Abstract
The reaction N d20 3 -I-C uO -*■ N d 2C u 0 4 was investigated on the basis of the Wagner-Schmalzried theory of diffusion-driven reactions between crystalline solids, with the help of monodimen sional chemical diffusion measurements. It was found that the kinetics of the process is governed by C u2+ diffusion, and the chemical diffusion coefficient Z)(Cu2+) in the N d 2C u 0 4 matrix was obtained (9.9 • 10-13, 1.97 • 10-12 and 1.98 • 10" 11 cm 2 s -1 at 1223, 1273 and 1323 K, respectively). The rela tively low value of D (C u2+) was interpreted on the basis of the crystal structures of the compound and related materials, which in addition may suggest an alternative rationale for the interpretation of neodymium immobility.
Introduction
As found by Tokura et al. [1] in 1989, Ce doped N d2C u 0 4 is a superconductor with a critical temper ature not much above 20 K. The N d 2 _,cCe;(.C u 0 4 solid solutions were then believed to be n-type con ductors, but so far no general agreement is attained about the electronic structure and transport proper ties of these materials [2] [3] [4] , Very broadly speaking, the electronic properties of superconducting oxides at low temperatures can dramatically change depending on the conditions used in the preparation, which is usually carried out through the sequence: i) solid state reaction, ii) sintering and iii) annealing. However, little efforts have been made until now to understand the elementary mechanisms of the solid state synthesis of superconducting oxides.
Introduction
As a part of a research program devoted to these aspects [5] [6] [7] [8] , here we present a kinetic study on the formation of N d 2C u 0 4 from N d 20 3 and CuO.
Introduction
From a chemical point of view, this system seems to be ideal for the application of the current WagnerSchmalzried theory of diffusion-driven reactions be tween crystalline solids [9] for the following main rea sons: i) the compound has a relatively simple tetragonal crystal structure [1] , copper and neodymium atoms occupy equivalent positions, whilst only two distinct positions exist for oxygens: as all the atom s are in special positions, we are left with a very small num ber of positional degrees of freedom in describing the con tent of the unit cell; ii) data concerning the free energy of formation [10] , and the diffusivities of some ionic species [11] are available in the current literature, iii) N d 2C u 0 4 represents a true ternary system, so that only three independent diffusion coefficients exist [12] .
Experimental
CuO (Aldrich 99.99%) and N d 20 3 (Aldrich 99.9%) were used as supplied or sintered at 1173 and 1773 K, respectively, in reagent grade oxygen flux (21/h) at 1 • 105 Pa. The monodimensional diffusion couple measurements were performed in a horizontal tube furnace under isothermal ( ± 1 K) conditions, under a flux (2 1/h) of reagent grade oxygen (or, as indicated in the text, nitrogen) at 1 • 105 Pa, on couples of sintered pellets (8 mm wide, ~ 1.5 mm thick, obtained by pressing uniaxially the powders of the reagents at 103 kg cm " 2). The experimental apparatus and proce dure were previously described [8] .
Experimental
A TA Instruments Thermal Analyst 2000, equipped with an attachm ent for differential thermal analysis (DTA) from Du Pont were used for thermo-analytic characterisations; alumina crucibles were used as sam ple holders, with a scan rate of, usually, 5 K m in-1 .
Experimental
X-ray Powder Diffraction (XRPD) patterns were taken on a Philips 1710 diffractometer equipped with a copper anode operating at 45 KV and 35 mA, ad justable divergence slit, graphite m onochrom ator on the diffracted beam, and proportional detector. further details on the lattice constant determination procedure, reference is made to a previous paper [8] , and to the citations therein.
Experimental
Optical microscopic observations were made with a Zeiss Axioplan microscope, on couples reacted for a certain time t, and then included in epoxy resin (Struers), cut along a plane parallel to the diffusion direction, and polished to optical quality. 
TIK

Results
For an equim olar mixture of CuO and N d20 3, the DTA trace (see Fig. 1 ) shows that the reaction starts at 1223 K; the endothermic peak at 1343 K has been ascribed to the eutectic between N d 2C u 0 4 and CuO, in agreement with the equilibrium phase diagram [13] , Accordingly, the diffusion couples method has been applied at three different temperatures, 1223,1273 and 1323 K, in a pure oxygen flux. In every case, after a suitable reaction time, a single black reaction layer is generally obtained, on the N d 20 As an example, in Fig. 3 the data obtained at 1223 K are reported. The (practical) kinetic constants k were obtained by measuring the thickness Ax of the product layer at different reaction times, t, through the equation: Ax2 = 2 kt. The values are summarised in Table 1 . An activation energy of the reaction of 401 kJ m o l-1 was estimated by Arrhenius plot.
