This section is dealing with specific types of analysis performed by MDANSE. If you are not sure where these fit into the general workflow of data analysis, please read MDANSE Workflow.

Analysis: Scattering

This section contains background theory for following plugins:

This section discusses plugins used to calculate neutron spectroscopy observables from the trajectory. These plugins will be explored in depth in further sections, however, before that, it is important to understand how MDANSE performs these analyses. A part of that depends on how the \(\mathbf{q}\)-vectors are used to perform these analyses, see Section q-Vector Generation for details.

Scattering Background

Dynamic Structure Factor \(S(\mathbf{q}, \omega)\): This is a central concept in neutron scattering experiments. This factor characterizes how scattering intensity changes with alterations in momentum \(\mathbf{q}\) and energy \(\hbar\omega\) during scattering events. It is instrumental in unraveling the atomic and molecular structures of materials.

Double Differential Cross-Section: The dynamic structure factor is closely related to the double differential cross-section, which is a vital measurement in neutron scattering. The double differential cross-section, \({\mathrm{d}^{2}{\sigma/\mathit{\mathrm{d}\Omega \mathrm{d}E}}}\), is defined as the number of neutrons scattered per unit time into the solid angle interval \({\left\lbrack {\Omega, {\Omega + \mathrm{d}}\Omega} \right\rbrack}\) with an energy in the interval \({\left\lbrack {E, {E + \mathrm{d}}E} \right\rbrack}\). To make meaningful comparisons, the double differential cross-section is normalized by \(\mathrm{d}\Omega\), \(\mathrm{d}E\), and the flux of the incoming neutrons. The relationship between the double differential cross-section and the dynamic structure factor is given by

(76)\[{{\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E} = N}\frac{k_{\mathrm{f}}}{k_{\mathrm{i}}}S\left( {\mathbf{q},\omega} \right).}\]

This equation relates the double differential cross-section, which represents the number of neutrons scattered per unit time into specific solid angle and energy intervals, to the dynamic structure factor, \(S(\mathbf{q}, \omega)\). It includes terms related to the number of atoms \(N\) and wavenumbers of scattered \(k_{\mathrm{f}}\) and incident \(k_{\mathrm{i}}\) neutrons. They are related to the corresponding neutron energies by

(77)\[E_{\mathrm{f}} = \hbar^{2} k_{\mathrm{f}}^{2} / 2m_{\mathrm{n}} \qquad \text{and} \qquad E_{\mathrm{i}} = \hbar^{2}k_{\mathrm{i}}^{2} / 2m_{\mathrm{n}}.\]

These equations relate the neutron energies, \(E_{\mathrm{f}}\) and \(E_{\mathrm{i}}\), to their respective wavenumbers, \(k_{\mathrm{f}}\) and \(k_{\mathrm{i}}\), using the mass of the neutron \(m_{\mathrm{n}}\). They are fundamental for connecting energy and momentum in neutron scattering.

Momentum and Energy Transfer: These equations below define the momentum \(\mathbf{q}\) and energy transfer \(\hbar\omega\) based on the incident and scattered wavevectors and energies

(78)\[\mathbf{q} = \mathbf{k}_{\mathrm{i}} - \mathbf{k}_{\mathrm{f}} \qquad \text{and} \qquad \hbar\omega = E_{\mathrm{i}} - E_{\mathrm{f}}.\]

The square modulus of the momentum transfer can be expressed in terms of a scattering angle and the energies of the incident and scattered neutrons

(79)\[q^2 = \frac{2 m_{\mathrm{n}}}{\hbar^2} \left\{ E_{\mathrm{i}} + E_{\mathrm{f}} - 2\left(E_{\mathrm{i}}E_{\mathrm{f}} \right)^{1/2} \cos(\phi) \right\}.\]

Intermediate Scattering Function \(F(\mathbf{q}, t)\): This equation defines the dynamic structure factor \(S(\mathbf{q}, \omega)\) as a Fourier transform of the intermediate scattering function \(F(\mathbf{q}, t)\) with respect to time \(t\). It captures information about the structure and dynamics of the scattering system [Ref16] and can be written as

