Partial structure factors

There are a few, somewhat different definitions of partials S(q) used in practice, and computed by I.S.A.A.C.S.

Faber-Ziman definition/formalism

One way used to define the partial structure factors has been proposed by Faber and Ziman [11]. In this approach the structure factor is represented by 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:

$$\int_{{0}}^{{\infty}} {\frac{{\sin qr}}{{qr}}} \left(\vphantom{g_{\alpha \beta}(r) - 1}\right. \left.\vphantom{g_{\alpha \beta}(r) - 1}\right)$$ (5.15)

where the g_αβ(r) are the partial radial distribution functions [Eq. 5.4].
The total structure factor is then obtained by the relation:

$$\sum_{{\alpha,\beta}}^{} \left[\vphantom{S^{FZ}_{\alpha \beta}(q) - 1}\right. \left.\vphantom{S^{FZ}_{\alpha \beta}(q) - 1}\right]$$ (5.16)

Ashcroft-Langreth definition/formalism

In a similar approach, based on the correlation between the chemical species, and developped by Ashcroft et Langreth [12,13,14], the partial structure factors S^AL_αβ(q) are defined by:

$$\left(\vphantom{{c_\alpha c_\beta}}\right. \left.\vphantom{{c_\alpha c_\beta}}\right)^{{1/2}}_{} \int_{{0}}^{{\infty}} {\frac{{\sin qr}}{{qr}}} \left(\vphantom{g_{\alpha \beta}(r) - 1}\right. \left.\vphantom{g_{\alpha \beta}(r) - 1}\right)$$ (5.17)

where δ_αβ is the Kronecker delta, c_α = $${\frac{{N_\alpha}}{{N}}}$$, and the g_αβ(r) are the partial radial distribution functions [Eq. 5.4].
Then the total structure factor can be calculated using:

$${\frac{{\displaystyle{\sum_{\alpha, \beta}} b_\alpha b_\beta \l... ...ha \beta}(q) + 1\right]}}{{\displaystyle\sum_{\alpha} c_\alpha b_\alpha^2}}}$$ (5.18)

Bhatia-Thornton definition/formalism

In this approach, used in the case of binary systems AB_x [15] only, the total structure factor S(q) can be express as the weighted sum of 3 partial structure factors:

$${\frac{{\langle b \rangle^2 S_{NN}(q) + 2\langle b \rangle(b_\tex... ...- (c_\text{A} b_\text{A}^2 + c_\text{B} b_\text{B}^2)}}{{\langle b \rangle^2}}}$$ (5.19)

where 〈b〉 = c_Ab_A + c_Bb_B, with c_A and b_A reprensenting respectively the concentration and the 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:

$$\sum_{{\text{A}=1}}^{{2}} \sum_{{\text{B}=1}}^{{2}}$$ (5.20)

$$\left[\vphantom{ c_\text{A}\times\left(S^{FZ}_{\text{A}\text{A}}... ...t(S^{FZ}_{\text{B}\text{B}}(q) - S^{FZ}_{\text{A} \text{B}}(q)\right) }\right. \left(\vphantom{S^{FZ}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A} \text{B}}(q)}\right. \left.\vphantom{S^{FZ}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A} \text{B}}(q)}\right) \left(\vphantom{S^{FZ}_{\text{B}\text{B}}(q) - S^{FZ}_{\text{A} \text{B}}(q)}\right. \left.\vphantom{S^{FZ}_{\text{B}\text{B}}(q) - S^{FZ}_{\text{A} \text{B}}(q)}\right) \left.\vphantom{ c_\text{A}\times\left(S^{FZ}_{\text{A}\text{A}}... ...t(S^{FZ}_{\text{B}\text{B}}(q) - S^{FZ}_{\text{A} \text{B}}(q)\right) }\right]$$ (5.21)

$$\left[\vphantom{ 1 + c_{\text{A}} c_{\text{B}} \times \left[ \sum... ...Z}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A}\text{B}}(q) \right)\right] }\right. \left[\vphantom{ \sum_{\text{A}=1}^{2} \sum_{\text{B}\ne\text{A}}... ...ft( S^{FZ}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A}\text{B}}(q) \right)}\right. \sum_{{\text{A}=1}}^{{2}} \sum_{{\text{B}\ne\text{A}}}^{{2}} \left(\vphantom{S^{FZ}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A} \text{B}}(q)}\right. \left.\vphantom{S^{FZ}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A} \text{B}}(q)}\right) \left.\vphantom{ \sum_{\text{A}=1}^{2} \sum_{\text{B}\ne\text{A}}... ...ft( S^{FZ}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A}\text{B}}(q) \right)}\right] \left.\vphantom{ 1 + c_{\text{A}} c_{\text{B}} \times \left[ \sum... ...Z}_{\text{A}\text{A}}(q) - S^{FZ}_{\text{A}\text{B}}(q) \right)\right] }\right]$$ (5.22)

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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).

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S_CC(q) is the Concentration-Concentration partial structure factor.
Its Fourier transform allows to obtain an idea of the distribution 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^5.2, S_CC(q) is constant and equal to c_Ac_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_Ac_B: radom distribution.
• S_CC(q) > c_Ac_B: homopolar atomic correlations (A-A, B-B) prefered.
• S_CC(q) < c_Ac_B: heterpolar atomic correlations (A-B) prefered.
• 〈b〉 = 0: S_CC(q) = S(q).

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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):

$${\frac{{S_{CC}(q)}}{{c_A c_B}}} {\frac{{S_{NC}(q)}}{{c_B/Z_A}}}$$ (5.23)

c_A et Z_A represent the concentration and the charge of the chemical species A, the global neutrality of the system must be respected therefore c_AZ_A + c_BZ_B = 0.

Figure [Fig. 5.6] illustrates, and allows to compare, the partial structure factors of glassy GeS₂ at 300 K calculated in the different formalisms Faber-Ziman [11], Ashcroft-Langreth [12,13,14], and Bhatia-Thornton [15].

Figure 5.6: Partial structure factors of glassy GeS₂ at 300 K. A Faber-Ziman [11], B Ashcroft-Langreth [12,13,14] et C Bhatia-Thornton [15].

I.S.A.A.C.S. can compute the following partial structure factors:

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Faber-Ziman S^FZ_αβ(q)

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Ashcroft-Langreth S^AL_αβ(q)

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Bhatia-Thornton S_NN(q), S_NC(q), S_CC(q) and S_ZZ(q)