe 2θ-angle by Bragg’s Law[32]

d = λ

2 sinθ(3)

Fig. 1. Scanning electron micrograph of 1.5 wt.% Nd doped TiO2.

whereλ is the X-ray wavelength equal to 0.15405 nm for CuK� radiation andθ is in radians. For this analysis, the peakpositions (2θ) of the anatase (1 0 1) and (2 0 0) reflectionswere used to calculate the lattice parameters.

Grain sizes were calculated by Scherrer’s Formula shownbelow

t = 0.9λ

B cos(θB)(4)

whereλ is the X-ray wavelength (0.154178 nm for Cu K�),B the peak width (FWHM), andθB is the Bragg angle[32].

The activation energy (kJ/mol) of phase transformationcan be approximated based on the extrapolation of anataseand rutile weight fractions versus the annealing temperaturefrom XRD spectra. The relationship is given below

Ea = −∂ln(xr)

∂(1/T)

R

1000(5)

whereT is the temperature in Kelvin,R the universal gasconstant (8.314 J/mol K), andxr is the weight fraction ofrutile as previously defined inEq. (1).

3. Results and discussion

Fig. 1 shows a scanning electron micrograph of the0.1 mol% Nd-doped sample. Grain sizes, ranging from 12to 25 nm, were obtained from XRD usingEq. (4). The grainsize of the sample remained essentially unaffected by thedoping in the range discussed in this paper, though therewas considerable variation as a function of temperature.

To draw useful observations from the collected data, anaccurate assessment of the dopant concentration and distri-

A. Burns et al. / Materials Science and Engineering B 111 (2004) 150–155 153

Fig. 2. Rutherford back-scattering spectrum of 3.0 wt.% Nd-doped titania.

bution in the TiO2 lattice was necessary. The XRD spectradid not show any evidence of secondary phases related to Nd.To ascertain successful intermixing of the Nd dopant withthe titania during processing, RBS was employed. ThroughRBS, the relative concentrations of Ti and Nd in the sampleswere determined. This technique is particularly useful be-cause of its direct analysis of the nuclei in the sample. RBSprobes to large depth in the sample, analyzing the sampleas a whole, rather than simply the surface. An RBS spec-trum for a 0.2 mol% doped sample is shown inFig. 2. Thesilicon peak in the spectrum is due to the substrate used inthe experiment. This data, in conjunction with results fromEDX, are used as the benchmarks in the results that follow.

In order to evaluate the activation energy and transitiontemperature of sol–gel synthesized doped and undopednanostructured TiO2, we carried out sequential annealingexperiments followed by XRD analyses. The as-depositedsamples were calcined at temperatures ranging from 700to 900◦C. The XRD patterns obtained after calcination ofundoped, 0.1 and 0.2 mol% Nd-doped samples are shownin Figs. 3–5.

All as-deposited doped and undoped samples were amor-phous. Since the focus of this study was the investigation ofthe anatase to rutile transformation, we did not concentrateon the amorphous to anatase transformation. Annealing re-sulted in diffraction peaks related only to the anatase andrutile polymorphs of TiO2. In the undoped samples (Fig. 3),the first XRD peaks to appear were all anatase-related. Pureanatase phase persisted up to 700◦C. Rutile-related peaksbegan to appear between 700 and 750◦C. Between 850 and900◦C, the sample converted completely to rutile, and noanatase-related peak was detectable. In the 0.1 mol% Nd3+doped samples (Fig. 4) the phase transformation began be-tween 750 and 800◦C. Likewise, in the 0.2 mol% Nd3+

Fig. 3. X-ray diffraction patterns as a function of annealing temperaturefor undoped nanostructured titania (A, anatase; R, rutile).

doped sample (Fig. 5) no rutile-related XRD peaks were de-tected until the annealing temperatures reached 800◦C.

Due to the small crystal size (15–30 nm) of the preparedsamples, the anatase to rutile conversion temperature fallswithin the range of nanoscale phase transformation.Fig. 6is a plot of the natural logarithm of rutile weight fractionversus inverse temperature for doped and undoped titaniasamples.

The calculated activation energies fromFig. 6 for theanatase–rutile phase transformations are listed inTable 1.The calculated activation energies of both the doped andundoped TiO2 nanoparticles are lower than that of the bulk(350–700 kJ/mol). This is expected because the nanosize

Fig. 4. X-ray diffraction patterns as a function of annealing temperaturefor 1.5 wt.% Nd(III)-doped nanostructured titania (A, anatase; R, rutile).

154 A. Burns et al. / Materials Science and Engineering B 111 (2004) 150–155

Fig. 5. X-ray diffraction patterns as a function of annealing temperaturefor 3.0 wt.% Nd(III)-doped nanostructured titania (A, anatase; R, rutile).

Fig. 6. ln of the rutile percentage (xr) vs. inverse temperature to determinethe anatase to rutile phase transformation activation energy.

grains cause higher surface area to volume ratios comparedto bulk, which increase the total surface energy. Therefore,less energy is required to induce the phase transformation.The activation energy for the anatase to rutile phase trans-formation increases with the addition of the dopants.

