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Simulation of neutrons and X-rays scattering

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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):

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 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 bj = fj (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 $ bf {r}_{j}^{}$ - $ bf {r}_{k}^{}$.
This average on the orientations of the q vector lead to the Debye result:

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 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:

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 thru the Fourier transform:

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 scattering length of species α.
G(r) towards to the value - $displaystyle sum_{{alpha,beta}}^{}$ cαbα cβbβ 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:

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) is defined using the partial radial distribution functions :

$displaystyle bf {g}$(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.
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.


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 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 GeS2 at 300 K obtained using ab-initio molecular dynamics.
In several cases the neutron structure factor S(q) and the radial distribution function $ bf {g}$(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:

D(r) = 4πrρ G(r)
G(r) =D(r)/〈b2
T(r) = D(r) + 4πrρb2
(9)


$ bf {g}$(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 GeS2 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.

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

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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