(80)\[{S{\left( {\mathbf{q},\omega} \right) = \frac{1}{2\pi}}{\int\mathrm{d}t \, } F\left( {\mathbf{q},t} \right) e^{-i \omega t}}\]

and

(81)\[{\text{F}{\left( {\mathbf{q},t} \right) = \frac{1}{N} {\sum\limits_{jk}{\Gamma_{jk}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\hat{\mathbf{r}}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\hat{\mathbf{r}}_{k}\left( t \right)} \right\rbrack} \right\rangle}}},}\]
(82)\[{{\Gamma_{jk} = }{\overline{b_{j}^{\dagger}}{\overline{b_{k}} + \delta_{jk}}( {\overline{\vert b_{j}\vert^{2}} - {\vert\overline{b_{j}}}\vert^{2}} )}}\]

where \(\hat{\mathbf{r}}_{j}(t)\) are the position operators of the nuclei in the Heisenberg picture. The quantities \(b_{j}\) are the scattering lengths of the nuclei which depend on the isotope and the relative orientation of the spin of the neutron and the spin of the scattering nucleus. If the spins of the nuclei and the neutron are not prepared in a special orientation one can assume a random relative orientation and that spin and position of the nuclei are uncorrelated. The overlines in Eq. (82) denotes that an average over isotopes and relative spin orientations of neutron and nucleus is made.

Coherent and Incoherent Scattering: Usually, one splits the intermediate scattering function and the dynamic structure factor into their coherent and incoherent parts which describe collective and single particle motions, respectively. By defining

(83)\[b_{\mathrm{coh},j} = \overline{b_{j}} \qquad \text{and} \qquad b_{\mathrm{inc},j} = \sqrt{\overline{\vert b_{j}\vert^{2}} - {\vert\overline{b_{j}}}\vert^{2}}\]

the coherent and incoherent intermediate scattering functions can be written. They are expressed as sums over pairs of nuclei, with different treatments for coherent and incoherent scattering lengths

(84)\[{\text{F}_{\text{coh}}{\left( {q,t} \right) = \frac{1}{N}}{\sum\limits_{jk}b_{\mathrm{coh},j}}b_{\mathrm{coh},k}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\hat{\mathbf{r}}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\hat{\mathbf{r}}_{k}\left( t \right)} \right\rbrack} \right\rangle,}\]
(85)\[{\text{F}_{\text{inc}}{\left( {q,t} \right) = \frac{1}{N}}{\sum\limits_{j}{b_{\mathrm{inc},j}^{2}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\hat{\mathbf{r}}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\hat{\mathbf{r}}_{j}\left( t \right)} \right\rbrack} \right\rangle}}.}\]

Classical Framework and Corrections: In the classical framework the intermediate scattering functions are interpreted as classical time correlation functions. The position operators are replaced by time-dependent vector functions and quantum thermal averages are replaced by classical ensemble averages. It is well known that this procedure leads to a loss of the universal detailed balance relation

(86)\[S(\mathbf{q},\omega) = \exp(\beta\hbar\omega) S( - \mathbf{q} , - \omega)\]

and also to a loss of all odd moments

(87)\[{\left\langle \omega^{2{n + 1}} \right\rangle = {\int{\mathrm{d}\omega}}\, \omega^{2{n + 1}}S\left( {\mathbf{q},\omega} \right).}\]

The odd moments vanish since the classical dynamic structure factor is even in \(\omega\), assuming invariance of the scattering process with respect to reflections in space. The first moment is also universal. For an atomic liquid, containing only one type of atom

(88)\[{{\left\langle \omega \right\rangle = \frac{\hbar q^{2}}{2M}}}\]

where \(M\) is the mass of the atom.

Recoil Moment: Eq. (88) shows that the first moment is given by the average kinetic energy (in units of \(\hbar\)) of a particle which receives a momentum transfer \(\hbar q\). Therefore, \(\langle\omega\rangle\) is called the recoil moment. A number of β€œrecipes” has been suggested to correct classical dynamic structure factors for detailed balance and to describe recoil effects in an approximate way. The most popular one has been suggested by Schofield [Ref17]

(89)\[{{S\left( {\mathbf{q},\omega} \right)\approx\exp(\beta\hbar\omega / 2)} S_{\mathrm{cl}}\left( {\mathbf{q},\omega} \right)}.\]

One can easily verify that the resulting dynamic structure factor fulfils the relation of detailed balance. Formally, the correction (89) is correct to first order in \(\hbar\) and cannot be used for large \(\mathbf{q}\)-values which correspond to large momentum transfers \(\hbar q\). This is actually true for all correction methods which have suggested so far. For more details we refer to [Ref18].