Table 1Activation energies of anatase–rutile transformation vs. dopant concen-tration

Sample ∂ ln(xr)/∂(1/T) Ea (kJ/mol)

Undoped TiO2 −1.86 ± 0.05 × 104 154 ± 40.1 mol% doped TiO2 −2.28 ± 0.06 × 104 190 ± 50.2 mol% doped TiO2 −2.22 ± 0.06 × 104 185 ± 5

Fig. 7. Lattice constants of titania as a function of neodymium doping.

The anatase-to-rutile phase transformation is known tobe a nucleation and growth process during which rutilenuclei form within the anatase phase and grow in sizewith increasing temperature, eventually consuming thesurrounding anatase. The addition of dopant ions causesgrain-boundary pinning in which grain growth is limited bythe symmetry-breaking effects of dopants at the boundarywhich slows the growth of rutile phase, resulting in higheractivation energies for the phase transformation. It is unclearwhy the increase in the activation energy is independent ofthe total dopant concentration, which offers room for fur-ther study. The TiO2 anatase unit crystal is tetragonal, withlattice parameters ‘a’ and ‘c’. The rutile crystal structure isalso tetragonal with a smallerc/a ratio than that of anatase.

In order to measure the effects of neodymium-ion additionto the TiO2 lattice, the lattice constants of samples contain-ing varying amounts of neodymium dopant were determinedvia XRD. This analysis was performed on samples calcinedat 700◦C (below the anatase to rutile transformation tem-perature) to avoid the effects of rutile crystals segregatingout of the anatase system. Samples containing 0.0, 0.1, 0.2,and 0.35 mol% neodymium were annealed for 30 min beforeanalysis by XRD. For this analysis, the peak values of theanatase (1 0 1) and (2 0 0) reflections were used to calculatethe lattice parameters viaEqs. (2) and (3).

From the data collected, it was determined that the latticeconstant ‘c’ increases with dopant addition, up to 0.1 mol%,and becomes constant at approximately 9.675 Å for furtherincreases in dopant concentration, while the value of ‘a’remains essentially unchanged, as shown inFig. 7.

The effective ionic radii of Ti4+ and Nd3+ are 0.605 and0.983 Å, respectively[34,35]. Therefore, any substitution ofan neodymium ion for a titanium ion in the TiO2 latticewould introduce a distortion. Because only the ‘c’ dimen-sion is changing, while ‘a’ remains constant for the rangeof dopant stoichiometries studied, it can be speculated thatNd3+ is substituting for Ti4+ preferentially on the body cen-

A. Burns et al. / Materials Science and Engineering B 111 (2004) 150–155 155

tered and face centered lattice sites in the anatase structure.The existence of a maximum ‘c’ value, which remains con-stant for higher dopant concentrations, also suggests that theNd3+ may incorporate interstitially as well as substitution-ally. If Nd3+ were only incorporated substitutionally, the lat-tice constants would continue to increase with higher dopantconcentrations, but as evinced byFig. 7, this is not the case.As previously discussed, XRD analysis showed no evidenceof a secondary phase related to neodymium, therefore, theonly place for it to be incorporated is in the interstices. Withonly XRD, it is difficult to determine the exact fraction ofsubstitutionally incorporated Nd3+. Other techniques, suchas extended X-ray absorption fine structure (EXAFS) mustbe used to determine the exact percentage of substitutionallyand interstitially accommodated neodymium. This is amongthe future goals of this research. Corroborative evidence forinterstitial accommodation of Nd3+ is seen in the thermo-dynamic analysis of the anatase to rutile phase transforma-tion in Table 1, where no further increase inEa was seen forNd3+ concentrations above 0.1 mol%.

Interstitial neodymium does not affect the charge balancein the anatase lattice, as a substitutional neodymium does.Thus, interstitial dopants do not affect the photocatalyticproperties of the nanoparticles since they cannot act directlyas trapping sites to enhance the carrier lifetime. Furthermore,other results in the group working on MOCVD-synthesizedTiO2 [15] showed a maximum in the photocatalytic activityaround 0.07 mol% Nd3+.

4. Conclusions

Doped and undoped nanostructured TiO2 was synthesizedby a sol–gel method using TiCl4 as titanium precursor. Sam-ples were characterized for their chemical composition byEDX and RBS and for their structure by XRD. The pur-pose of this study was to understand the effect of dopants,in particular neodymium, on the anatase to rutile structuraltransformation in nanostructured TiO2. Both the transforma-tion energy and the transformation temperature, measuredfrom the annealing and XRD experiments, were found toincrease with doping, with a maximum increase in activa-tion energy occurring at 0.1 mol%. Higher concentrationsof Nd3+ had no further effect on the phase transforma-tion temperature or enthalpy. The anatase lattice was shownto deform predominantly along thec-axis to accommodatesubstitutionally-incorporated Nd3+. The maximum in elon-gation of thec-axis also occurs at 0.1 mol%. This suggestedsome incorporation of Nd on the interstitial sites. This pos-sibility must be confirmed by further analyses by techniquessuch as extended X-ray absorption fine structure, etc.

