Static structure factors S(q) may be compared to experimental scattering data and that is why are useful structural characteristics computed by I.S.A.A.C.S.
Thereafter we describe the theoretical background of S(q)s computed by I.S.A.A.C.S.

Total scattering

Neutron or X-ray scattering static structure factor S(q) is defined as:

S(q) = $\displaystyle {\frac{{1}}{{N}}}$$\displaystyle \sum_{{j,k}}^{}$bj bk$\displaystyle \left<\vphantom{ e^{\displaystyle{iq[{\bf {r}}_j-{\bf {r}}_k]}} }\right.$eiq[$\scriptstyle \bf {r}_{j}$ - $\scriptstyle \bf {r}_{k}$]$\displaystyle \left.\vphantom{ e^{\displaystyle{iq[{\bf {r}}_j-{\bf {r}}_k]}} }\right>$ (1)

where bj and $ bf {r}_{j}^{}$ represent respectively the neutron or X-ray scattering length, and the position of the atom j. N is the total number of atoms in the system studied. To take into account the inherent/volume averaging of scattering experiments it is necessary to sum all possible orientations of the wave vector q compared to the vector $ bf {r}_{j}^{}$ - $ bf {r}_{k}^{}$.
This average on the orientations of the q vector leads to the famous Debye's equation:

S(q) = $displaystyle {frac{{1}}{{N}}}$$displaystyle sum_{{j,k}}^{}$bj bk$displaystyle {frac{{sin (qvert{bf {r}}_j-{bf {r}}_kvert)}}{{qvert{bf {r}}_j-{bf {r}}_kvert}}}$ (2)


Nevertheless the instantaneous individual atomic contributions introduced by this equation [Eq. 2] are not easy to interpret, It is more interesting to express these contributions using the formalism of radial distribution functions.

In order to achieve this goal it is first necessary to split the self-atomic contribution (j = k), from the contribution between distinct atoms:

S(q) = $displaystyle sum_{{j}}^{}$ cjbj2 + $displaystyle underbrace{{frac{1}{N} sum_{jne k} b_j,b_k frac{sin (qver... ...r}}_kvert)}{qvert{bf {r}}_j-{bf {r}}_kvert}}}_{{displaystyle{I(q)}}}^{},$ (3)

with cj = $displaystyle {frac{{N_j}}{{N}}}$.
4π $displaystyle sum_{{j}}^{}$ cjbj2 represents the total scattering cross section of the material.
The function I(q) which describes the interaction between distinct atoms is related to the radial distribution functions through a Fourier transformation:
I(q) = 4πρ $displaystyle int_{{0}}^{{infty}}$ dr r2 $displaystyle {frac{{sin qr}}{{qr}}}$ G(r) (4)

where the function G(r) is defined using the partial radial distribution functions :
G(r) = $displaystyle sum_{{alpha,beta}}^{}$ cαbα cβbβ (gαβ(r) - 1) (5)

where cα = $displaystyle {frac{{N_alpha}}{{N}}}$ and bα represents the neutron or X-ray scattering length of species α.
G(r) approaches - $displaystyle sum_{{alpha,beta}}^{}$ cαbα cβbβ for r = 0, and 0 for r→∞.
Usually the self-contributions are substracted from equation [Eq. 3] and the structure factor is normalized using the relation:
S(q) - 1 = $displaystyle {frac{{I(q)}}{{displaystyle{langle b^{2} rangle}}}}$ with 〈b2〉 = $displaystyle left(vphantom{sum_{alpha} c_{alpha} b_{alpha} }right.$$displaystyle sum_{{alpha}}^{}$cαbα$displaystyle left.vphantom{sum_{alpha} c_{alpha} b_{alpha} }right)^{{2}}_{}$ (6)

