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Partial structure factors

The total structure factor can be decomposed as a sum of partial contributions. These contributions are functions of the various properties which characterize the material.
Several methods exist to decomposed the total S(q), the main goal of each of them is to explain the origin of the First Sharp Diffraction Peak F.S.D.P. in the total neutrons and X-rays 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 SFZαβ(q) is defined by:

SFZαβ(q) = 1 + 4πρ$displaystyle int_{{0}}^{{infty}}$ dr r2 $displaystyle {frac{{sin qr}}{{qr}}}$ $displaystyle left(vphantom{g_{alpha beta}(r)-1}right.$gαβ(r) - 1$displaystyle left.vphantom{g_{alpha beta}(r)-1}right)$ (10)


where the gαβ(r) are the partial radial distribution functions.
The total structure factor is then obtained by the relation:

S(q) = $displaystyle sum_{{alpha,beta}}^{}$ cαbα cβbβ $displaystyle left[vphantom{S^{FZ}_{alpha beta}(q) - 1}right.$SFZαβ(q) - 1$displaystyle left.vphantom{S^{FZ}_{alpha beta}(q) - 1}right]$ (11)

 


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 SALαβ(q) are defined by:

SALαβ(q) = δαβ + 4πρ$displaystyle left(vphantom{{c_alpha c_beta}}right.$cαcβ$displaystyle left.vphantom{{c_alpha c_beta}}right)^{{1/2}}_{}$ $displaystyle int_{{0}}^{{infty}}$ dr r2 $displaystyle {frac{{sin qr}}{{qr}}}$ $displaystyle left(vphantom{g_{alpha beta}(r)-1}right.$gαβ(r) - 1$displaystyle left.vphantom{g_{alpha beta}(r)-1}right)$ (12)


where δαβ is the Kronecker delta, cα = $displaystyle {frac{{N_alpha}}{{N}}}$, and the gαβ(r) are the partial radial distribution functions..
Then the  total structure factor can be calculated using:

S(q) = $displaystyle {frac{{displaystyle{sum_{alpha, beta}} b_alpha b_beta l... ...ha beta}(q) + 1right]}}{{displaystylesum_{alpha} c_alpha b_alpha^2}}}$ (13)

 


Bhatia-Thornton formalism

In this approach, reserved to the case of binary systems ABx [f], the total structure factor S(q) can be express as the weighted sum of 3 partial structure factors:

S(q) = $\displaystyle {\frac{{\langle b \rangle^2 S_{NN}(q) + 2\langle b \rangle(b_\tex... - (c_\text{A} b_\text{A}^2 + c_\text{B} b_\text{B}^2)}}{{\langle b \rangle^2}}}$ +  1 (14)


where b〉 = cAbA + cBbB, with cA and bA reprensenting respectively the concentration and the neutron or X-rays scattering length of species A.
SNN(q), SNC(q) and SCC(q) represent  combinaisons of the partial structure factors calculated using the Faber-Ziman formalism and weighted using the concentrations of the 2 chemical species:

SNN(q) = $displaystyle sum_{{text{A}=1}}^{{2}}$$displaystyle sum_{{text{B}=1}}^{{2}}$cAcBSFZAB(q) (15)

 

SNC(q) = cAcB×$displaystyle left[vphantom{ c_text{A}timesleft(S^{FZ}_{text{A}text{A}}... ...t(S^{FZ}_{text{B}text{B}}(q) - S^{FZ}_{text{A} text{B}}(q)right) }right.$ cA×$displaystyle left(vphantom{S^{FZ}_{text{A}text{A}}(q) - S^{FZ}_{text{A} text{B}}(q)}right.$SFZAA(q) - SFZAB(q)$displaystyle left.vphantom{S^{FZ}_{text{A}text{A}}(q) - S^{FZ}_{text{A} text{B}}(q)}right)$ - cB×$displaystyle left(vphantom{S^{FZ}_{text{B}text{B}}(q) - S^{FZ}_{text{A} text{B}}(q)}right.$SFZBB(q) - SFZAB(q)$displaystyle left.vphantom{S^{FZ}_{text{B}text{B}}(q) - S^{FZ}_{text{A} text{B}}(q)}right)$ $displaystyle left.vphantom{ c_text{A}timesleft(S^{FZ}_{text{A}text{A}}... ...t(S^{FZ}_{text{B}text{B}}(q) - S^{FZ}_{text{A} text{B}}(q)right) }right]$ (16)

 

SCC(q) = cAcB×$displaystyle 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.$1 + cAcB×$displaystyle left[vphantom{ sum_{text{A}=1}^{2} sum_{text{B}netext{A}}... ...ft( S^{FZ}_{text{A}text{A}}(q) - S^{FZ}_{text{A}text{B}}(q) right)}right.$$displaystyle sum_{{text{A}=1}}^{{2}}$$displaystyle sum_{{text{B}netext{A}}}^{{2}}$$displaystyle left(vphantom{S^{FZ}_{text{A}text{A}}(q) - S^{FZ}_{text{A} text{B}}(q)}right.$SFZAA(q) - SFZAB(q)$displaystyle left.vphantom{S^{FZ}_{text{A}text{A}}(q) - S^{FZ}_{text{A} text{B}}(q)}right)$$displaystyle left.vphantom{ sum_{text{A}=1}^{2} sum_{text{B}netext{A}}... ...ft( S^{FZ}_{text{A}text{A}}(q) - S^{FZ}_{text{A}text{B}}(q) right)}right]$$displaystyle 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]$ (17)



  • SNN(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 bA = bB then SNN(q) = S(q).
  • SCC(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 SNN(q). Therefore the SCC(q) describes the chemical order in the material. In the case of an ideal binary mixture of 2 chemical species A and B5.1, SCC(q) is constant and equal to cAcB. 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 SCC(q) to the product of the concentrations of the 2 chemical species of the mixture:

    • SCC(q) = cAcB: radom distribution.
    • SCC(q) > cAcB: homopolar atomic correlations (A-A, B-B) prefered.
    • SCC(q) < cAcB:heterpolar atomic correlations (A-B) prefered.
    • b〉 = 0: SCC(q) = S(q).
  • SNC(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 ( SAA(q)≠SBB(q)) and the more the oscillations are important in the SNC(q). In the case of an ideal mixture SNC(q) = 0, and all the information about the structure of the system is given by the SNN(q).

If we consider the binary mixture as an ionic mixture then it is possible to calculate the Charge-Charge SZZ(q) and the Number-Charge SNZ(q) partial structure factors using the Concentration-Concentration SCC(q) and the Number-Concentration SNC(q):

SZZ(q) = $displaystyle {frac{{S_{CC}(q)}}{{c_A c_B}}}$ and SNZ(q) = $displaystyle {frac{{S_{NC}(q)}}{{c_B/Z_A}}}$ (18)


cA and ZA représent the concentration and the charge of the chemical species A, the global neutrality of the system must be respected therefore cAZA + cBZB = 0.

Figure [Fig. 3] illustrates, and allows to compare, the partial structure factors of glassy GeS2 at  300 K calculated in the different formalisms Faber-Ziman [b], Ashcroft-Langreth [c,d,e], and Bhatia-Thornton [f].


Image allsqp
Figure 3: Partial structure factors of glassy GeS2 at 300 K. , A Faber-Ziman [b], B Ashcroft-Langreth [c,d,e] and C 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).


Last Updated on Monday, 23 June 2014 05:01  

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