Results
For the sake of completeness, however, thin layer form ation also w ithout contact between pellets must be mentioned: this was probably due to a slow copper oxide evaporation, which could give a small but effec tive activity of Cu in the gas phase.
Results
To gain a deeper understanding of the reaction mechanism (see the discussion), diffusion experiments were performed at 1273 K also in pure nitrogen flux: in these conditions, the actual reaction occurs between N d20 3 and C u20 . In any case, no significant varia tion in the product layer thickness was found.
Discussion
As previously mentioned, in a ternary system there are only three independent diffusion coefficients [12] ; let us denote them with D(a), D(b), and D(c), and suppose, without any loss of generality that
D(a)> D (b) > D(c).
(1)
D(a)> D (b) > D(c).
In the Wagner-Schmalzried theory of solid state reac tions in ternary systems [9] , the kinetics of the chemi cal process is determined by the transport of the inter mediate diffusivity constituent (b), since the con stituent with the smallest diffusion coefficient (c) can be taken as immobile because of the second inequality in (1) , and the fastest diffusing constituent (a) is cou pled by a local electroneutrality condition with the intermediate one (b) .
D(a)> D (b) > D(c).
From the position of the reaction layer with respect to the original reaction interface one can argue that in this case neodymium is immobile; nevertheless, for the reasons explained below, the explicit identification c = N d 3+ is not performed, and it is only observed that the reaction mechanism involves the concurrent diffusion of oxygen and C u2 + . In the literature [11] the 0 2~ ion is reported to have a very high chemical diffusion coefficient in the N d 2C u 0 4 matrix (of the order of 10" 5 cm 2 s -1 at 1173 K). Moreover, by the experimental finding that a couple reacted at 1273 K in a pure nitrogen flux shows no variation in the layer thickness, one can exclude gas phase transport of ox ygen. Thus the rate determining constituent, if atten tion is given to the very small values of the practical kinetic constants (see Table 1 ), is to be identified with C u 2 + .
D(a)> D (b) > D(c).
The local electroneutrality condition implies the equality of the fluxes (j) of copper and oxide ions,
D(a)> D (b) > D(c).
and, according to the theory [9] , the practical rate constant becomes
D(a)> D (b) > D(c).
where AG°(Nd2C u 0 4) is the standard Gibbs free en ergy variation associated with the reaction, which can be obtained from literature data *: thus the chemical diffusion coefficient of the copper ion D (C u2 + ) was evaluated from the kinetic determinations. The results are summarised in Table 1 : the activation energy of diffusion could be estimated as 400 kJ m ol-1 . At this point two remarks must be made. The first one concerns the reliability of the diffusion coeffi cients) determination: unfortunately, literature data on the diffusivity of C u2 + in neodymium cuprate are, until now, not available, so that the validity of this approach cannot be directly checked. Nevertheless, a useful even if indirect estimate of these results can be obtained by comparing the D (C u2 + ) so obtained with the self-diffusion coefficient Z)*(Cu2 + ) in YBCO su perconductor [15] , namely a compound in which the chemical environm ent of copper is very similar to that of N d 2C u 0 4 **. Figure 4 compares the Arrhenius plots for the two diffusion coefficients, together with the same plot for the chemical diffusion coefficient of the oxide ion in N d 2C u 0 4 [11] . It is readily seen that copper diffusion, at a given temperature, is easier in YBCO than in neodymium cuprate. In structural terms, this is nicely explained taking into account the coordination of copper in the two unit cells (see Fig  ure 5 ). As in N d 2C u 0 4 the oxygen non-stoichiometry is not very im portant, almost every copper atom is in a square planar coordination; in the orthorhom bic YBa2C u 30 7_JC , even if x = 0, the O j site is only half occupied; moreover, this compound is heavily nonstoichiometric, and when x = 1, every O x is empty, the cell symmetry becomes tetragonal, and YBCO under goes a phase transition. It is clear that an oxygen vacancy in the O x site lowers the coordination num ber of two copper atoms: therefore, even if only steric hindrance is taken into account, a greater diffusivity of copper in YBCO than in N d 2C u 0 4 is easily expected. The second remark regards the influence on the reaction mechanisms of the structural relationships between the product and the parent materials. When dealing with a solid state reaction, as long as the kinet ics can be considered diffusion limited, only the trans port properties (i.e. the proper diffusion coefficients) of the product are usually taken into consideration. Al though this approach is quite reasonable, it must be pointed out that for a given negligible flux there may be: i) a negligible diffusion coefficient and/or ii) a neg ligible chemical potential gradient.
