Les cristaux d'apatite proviennent de la région de Durango ; leur composition est donnée par Young et al. [20]. Les cristaux sont orientés, sciés et polis. Les sections obtenues sont introduites dans des chlorures de TR (La, Ce, Nd et Eu) et portées dans un four à des températures comprises entre 900 et 1150 °C, pour une durée maximale de 35 jours. Après, les cristaux sont enrobés par la résine, polis et passés à l'analyse. Les analyses sont effectuées dans les deux directions cristallographiques (a et c), à l'aide d'une microsonde électronique type CAMECA SX 50.

Dans toutes les expériences, à l'interface sel fondu–apatite, les résultats montrent la cristallisation d'une mince couche, composée d'un mélange de chlorures de TR et d'apatite. La source de diffusion des TR est considérée comme plane et infinie. Selon la direction cristallographique a, les distances de diffusion sont faibles et les écarts entre les TR diffusantes négligeables. Pour la direction c, elles sont beaucoup plus grandes et les écarts entre les TR plus importants. Cependant, le même mécanisme d'échange opère dans les deux directions : $2{\mathrm{TR}}^{3+}\iff 3{\mathrm{Ca}}^{2+}$ (exemple du Nd, Fig. 1). À la surface de l'apatite, les profils de diffusion montrent une concentration maximale en TR correspondant à 2 atomes par formule (Fig. 2).

Les profils de diffusion montrent un processus complexe de diffusion. La concentration initiale de TR à la surface de l'apatite augmente avec la diffusion. Ceci provoque un changement de la forme des profils qui, initialement concave à faible concentration, devient convexe quand cette concentration avoisine le taux maximum de 2 atomes de TR (Fig. 3). L'échange TR/Ca s'effectue selon la réaction : $2{\mathrm{TR}}^{3+}+\iff 3{\mathrm{Ca}}^{2+}$, avec un taux maximum de lacunes de 10 % des sites cationiques. C'est le même taux observé dans des apatites synthétiques [14]. La structure de l'apatite semble se déstabiliser au-delà de ce seuil de lacunes, ce qui s'opposerait à une substitution totale du Ca par les TR et limiterait leur concentration à la surface du cristal à 2. Dans l'apatite, Ca occupe deux types de sites : CaI et CaII. Autour de CaII, le diamètre des tunnels coı̈ncidant avec l'axe sénaire étant plus important : entre 3 et 4,5 Å [12,13,15–17], l'échange TR/Ca s'effectue plutôt dans les sites CaII. Cette préférence pour ce type de sites a déjà été observée dans de nombreuses investigations naturelles et expérimentales [7–11] où les concentrations élevées en TR observées selon la direction c. L'existence de tunnels le long de l'axe c explique également la diffusion plus rapide des TR selon cette direction.

Les distances de diffusion sont calculées à partir des profils de diffusion qui présentent deux parties distinctes : la zone de diffusion dont la concentration en TR est supérieure à la concentration initiale dans l'apatite ; la deuxième zone correspond à celle où la concentration reste inchangée. Des droites de régression permettent de limiter clairement les deux zones (Fig. 4) et de calculer les distances de diffusion (Tableau 1). Selon l'axe a, les distances de diffusion des TR sont proches et les écarts faibles.

Selon l'axe c, les distances de diffusion augmentent avec le rayon ionique des TR, ce qui est en contradiction avec les observations faites sur les verres et les milieux isotropes où ce sont les ions les plus petits qui diffusent le plus loin [1,8]. La Fig. 5 montre de bonnes corrélations entre les masses atomiques des TR et les distances de diffusion. L'existence de tunnels le long de l'axe c favorise la diffusion rapide des TR selon cette direction. Leur diamètre étant supérieur à celui des TR, l'effet de la taille semble ne plus affecter la diffusion, et ce sont, selon toute logique, les cations les moins lourds qui migrent le plus loin. Il est évident que, dans ce cas, les coefficients de diffusion ne sont pas constants. Cependant, nous utiliserons la relation de base $x^2=\mathrm{D}t$ pour le calcul des diffusivités (Tableau 1). Les énergies d'activation sont calculées à partir de la relation $\mathrm{D}=\mathrm{k}\exp (-\mathrm{E}/\mathrm{R}\mathrm{T})$ [4]. Les relations d'Arrhenius obtenues selon l'axe c (Fig. 6) sont :

La comparaison de nos résultats avec ceux des études antérieures [2,3,18,19] fait apparaı̂tre des diffusivités plus grandes et des énergies d'activation plus faibles dans notre cas. Les modes opératoires n'étant pas les mêmes, il est logique que les résultats soient différents. En effet, la concentration initiale et le type de cations (couplés aux TR et impliqués dans l'échange TR/Ca) contrôlent le mode de substitution et affectent le processus et les paramètres de la diffusion.

