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Latin American applied research

versión impresa ISSN 0327-0793versión On-line ISSN 1851-8796

Lat. Am. appl. res. vol.43 no.2 Bahía Blanca abr. 2013

 

New proposal for CO32- position in b-type carbonated apatite

J.C. De Araujo , E.L. Moreira§ , G.D.A. Soares# and I.N. Bastos

Universidade do Estado do Rio de Janeiro, Brazil, jcaraujo@iprj.uerj.br
§Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro RJ Brazil, beth@cbpf.br
#Universidade Federal do Rio de Janeiro, Rio de Janeiro RJ Brazil, gloria@metalmat.ufrj.br
Instituto Politécnico, Universidade do Estado do Rio de Janeiro, Nova Friburgo RJ 28.630-050 Brazil inbastos@iprj.uerj.br

Abstract— The crystal structure of two B-type carbonated apatite samples was re-analyzed, using the data of Wilson et al (2006). By applying the least squares method for the determination of carbon and oxygen atoms, a slightly different result was obtained, suggesting that the carbonate ion is not rigorously planar.

Keywords— Carbonated Apatite; Carbonate; X-ray Diffraction.

I. INTRODUCTION

Apatites are a material of huge importance in biomedical and mineral fields; therefore, a vast number of works is devoted to their studies (López et al., 2008; Araújo et al., 2007; Lima et al., 2006). In spite of these studies, some questions are not a consensus, and one reason for that is the large capability of atom substitution in apatite lattice.

Low content carbonate (CO32-) is present in many minerals and in synthetic and biological apatites. The precise location of carbonate in the apatite lattice, however, is still controversial. Thus, some authors (Ivanova et al., 2001; Wilson et al., 2004; 2006) have been dedicated their studies to this subject.

Ivanova et al. (2001) analyzed by X-ray powder diffraction and Rietveld refinement the crystal structure of a synthetic calcium-deficient carbonate apatite of B-type with 4.4 wt (%) CO32-. They showed that there are two orientations for the carbonate triangles and that the carbon atom can occupy randomly the adjacent faces (parallel to the axis c) of the tetrahedron PO43-.

Wilson et al. (2004) proposed in their studies using X-ray and time-of-flight neutron diffraction data collected from a sodium-containing carbonate apatite (CO32- content 12.5(7) wt%) a structural model known as face model with the ion in disorder with the mirror symmetry related to the faces of vacant PO43- ion site. The calculated angle between the normal to the plane of CO32- ion and c-axis was 30°(3). This configuration for the CO32- ion corresponded to the best fitting in refinement. In this "face model", the oxygen coordinates were labeled as Oa, Ob and Oc. The oxygen Oa was no longer constrained to occupy the vertex 1 of carbonate triangle, Ob was placed 2.241Å from vertex 1 towards vertex 2, and close to vertex 2, while Oc was in the face formed by vertices 1-3 of the phosphate tetrahedron at a distance 2.241Å from both Oa and Ob. Thus Oa and Ob lie in the mirror plane at z=1/4 while Oc and C were in two folder disorder about the mirror plane.

Later, Wilson et al. (2006) studied seven samples of Na-free CO3A, modeling the carbonate ion with soft constraints. In this sense, a good approximation of calcite structure (C-O and O-O distances with 1.294 and 2.413 Å, respectively) was considered. In this geometry, the hard constrained consideration of ion position, the Oa was taken as O1; Ob was placed on the O1-O2 line, 2.413 Å away from the O1/Oa site, and still constrained by the symmetry to remain in the mirror plane. Oc and C were located in the face formed by vertices 1-3 of the phosphate tetrahedron at an equal distance of 2.241 Å from both Oa and Ob. Besides, with this geometry, the C carbon atom is located on the center of an equilateral triangle of oxygen atoms, maintaining the configuration proposed on the previous paper Wilson et al. (2004).

The proposal of this work was to analyze part of the data published by Wilson et al. (2006) using a different approach to determine the positions formed by the carbon atom and the oxygen coordination.

II. METHODS

From Wilson et al. (2006) data, two samples were chosen: Mon 8 and Mon 10. Table 1 shows the positional parameters of these two samples and the mean of seven samples analyzed by the authors. The results show a slightly difference for the C atom position between Mon 8 and Mon 10. Therefore, Mon 8 was chosen due to the high deviation of positional parameters, while Mon 10 can represent all other samples. The nomenclature of these samples was maintained in order to facilitate the interpretations.