Current Correlation Function

The current correlation functions \(J_{\mu\nu}(\mathbf{q}, t)\) and its Fourier transform \(J_{\mu\nu}(\mathbf{q}, \omega)\) are closely related to the intermediate scattering function \(F(\mathbf{q}, t)\) and the dynamics structure factor \(S(\mathbf{q}, \omega)\) respectively. The intermediate scattering function \(F(\mathbf{q}, t)\) is a correlation function of the Fourier components of particle density whereas the current correlation function \(J_{\mu\nu}(\mathbf{q}, t)\) is the correlation function of the Fourier components of the particle current

(90)\[ J_{\mu\nu}(\mathbf{q}, t) = \sum_{\alpha}\sum_{\beta \geq \alpha} W_{\alpha\beta} J_{\alpha\beta\mu\nu}(\mathbf{q}, t),\]
(91)\[ J_{\alpha\beta\mu\nu}(\mathbf{q}, t) = \frac{1}{N \sqrt{c_{\alpha} c_{\beta}}} \langle j_{\alpha\mu}(-\mathbf{q}, 0) j_{\beta\nu}(\mathbf{q}, t) \rangle\]

where \(\mu\) and \(\nu\) are the cartesian directions \(x\), \(y\) or \(z\) and

(92)\[j_{\alpha\mu}(\mathbf{q}, t) = \sum_{k \in \alpha} v_{k\mu}(t) \exp(i\mathbf{q} \cdot \mathbf{r}_{k}(t)).\]

The particle currents can be projected onto longitudinal and transverse components of the \(\mathbf{q}\)-vector. The longitudinal and transverse particle current are

(93)\[ \mathbf{j}^{\mathrm{L}}_{\alpha}(\mathbf{q}, t) = \sum_{k \in \alpha} \hat{\mathbf{q}} \left[\mathbf{v}_{k}(t) \cdot \hat{\mathbf{q}}\right]\exp(i \mathbf{q}\cdot \mathbf{r}_k(t)),\]
(94)\[ \mathbf{j}^{\mathrm{T}}_{\alpha}(\mathbf{q}, t) = \sum_{k \in \alpha} \left\{\mathbf{v}_{k}(t) - \hat{\mathbf{q}}\left[\mathbf{v}_{k}(t) \cdot \hat{\mathbf{q}}\right] \right\}\exp(i \mathbf{q}\cdot \mathbf{r}_k(t))\]

where \(\hat{\mathbf{q}}\) are unit vectors of \(\mathbf{q}\). The partial longitudinal and transverse current correlation functions are

(95)\[ J^{\mathrm{L}}_{\alpha\beta}(\mathbf{q}, t) = \frac{1}{N \sqrt{c_{\alpha} c_{\beta}}} \langle \mathbf{j}^{\mathrm{L}}_{\alpha}(-\mathbf{q}, 0) \cdot \mathbf{j}^{\mathrm{L}}_{\beta}(\mathbf{q}, t) \rangle,\]
(96)\[ J^{\mathrm{T}}_{\alpha\beta}(\mathbf{q}, t) = \frac{1}{N \sqrt{c_{\alpha} c_{\beta}}} \langle \mathbf{j}^{\mathrm{T}}_{\alpha}(-\mathbf{q}, 0) \cdot \mathbf{j}^{\mathrm{T}}_{\beta}(\mathbf{q}, t) \rangle.\]

From the continuity equation we can obtain a relation between the longitudinal current correlation and the dynamic structure factor

(97)\[ J^{\mathrm{L}}_{\alpha\beta}(\mathbf{q}, \omega) = \frac{\omega^2}{q^2} S_{\alpha\beta}(\mathbf{q}, \omega).\]

Dynamic Coherent Structure Factor

In MDANSE the dynamic coherent structure factor (DCSF) and coherent intermediate scattering function is a weighted sum of the partial term