À hautes températures, la diffusion des TR dans l'apatite est régie par l'existence d'un contrôle structural responsable de la non-uniformité des processus de diffusion et par la concentration initiale des cations diffusants. Le type de substitution impliqué dans l'échange TR/Ca est aussi un facteur dominant pouvant affecter le processus et les paramètres de la diffusion.

Experiments in the a crystallographic directions showed little diffusion and almost no detectable difference for the different diffusing species at different temperatures. Therefore, only results for the c crystallographic direction are reported here.

The exchange of light REEs for calcium in the apatites is clearly shown in Fig. 1, where the REE diffusing ion Nd contents are plotted against Ca for one experiment at 1100 °C.

It is easily seen that the replacement, and hence diffusion, of trivalent REE cations into apatite scrupulously respects electronic balance, where $2{\mathrm{R}}^{3+}=3{\mathrm{R}}^{2+}$ (slope of 0.64 for the Nd/Ca relations). Electronic balance is maintained then in the crystal structure at the concentration level investigated. The important point is that the introduction of trivalent REE elements into apatite in this study is accomplished by a two-ion-plus vacancy for three-ion substitution. This means that not only there is inter-diffusion, Ca for REE, one entering and the other one leaving the crystal, but also that there is a difference in site occupancy, leaving a vacancy in the structure after diffusion has been accomplished. There are more Ca ions leaving the crystal than REE ions entering.

The profiles of the exchanged elements in the apatite crystals are given in Fig. 2.

The concentration profiles of light REEs observed in the apatites after the experiments indicate that there is continuous exchange of REEs for Ca up to a certain limit of ionic concentration. The upper limit of substitution seems to be near two REE-substituted ions for the ten possible Ca ions in apatite. In the experiments where a high amount of exchange was effectuated, the profiles show a decided maximum value, which is similar for different ionic species. This maximum value is seen to penetrate into the crystal.

The diffusion profiles in Fig. 2 show that the diffusion process is complex. The shapes of the curves are different under different conditions of time and temperature, depending upon the diffusing ion. Analysis of the curves shows that two points are very important to note initially. The first is that the initial concentration of diffusing ion into the apatite changes as diffusion increases. This is clearly seen for Nd in Fig. 2c. The initial maximum, exterior face value is 0.2, increasing to 2. This is not the classic situation for diffusion. Normally, the outside face is initially and continually in equilibrium with the outside-diffusing medium concentration.

A second very important observation is that the shape of the profiles changes after a maximum concentration of two ions is reached. The initial, low concentration profile is convex downward for diffusing ions (Eu for example, Fig. 2d) and convex upward near the crystal edge when more exchange has taken place (La for example, Fig. 2a). High exchange produces a reversed S-shaped concentration profile. Nd and Ce show both types of profile in the experiments conducted here (Figs. 2b and c): one at low temperature, one at high temperature. When this value of two ions is reached, a diffusion front is established which advances into the crystal. The concentration vs. distance curve has a plateau.

As a result, one finds convex and concave concentration profiles of light REEs in the apatite experiments depending upon the amount of exchange that has occurred. Profiles with a high amount of exchange tend to show an abrupt decrease in diffusing ion concentration with a steep diffusion front. Diffusion in the a crystallographic direction gives only C-shaped profiles (Nd for example, Fig. 2e), indicating slower and more incomplete diffusion for the temperature-time conditions of the experiments compared to the c crystallographic direction. The maximum penetration depth of the diffusing ion in the a direction is half to one third that of the c direction in the same experiment. We will therefore limit our discussion to the c crystallographic direction. We will now consider the diffusion profiles according to their shape.

4.1.1 C shaped, convex downward profile

4.1.2 S shaped curves

The diffusion profile, once it attains a maximum saturation value advances into the crystal with time, showing a more and more abrupt profile as the process advances.

In order to discuss REE diffusion mechanism in apatite crystals, first of all, the apparent maximum of REE substitution in the apatite crystal of two in ten possible exchange sites indicates that there is a structural limitation to the diffusion. The maximum rate of vacancies created by the substitution: $3{\mathrm{Ca}}^{2+}\iff 2{\mathrm{REE}}^{3+}+$ corresponds to 10% of cationic sites. That is to say a vacancy on the ten occupied atoms initially by the Ca. It is the same order of concentration observed in the synthetic apatites [14]. The structure of the apatite seems to destabilise itself beyond this level of vacancies, which oppose a total substitution of the Ca by REE and would limit the maximum concentration at the surface of the apatite crystal to 2.

In the apatite crystal, one can find two types of co-ordination sites for the calcium ions, where four sites have nine nearest oxygen neighbours (CaI) and six have seven (CaII). The seven fold coordinated calcium ions are situated around helical ‘tunnels’ or void spaces in the apatite, where the (OH), Cl and F cationic groups are found. Around CaI sites, the ternary axes have a diameter of 2.5 Å. The senary axes that surround CaII sites have a more important diameter: between 3 and 4.5 Å [12,13,15–17]. Given that one could imagine that the REEs diffuse quickly in the largest channels and will substitute easily in CaII sites. This appears to be the case in various experimental studies as well as found in observations of natural minerals [6,7,9–11].

