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Partial structure factors |
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An alternative way to analyze the results of molecular dynamics simulations is to compute the static neutrons and X-rays structure factors S(q).
S(q) allows to study the organization of the particles in the material and can be directly compared to neutrons and X-rays scattering experiments.
Thereafter you will find the theoretical elements that will help you to understand theses techniques, and to obtain an overview of the formalisms implemented in the R.I.N.G.S. code.
Total scattering
The pair correlation function g(r) cannot be measured experimentally, therefore the g(r) obtained using simulations cannot be compared directly to experimental data. Nevertheless it is possible to compare the results of the numerical simulations to neutron and X-rays scattering experiments using the static neutron structure factor S(q):
where b_{j} and represent respectively the neutron or the X-rays scattering length and the position of the atom j. N represents the total number of atoms in the system studied. The X-rays scattering is q point dependent with b_{j} = f_{j} (q) this is implemented in the rings-code from version 1.2.5, see "X-ray scattering factors computed from numerical Hartree-Fock wave functions", in [a] for details.
Equation [Eq. 1] defines for the atoms of the system the dispersion of a beam radiation at a given q vector. In neutron scattering experiment it is necessary to sum all possible orientations of q compared to the vector - .
This average on the orientations of the q vector lead to the Debye result:
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 introducing the 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:
with c_{j} = .
4π c_{j}b_{j}^{2} 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 thru the Fourier transform:
where the function G(r) is defined using the partial radial distribution functions
:
where c_{α} = and b_{α} represents the neutron scattering length of species α.
G(r) towards to the value - c_{α}b_{α} c_{β}b_{β} for r = 0 and to 0 for r→∞.
Usually the self-contributions are substracted from equation [Eq. 3] and the structure factor is normalized using the relation:
It is therefore possible to write the structure factor [Eq. 4] in a more standard way:
where(r) is defined using the partial radial distribution functions :
In the case of a single atomic species system the normalization allows to obtain values of S(q) and(r) which are independent of the scattering factor/length and therefore independent of the measurement technique.
Nevertheless in the other cases the total functions are combinations of the partial functions weighted using the scattering factor and therefore depend on the measurement technique (Neutron, X-rays ...) used or simulated.
Figure [Fig. 1] presents a comparison bewteen the calculations of the total neutron structure factor done using on the one hand the atomic correlations [Eq. 2] and on the other hand the pair correlation functions [Eq. 7].
The material studied is a sample of glassy GeS_{2} at 300 K obtained using ab-initio molecular dynamics.
In several cases the neutron structure factor S(q) and the radial distribution function (r) [Eq. 8] can be compared to experimental data.
Nevertheless depending on the neutron scattering formalism used other function can be used to be compared to the experiment. Among these function the G(r) has already been defined [Eq. 5], the differential correlation function D(r), G(r) and the total correlation function T(r) are defined by:
(r) is equal to zero for r = 0 and towards to 1 for r→∞.
D(r) is equal to zero for r = 0 and towards to 0 for r→∞.
G(r) is equal to zero for r = 0 and towards to 0 for r→∞.
T(r) is equal to zero for r = 0 and towards to ∞ for r→∞.
The result of the evaluation of these different neutron weighted functions for a glassy GeS_{2} sample at 300 K obtained using ab-intio molecular dynamics is presented in figure [Fig. 2].
This allows to illustrate the differences between each of these functions.
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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