The weight of a phase in a mixture is proportional to the product of the scale factor, as derived in a multicomponent Rietveld analysis of the powder diffraction pattern, with the mass and volume of the unit cell. If all phases are identified and crystalline, the weight fraction W of phase p is given by Wp = Sp( Z M IO p/~ S,( Z M IO,, iwhere S, Z, M and V are, respectively, the Rietveld scale factor, the number of formula units per unit cell, the mass of the formula unit and the unit-cell volume. This is the basis of a method providing accurate phase analyses without the need for standards or for laborious experimental calibration procedures. The method is demonstrated by measurements on binary mixtures of rutile, corundum, silicon and quartz.

Neutron powder diffraction patterns from rutile (TiO2, P42/mnm, a = 4.594, c = 2.959/k) and anatase (TiO2, I41/amd, a= 3-785, c=9-514/k) have been analysed by the Rietveld method. The positional parameters were determined to be xo=Yo= 0.30478 (6) in rutile and Zo = 0.20806 (5) in anatase. The anisotropic thermal parameters were also determined. The results from this constant-wavelength neutron diffraction study are in remarkably good agreement with the results from a recently published analysis of time-of-flight neutron data from the same phases. A comparison is made with results from earlier X-ray single-crystal investigations of these polymorphs of titania, and again the agreement is good. In particular, there is no evidence for any significant difference between X-ray and neutron determinations of the oxygen position such as might have resulted from polarization effects. The thermal vibrations show marked anisotropy, which appears to be determined by the stereochemistry of the crystal structure. The Debye temperatures esti-* Present address: Bristows Cooke and Carpmael, 10 Lincoln's Inn Fields, London WC2A 3BP, England.0108-7681/91/040462-07503.00 mated from the diffraction data are 600 (10)K for rutile and 520(10)K for anatase at room temperature.

The structures of the four metal dioxides GeO2, SnO2, RuO2 and IrO2 (germanium, tin, ruthenium and iridium dioxides, respectively) have been redetermined by Rietveld refinement from neutron diffraction powder data. The four dioxides all adopt the rutile-type structure, space group P42/mnm (no. 136), with a = 4.4066 (1), 4.7374 (1), 4.4968 (2) and 4.5051 (3), c = 2.8619 (1), 3.1864 (1), 3.1049 (1) and 3.1586 (2) Å, and x = 0.3060 (1), 0.3056 (1), 0.3053 (1) and 0.3077 (3), respectively. These results are compared with those for other metal dioxides that adopt the rutile structure and trends in structural and thermal vibrations for a series of 11 metal dioxides which adopt the rutile-type structure are described.

The crystal structures of monoclinic ZrO2 [P21/c, a=5.1505 (1) [Fm3m, a=5.0858(1)./k, V= 131.55(1),/k 3, Z=4, Rwp= 0.083] have been refined by Rietveld analysis of 1.377 A neutron powder diffraction data collected at 295 K. In both tetragonal and cubic ZrO2, the stabilizer atoms randomly occupy the Zr site and charge balance is achieved by an appropriate number of vacancies on the O site. In cubic ZrO 2, the anions are displaced from their ideal fluorite positions by 0.025a in the [111] direction and there is evidence for the presence of either a small quantity of a tetragonal impurity phase, or a slight tetragonal distortion. IntroductionPure zirconia, ZrO2, is monoclinic at room temperature, tetragonal between ~ 1440 and ~2640 K, and cubic up to the melting point at ~2950K. The monoclinic phase is a distortion of the fluorite (CaF 2) structure with the Zr atom in seven coordination. In both high-temperature phases, the Zr atom assumes eight coordination, as in fluorite, but in the tetragonal form the O atom is substantially displaced from its ideal fluorite 1 1 ~ position. The tetragonal and cubic phases ~,~,~ of pure zirconia can be stabilized at room temperature by the addition of suitable oxides, namely MgO, CaO, 0108-7681/88/020116-05503.00Sc203, Y203 and certain rare-earth oxides. An orthorhombic form has also been prepared by quenching from high pressure and temperature (Suyama, Ashida & Kume, 1985) but this phase will not be considered further here.The crystal structures of, and mechanisms of the transformations between, the monoclinic, tetragonal and cubic phases are of considerable technical interest since they can be manipulated to provide optimized physical and chemical properties of the materials fabricated from the stabilized zirconia (Garvie, Hannink & Pascoe, 1975;Roth, 1975;Claussen, Ruhle & Heuer, 1984;Fisher, 1986). The so-called partially stabilized zirconias (PSZ), which are typically twophase cubic and tetragonal or single-phase tetragonal, are of importance for mechanical and structural applications. The fully stabilized zirconias (FSZ), which are normally single-phase cubic, are of interest for heating elements, oxygen sensors and fuel-cell applications.Crystal structure determinations have been performed on tetragonal ZrO 2 using X-ray powder diffraction intensities collected at 1470 to 2230 K by Teufer (1962). Monoclinic ZrO 2 (baddeleyite) has been studied at room temperature using X-ray single-crystal methods by McCullough & Trueblood (1959) and Smith & Newkirk (1965). Cubic Zr(Ca,Y)O2_ x solid solutions have been analyzed at various temperatures from both X-ray and neutron data by Carter & Roth (1968), Steele & Fender (1974), Faber, Mueller & Cooper (1978, Morinaga, Cohen & Faber (1979) and Horiuchi, Schultz, Leung & Williams (1984).© 1988 International Union of Crystallography C. J. HOWARD, R. J. HILL AND B. E. REICHERT 117The present study gives new refinements of the crystal structures and compositions of the three ZrO 2 polymorphs from high-resolution neutron pow...

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