One can note that all of the low concentration parts of the curves for diffusing ion concentration approach the zero concentration level in an asymptotic manner. In order to estimate the greatest distance of cation exchange in the crystal, we found that a plot of this low concentration portion of the curve as the log of the distance against concentration scale was efficient for estimating the position of zero concentration, and hence maximum diffusion distance in the crystal. The intersection of these curves with background values was used to estimate the point of maximum diffusion distance (Fig. 4).

These values of maximum diffusion distance determined in this manner are listed in Table 1.

The precise total distance of ionic migration cannot be determined accurately, because there was a small amount of apatite dissolution during the experiment as indicated by the 2–5-μm-thick outer mixed REE chlorides and apatite zone. This zone indicates phosphate reaction with the molten salt and hence some dissolution of apatite into the salt solution. If significant apatite were lost, the distance of ionic migration would not be accurately estimated by a simple measurement of the concentrations in the remaining apatite crystal. We assume that the dissolution of apatite was not great, the mixed REE chlorides and apatite zone is of the order of the 2–5-μm thick and hence much less than the differences in distance of ionic migration noted for the different light REE elements in the different experiments. At any rate, the apatites will have dissolved at about the same rate for all of the experiments done at the same temperature and time span and hence the differences in migration of each REE at a given temperature will have significance for differential migration rates.

The parameters that we can derive then from the experiments are the variables of the experiment: time, temperature and the maximum distance of migration of the elements from a high-concentration zone in the chloride solution into the essentially light REE-free apatite.

In the range of REE ionic radii studied, one sees that the larger elements diffuse more rapidly in the apatite crystals. The relation of increasing diffusion distance in the c crystallographic direction with an increase of ionic radius is in apparent contradiction with diffusion of ions in equant or isotropic media where it has generally been observed that the smallest ion of a given charge will diffuse the fastest or furthest in a given amount of time [1,8]. By contrast, the lighter the element, the faster it diffuses. This relation is clearly shown in Fig. 5, where one observes good interrelationships between the REE atomic masses and distances of diffusion. This relation is true in all ranges of temperatures.

It appears therefore that along tunnels whose diameter (3–4.5 Å) extensively exceeds that of the REEs (1.06 to 1.16 Å), the effect of the atomic mass on the diffusion dominates that of the ionic radius, which becomes negligible.

In order to explain the relations between diffusion distances of the different ions in the experiments, it is necessary to rank these distances as a function of the physical forces acting upon them. Ln distance squared is used because the relationship of $x^2=\mathrm{D}t$ is the basic relation between distance and time. D relates these variables to temperature as $\mathrm{D}=\mathrm{k}\exp (-\mathrm{E}/\mathrm{R}\mathrm{T})$ [4]. The relation between temperature and distance are those used to establish the activation energy of diffusion, the essential parameter in chemical diffusion.

Fig. 6 shows the maximum diffusion distances and temperature related as a function of distance squared and reciprocal temperature. It is evident that there is a regular relationship.

The following Arrhenius relation was determined:

In previous studies of diffusion in apatite [2,3,18,19], diffusing elements moving into the apatite crystals are at low concentrations in the surrounding medium, about two orders of magnitude lower than those in our experiments. The penetration depth observed in the apatite is roughly ten times smaller than ours, for the same experimental conditions of time and temperature. The concentration profiles in these studies have a concave downward profile for the most part. A tendency to form an S-shaped curve can be noted in only certain experiments. Watson and Green [19] calculate a diffusion constant of the order of $5\times {10}^{-14}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$ for La at 1120 °C, assuming C-shaped profiles where the outside concentration in the crystal is assumed to be in equilibrium with the surrounding medium, a silicate melt. They assume that the edge concentration of the diffusing element in the apatite is constant during the whole experiment. However, it is most likely not constant; their edge concentration at the end of the experiment is most likely lower than the equilibrium value between apatite and the surrounding medium. Thus the calculation method used is not valid if their system behaves as that observed in the experiments reported here. If one assumes that the edge concentration changes linearly with time, one will calculate a diffusion constant five times larger. However, this variation law is hypothetical and has to be checked by experiments.

In any event, the overall diffusion coefficients that could be calculated using our experiments (maximum diffusion distance) would in any event differ from the ones from [2,3,19] by a minimum of one order of magnitude. Further, these authors do not distinguish between c and a crystallographic directions in their apatite crystals, which we have observed to give very different results. The values which they reported are of the order magnitude of the ones that one would obtain along the a crystallographic directions using the data from our experiments. The observed variations are likely a consequence of differences in the exchange mechanism operating in our and previous studies. Diffusion would be rate-limited by REE and REE cation decoupling.