D(a)> D (b) > D(c).
Consequently, it can be inferred that the immobility of neodymium may admit another explanation, exter nal to the Wagner-Schmalzried theory, if the structures of neodymium cuprate [1] and hexagonal neodymium oxide [17] (which is stable at the reaction tem perature [18] ) are taken into account. For this purpose it is so that they are equal only in the limit of ideal defects be haviour. This assumption is here reasonably justified, since we are far enough from the melting point.
D(a)> D (b) > D(c).
necessary to consider only the neodymium sub-lattices (Fig. 6) , which are easily seen as different modifi cations of the neodymium sub-lattice of an ideal N d (0 , V0 )2 fluorite structure (note that the cubic N d 20 3, stable at low temperature [18] , has the latter structure, also shown in Figure 6 ). In the fluorite structure (space group Fm3m), along a <001) direc tion, two different neodymium planes are found. marked with a and ß in Figure 6 : the stacking se quence is ocßocßocßocß__ In the hexagonal neodymium oxide (space group P3m l ***), along the <001) direc tion the same planes are found (oc' and ß' in Fig. 6 ), with the same stacking sequence c l 'ß'c l 'ß'c l 'ß'c l 'ß'. .. , a rectangular deform ation and a small shifting in the ßr planes. Finally, also in N d 2C u 0 4 (space group *** For the sake of semplicity, use was here m ade of an approxim ated structural model proposed by L. Pauling [17] ; this model was revised in some details by Miiller-Buschbaum et al. [19, 20] , who attributed the structure to a different space group (P63/mmm). I4/mmm) along the <001) direction Nd atoms build up the same fluorite-like planes (a" and ß" in Fig. 6 ) with a stacking sequence c l "ß"ß"c l "c l "ß"ß"c l ". . . ; as clues, the facts may be quoted that: i) the aT (aT = 0.3944 nm [21] ) constant of N d 2C u 0 4 is almost n /2 . , 4 s f l ac = 1.108 nm [22] ,------ac = 0.3917 nm I, ii) the aT exactly times the ac constant of the cubic N d 20 3 nm in hexagonal neo-1 0 " 5 n m -3 in neodymium constant of N d 2C u 0 4 is almost equal to the aH (aH = 0.3831 nm [23] ) constant of the hexagonal N d20 3.
D(a)> D (b) > D(c).
As an incidental observation, the conversion of the hexagonal neodymium oxide into the neodymium cuprate can be seen as the occurrence of i) some small rearrangement of the neodynium and oxygen setting up, ii) filling up of all the oxygen vacancies, and iii) insertion of copper ions in some interstitial sites.
D(a)> D (b) > D(c).
These structural relationships suggest that the ac tivity of neodymium does not change remarkably in the three structures. Moreover, the concentration of neodymium, measured in atoms per unit volume, is practically the same: 2.4 • 10-5 n m -3 in cubic neo dymium oxide, 2.6 1 0 " 5 dymium oxide and 2.2 cuprate. Therefore it is not unreasonable to suppose that, both at the N d 20 3-N d 2C u 0 4 interface and in the whole reaction layer, the neodymium chemical potential gradient is negligible. Thus the neodymium flow might be near zero, neglecting the actual value of the neodymium diffusion coefficient. It is here perhaps worth noting that, in the YBCO superconductor, above 1200 K, the self-diffusion coefficient of yttrium (which has a chemical environment very similar to that of neodymium in N d 2C u 0 4) is at least com para ble with the self-diffusion coefficient of copper [15] .
D(a)> D (b) > D(c).
Finally the authors wish to point out that they are positively not proposing an atomistic reaction mecha nism; in this respect, this is only a tentative interpreta tion which ought to be experimentally verified, for example reacting properly oriented single crystals.

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