Table 1 - Wilson et al. (2006) data for positional parameters (standard deviation). (Values multiplied by 104). The C, Ob and Oc atom positions were multiplied by 103.

The matrix , where a and c are the lattice parameters, was used in the present work to turns the hexagonal crystal system (P63/m) into an orthogonal system where the calculus are simplified.

The use of least square method to fit the plane of a set of points was early reported by different authors such as Schomaker et al. (1959) and Hamilton (1961). We intend to find the best plane in a purely formal sense. The weight is taking as unit in the matrix given by Eq. (8). However, the theoretical treatment is remarkably simplified - as proposed by Giacovazzo et al. (2002) - if the Cartesian system is adopted because in this system the metric matrix G is equal the identities I. Therefore, the plane equation is defined by:

(1)

with the unit normal vector n=(A,B,C)T and D is the distance of the plane from the origin. In an orthogonal frame, the perpendicular distance of i-th atom from the plane is given by:

(2)

The goal is to find the plane that minimizes the weighted sum of squares of distances dj of points from the plane. The correspondent equation is expressed as:

(3)

We use the steady-state points of S and still keeping the condition that n is a unit vector as expressed in the linear functional F with Lagrange's multipliers λ.

The steady-state condition of S, which considers the normal vector unitary, can be expressed by the function F given by the Lagrange multipliers as below:

(4)

The partial derivative of Eq. (4) with respected to D parameter is given by:

(5)

The meaning of Eq. (5) is the centroid X0 of the set of atoms. The normal equations

(6)

leads to:

(7)

where X'i = Xi - X0 is obtained through the transformation of coordinates. Similar expressions are obtained to partial derivatives with respect the B and C parameters and leads to the linear system:

(8)

The system (8) can be expressed as:

(9)

Multiplying the left-hand side of Eq. (9) by nT results:

(10)

Thus, the best plane corresponds to the eigenvalue λ1, passes in the centroid and has the normal vector λ1. Furthermore:

(11)

where di-D is the distance of the i-th atom from the best plane.

III. RESULTS AND DISCUSSION

Table 2 shows the comparison of results obtained in this work (re-analyzed the Wilson et al., 2006) and the published results (using face model) for the PO43- bond lengths and volume. It is possible to observe the good agreement between data values.

Table 2 - PO43- bond lengths (Å) and volume (Å3) for analyzed samples.

a - Data published by Wilson et al. (2006).
b - Results obtained from the re-analysis of published parameters of crystal.

Table 3 showed the A, B, C and D values for best plane Ax+By+Cz=D, where x, y and z are the orthogonal coordinates. The small differences observed for Mon 8 and Mon 10 can be attributed to different positional parameters showed in Table 1.

Table 3 - Best least square planes in orthogonal coordinates. Displacements are expressed in Å.

The results of plane 1, obtained for sample Mon 8, show that the CO32- ion is not truly planar. Indeed, the CO32- ion in Mon 8 occupies a little tetrahedral volume (0.19 Å3). For the Mon 10 sample, the CO32- ion can be seen as having a non-significant tetrahedral volume (0.086Å3), being thus nearly planar with the face plane formed by Oa, Ob and Oc oxygen making ψc≈33.620 with the c-axis.

IV. CONCLUSIONS

From the data published by Wilson et al. (2006) related to the Rietveld refinement with soft restrictions of carbonate atomic positions in samples 8 and 10 of carbonated B-apatite, it was shown, in the present work, with the use of least square method, that the configuration of carbonate ions is not rigorously planar as previously considered, especially for the sample named 8.

We think that a rigorous planarity for the CO32- triangles, similar that of calcite structures can only be obtained with hard constrained for the C carbon atom position in the Rietveld refinement. In other words, carbon positions have to be fixed in the center of the oxygen coordination during all the Rietveld refinement as suggested by Ivanova et al. (2001). The soft constraint seems to be more realistic as it allows the C atom to move slightly out of the face plane formed by the oxygen atoms. Anyway, a further detailed investigation of Mon 8 crystal structure would be desirable.

ACKNOWLEDGMENT
The authors thank the financial support of Brazilian agencies such as CNPq, FAPERJ and CAPES.

REFERENCES
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Received: August 20, 2011
Accepted: May 29, 2012
Recommended by Subject Editor: Orlando Alfano

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