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Simulation of neutrons and X-rays scattering - Total scattering

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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

$ 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.

Last Updated on Monday, 23 June 2014 05:01  

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