Acknowledgements

We would like to thank the following for their contribu-tions to this research: Ernest Addo, Emily Peng, and Ger-

ald Poirier. We would also like to thank NSF-NIRT (Grantnumber: DMR-0210284), the University of Delaware Un-dergraduate Research Department American Vacuum Soci-ety for financial support of this research.

References

[1] D. Beydoun, R. Amal, G. Low, S. McEvoy, J. Nanoparticle Res. 1(1998) 439.

[2] W. Li, S.I. Shah, C.-P. Huang, O. Jung, C. Ni, PNAS 99 (2) (2002)6482.

[3] B. O’Regan, M. Gratzel, Nature 353 (1991) 737.[4] D. Ollis, H. Al-Ekabi (Eds.), Photocatalytic Purification and Treat-

ment of Water and Air, Elsevier, Amsterdam, 1993.[5] S. Rabaste, J. Bellessa, A. Brioude, C. Bovier, J.C. Plenet, R. Brenier,

O. Marty, J. Mugnier, J. Dumas, Thin Solid Films 416 (1) (2002)242.

[6] S.-H. Song, X. Wang, P. Xiao, Mater. Sci. Eng. B 94 (1) (2002) 40.[7] A. Henglein, Chem. Rev. 89 (1989) 1861.[8] U. Bach, D. Lupo, P. Comte, J.E. Moser, F. Weissortel, J. Salbeck,

H. Spreitzer, M. Gratzel, Nature 395 (1998) 583.[9] E. Pellizetti, M. Schiavello (Eds.), Photochemical Conversion and

Storage of Solar Energy, Kluwer Academic Publishers, Dordrecht,1991.

[10] A. Linsebigler, G. Lu, J. Yates, Chem. Rev. 95 (1995) 735.[11] J. Banfield, A. Navrotsky (Eds.), Nanoparticles and the Environment,

Mineralogical Society of America, 2001.[12] M. Ranade, A. Navrotsky, H. Zhang, J. Banfield, S. Elder, A. Zaban,

P. Borse, S. Kulkarni, G. Doran, H. Whitfield, Proc. Natl. Acad. Sci.U.S.A. 99 (2002) 6476.

[13] H. Zhang, J. Banfield, J. Mater. Chem. 8 (1998) 2073.[14] D. Huang, Z.-D. Xiao, J.-H. Gu, N.-P. Huang, C.-W. Yuan, Thin

Solid Films 305 (1997) 110.[15] W. Li, S.I. Shah, C.-P. Huang, O. Jung, C. Ni, Mater. Sci. Eng. B

96 (2002) 247.[16] R. Choi, A. Termin, M. Hoffmann, J. Phys. Chem. 98 (1994) 13669.[17] R. Berry, M. Meuller, Microchem. J. 50 (1994) 28.[18] A. Gribb, J. Banfield, Am. Mineral. 82 (1997) 717.[19] H. Zhang, J. Banfield, Am. Mineral. 84 (1999) 528.[20] K. Kumar, K. Keizer, A. Burggraaf, J. Mater. Chem. 3 (1993)

1141.[21] H. Zhang, J. Banfield, J. Phys. Chem. B. 104 (2000) 3481.[22] R. Arroyo, G. Cordoba, J. Padilla, V.H. Lara, Mater. Lett. 54 (2002)

397.[23] J. Arbiol, J. Cerda, G. Dezanneau, A. Cirera, F. Peiro, A. Cornet,

J.R. Morante, J. Appl. Phys. 92 (2002) 853.[24] Y.-H. Zhang, A. Reller, Mater. Sci. Eng. C 19 (2002) 13.[25] K. Okuyama, R. Ushio, Y. Kousaka, R. Flagan, J. Seinfeld, AIChE

J. 36 (1990) 409.[26] L. Palmisano, V. Augugliaro, A. Sclafani, M. Schiavello, J. Phys.

Chem. 92 (1998) 6710.[27] M. Akhtar, Y. Xiong, S. Pratsinis, AIChE J. 56 (1991) 13.[28] Y. Zhou, C. Wang, H. Liu, Y. Zhu, Z. Chen, Mater. Sci. Eng. B 67

(1999) 95.[29] K. Ranjit, B. Viswanathan, J. Photochem. Photobiol. A: Chem. 108

(1997) 79.[30] A. Burns, W. Li, C. Baker, S.I. Shah, Mater. Res. Soc. Symp. Proc.

703 (2002) V.5.2.1.[31] A. Coelho, R. Cleary,http://www.ccp14.ac.uk/tutorial/xfit-95/.[32] B. Cullity, Elements of X-Ray Diffraction, Addison-Wesley, Menlo

Park, CA, 1978.[33] R. Spurr, H. Myers, Anal. Chem. 29 (1957) 760.[34] R.D. Shannon, C.T. Prewitt, Acta Crystallogr. Sect. B 25 (1969) 925.[35] R.D. Shannon, Acta Crystallogr. Sect. A. 32 (1976) 751.