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Partial structure factors
Faber-Ziman formalism
The first way used to define the partial structure factors has been proposed by Faber and Ziman [b]. In this approach the structure factor is decomposed following the correlations between the different chemical species. To describe the correlation between the α and the β chemical species the partial structure factor S^{FZ}_{αβ}(q) is defined by:
where the g_{αβ}(r) are the partial radial distribution functions.
The total structure factor is then obtained by the relation:
Ashcroft-Langreth formalism
In a similar approach, based on the correlation between the chemical species, and developped by Ashcroft et Langreth [c,d,e], the partial structure factors S^{AL}_{αβ}(q) are defined by:
where δ_{αβ} is the Kronecker delta, c_{α} = , and the g_{αβ}(r) are the partial radial distribution functions..
Then the total structure factor can be calculated using:
Bhatia-Thornton formalism
In this approach, reserved to the case of binary systems AB_{x} [f], the total structure factor S(q) can be express as the weighted sum of 3 partial structure factors:
where 〈b〉 = c_{A}b_{A} + c_{B}b_{B}, with c_{A} and b_{A} reprensenting respectively the concentration and the neutron or X-rays scattering length of species A.
S_{NN}(q), S_{NC}(q) and S_{CC}(q) represent combinaisons of the partial structure factors calculated using the Faber-Ziman formalism and weighted using the concentrations of the 2 chemical species:
S_{NC}(q) = c_{A}c_{B}× cA×S^{FZ}_{AA}(q) - S^{FZ}_{AB}(q) - c_{B}×S^{FZ}_{BB}(q) - S^{FZ}_{AB}(q) | (16) |
- S_{NN}(q) is the Number-Number partial structure factor.
Its Fourier transform allows to obtain a global description of the structure of the solid, ie. of the repartition of the experimental scattering centers, or atomic nuclei, positions. The nature of the chemical species spread in the scattering centers is not considered. Furthermore if b_{A} = b_{B} then S_{NN}(q) = S(q).
- S_{CC}(q) is the Concentration-Concentration partial structure factor.
Its Fourier transform allows to obtain an idea of the repartition of the chemical species over the scattering centers described using the S_{NN}(q). Therefore the S_{CC}(q) describes the chemical order in the material. In the case of an ideal binary mixture of 2 chemical species A and B^{5.1}, S_{CC}(q) is constant and equal to c_{A}c_{B}. In the case of an ordered chemical mixture (chemical species with distinct diameters, and with heteropolar and homopolar chemical bonds) it is possible to link the variations of the S_{CC}(q) to the product of the concentrations of the 2 chemical species of the mixture:
- S_{CC}(q) = c_{A}c_{B}: radom distribution.
- S_{CC}(q) > c_{A}c_{B}: homopolar atomic correlations (A-A, B-B) prefered.
- S_{CC}(q) < c_{A}c_{B}:heterpolar atomic correlations (A-B) prefered.
- 〈b〉 = 0: S_{CC}(q) = S(q).
- S_{NC}(q) is the Number-Concentration partial structure factor.
Its Fourier transform allows to obtain a correlation between the scattering centers and their occupation by a given chemical species. The more the chemical species related partial structure factors are different ( S_{AA}(q)≠S_{BB}(q)) and the more the oscillations are important in the S_{NC}(q). In the case of an ideal mixture S_{NC}(q) = 0, and all the information about the structure of the system is given by the S_{NN}(q).
If we consider the binary mixture as an ionic mixture then it is possible to calculate the Charge-Charge S_{ZZ}(q) and the Number-Charge S_{NZ}(q) partial structure factors using the Concentration-Concentration S_{CC}(q) and the Number-Concentration S_{NC}(q):
c_{A} and Z_{A} représent the concentration and the charge of the chemical species A, the global neutrality of the system must be respected therefore c_{A}Z_{A} + c_{B}Z_{B} = 0.
Figure [Fig. 3] illustrates, and allows to compare, the partial structure factors of glassy GeS_{2} at 300 K calculated in the different formalisms Faber-Ziman [b], Ashcroft-Langreth [c,d,e], and Bhatia-Thornton [f].
- a
- D. T. Cromer and J. B. Mann
Acta. Cryst., A24:321 (1968). - b
- T. E. Faber and J. M. Ziman
Phil. Mag., 11(109):153-173 (1965). - c
- N. W. Ashcroft and D. C. Langreth.
Phys. Rev., 156(3):685-692 (1967). - d
- N. W. Ashcroft and D. C. Langreth.
Phys. Rev., 159(3):500-510 (1967). - e
- N. W. Ashcroft and D. C. Langreth.
Phys. Rev., 166(3):934 (1968). - f
- A. B. Bhatia and D. E. Thornton.
Phys. Rev. B, 2(8):3004-3012 (1970).
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