It is therefore possible to write the structure factor [Eq. 4] in a more standard way:
S(q) = 1 + 4πρ$displaystyle int_{{0}}^{{infty}}$ dr r2 $displaystyle {frac{{sin qr}}{{qr}}}$($displaystyle bf {g}$(r) - 1) (7)

where$ bf {g}$(r) ( the radial distribution function) is defined as :
$displaystyle bf {g}$(r) = $displaystyle {frac{{displaystyle{sum_{alpha,beta}} c_{alpha} b_{alpha}... ...beta} b_{beta} g_{alphabeta}(r) }}{{displaystyle{langle b^{2} rangle}}}}$ (8)


In the case of a single atomic species system the normalization allows to obtain values of S(q) and$ bf {g}$(r) which are independent of the scattering factor/length and therefore independent of the measurement technique. In most cases, however, the total S(q) and g(r) are combinations of the partial functions weighted using the scattering factor and therefore depend on the measurement technique (Neutron, X-rays ...) used or simulated.


Image sqsk
Figure 1: Total neutron structure factor for glassy GeS2 at 300 K - A Evaluation using the atomic correlations [Eq. 2], B Evaluation using the pair correlation functions [Eq. 7].

Figure [Fig. 1] presents a comparison bewteen the calculations of the total neutron structure factor done using the Debye relation [Eq. 2] and the pair correlation functions [Eq. 7]. The material studied is a sample of glassy GeS2 at 300 K obtained using ab-initio molecular dynamics.
In several cases the structure factor S(q) and the radial distribution function $ bf {g}$(r) [Eq. 8] can be compared to experimental data.
To simplify the comparison I.S.A.A.C.S. computes several radial distribution funcrions used in practice suh as G(r) defined [Eq. 5], the differential correlation function D(r), G(r), and the total correlation function T(r) defined by:

D(r) = 4πrρ G(r)
G(r) = D(r) / 〈b2〉 (9)
T(r) = D(r) + 4πrρ 〈b2〉


$ bf {g}$(r) equals zero for r = 0 and approaches 1 for r→∞.
D(r) equals zero for r = 0 and approaches 0 for r→∞.
G(r) equals zero for r = 0 and approaches 0 for r→∞.
T(r) equals zero for r = 0 and approaches ∞ for r→∞.

This set of functions for a model of GeS2 glass (at 300 K) obtained using ab-intio molecular dynamics is presented in figure [Fig. 2].

Image GTDgr
Figure 2: Exemple of various distribution functions neutron-weighted in glassy GeS2 at 300 K.


I.S.A.A.C.S. can compute, for the case of x-ray or neutrons, the following functions:
  • S(q) and Q(q) = q[S(q) -1] computed using the Debye equation
  • S(q) and Q(q) = q[S(q) -1] computed using the Fourier transform of the $ bf {g}$(r)
  • $ bf {g}$(r) and G(r) computed using the standard real space calculation
  • $ bf {g}$(r) and G(r) computed using the Fourier transform of the structure factor calculated using the Debye equation

Partial structure factors

There are a few, somewhat different definitions of 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 [a]. 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 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 definition/formalism

In a similar approach, based on the correlation between the chemical species, and developped by Ashcroft et Langreth [b,c,d], 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 definition/formalism

In this approach, used in the case of binary systems ABx [e] only, 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 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)


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 [a], Ashcroft-Langreth [b,c,d], and Bhatia-Thornton [e].


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


I.S.A.A.C.S. can compute the following partial structure factors:
  • Faber-Ziman
  • Ashcroft-Langreth
  • Bhatia-Thornton
a
T. E. Faber and Ziman J. M.
Phil. Mag., 11(109):153-173 (1965).
b
N. W. Ashcroft and D. C. Langreth.
Phys. Rev., 156(3):685-692 (1967).
c
N. W. Ashcroft and D. C. Langreth.
Phys. Rev., 159(3):500-510 (1967).
d
N. W. Ashcroft and D. C. Langreth.
Phys. Rev., 166(3):934 (1968).
e
A. B. Bhatia and D. E. Thornton.
Phys. Rev. B, 2(8):3004-3012 (1970).




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