(98)\[ S_{\text{coh}}(\mathbf{q},\omega) = \sum_{\alpha}\sum_{\beta \geq \alpha} W_{\alpha\beta} S_{\text{coh},\alpha\beta}(\mathbf{q},\omega),\]
(99)\[ F_{\text{coh}}(\mathbf{q},t) = \sum_{\alpha}\sum_{\beta \geq \alpha} W_{\alpha\beta} F_{\text{coh},\alpha\beta}(\mathbf{q},t)\]

where

(100)\[F_{\text{coh},\alpha\beta}{(\mathbf{q},t) = \frac{1}{N \sqrt{c_{\alpha} c_{\beta}}}}{\sum\limits_{j \in \alpha}{\sum\limits_{k \in \beta}\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{k}\left( t \right)} \right\rbrack} \right\rangle}}\]

and

(101)\[ S_{\text{coh},\alpha\beta}(\mathbf{q},\omega) = \int\mathrm{d}t \, F_{\text{coh},\alpha\beta}(\mathbf{q},t) e^{-i\omega t}.\]

To obtain results relevant to neutron scattering, the b_coherent weight setting should be used so that the weight will be generated using the coherent scattering lengths.

Dynamic Incoherent Structure Factor

In MDANSE the dynamic incoherent structure factor (DISF) and incoherent intermediate scattering function is a weighted sum of the partial term

(102)\[ S_{\text{inc}}(\mathbf{q},\omega) = \sum_{\alpha} W_{\alpha} S_{\text{inc},\alpha}(\mathbf{q},\omega),\]
(103)\[ F_{\text{inc}}(\mathbf{q},t) = \sum_{\alpha} W_{\alpha} F_{\text{inc},\alpha}(\mathbf{q},t)\]

where

(104)\[F_{\text{inc},\alpha}{(\mathbf{q},t) = \frac{1}{N c_{\alpha} }}{\sum\limits_{j \in \alpha}{\left\langle {\exp\left\lbrack {{- i}\mathbf{q}\cdot\mathbf{r}_{j}\left( 0 \right)} \right\rbrack\exp\left\lbrack {i\mathbf{q}\cdot\mathbf{r}_{j}\left( t \right)} \right\rbrack} \right\rangle}}\]

and

(105)\[ S_{\text{inc},\alpha}(\mathbf{q},\omega) = \int\mathrm{d}t \, F_{\text{inc},\alpha}(\mathbf{q},t) e^{-i\omega t}.\]

To obtain results relevant to neutron scattering, the b_incoherent weight setting should be used so that the weight will be generated using the incoherent scattering lengths.

Elastic Incoherent Structure Factor

The elastic incoherent structure factor (EISF) appears as the amplitude of the elastic line in the neutron scattering spectrum. Elastic scattering is only present for systems in which the atomic motion is confined in space, as for solids. To understand which information is contained in the EISF we consider for simplicity a system where only one sort of atoms is visible to the neutrons. To a very good approximation this is the case for all systems containing a large amount of hydrogen atoms, as for biological systems. Incoherent scattering from hydrogen dominates by far all other contributions.

Similar to the Dynamic Incoherent Structure Factor described above, b_incoherent weight setting is used and the weights are generated using the incoherent scattering lengths.

The Van Hove Function: The EISF is defined as the limit of the incoherent intermediate scattering function for infinite time

(106)\[{\mathrm{EISF}(\mathbf{q}) = \lim\limits_{t\rightarrow\infty} F_{\mathrm{inc}}( {\mathbf{q},t} ).}\]

Using the above definition of the EISF one can decompose the incoherent intermediate scattering function as follows

(107)\[{F_{\text{inc}}{( {\mathbf{q},t} ) = \mathrm{EISF}}{(\mathbf{q}) + F_{\text{inc}}'}( {\mathbf{q},t} )}\]

where \(F_{\mathrm{inc}}'(\mathbf{q}, t)\) decays to zero for infinite time. Taking now the Fourier transform it follows immediately that

(108)\[{S_{\text{inc}}{( {\mathbf{q},\omega} ) = \mathrm{EISF}}(\mathbf{q})\delta{(\omega) + S_{\text{inc}}'}( {\mathbf{q},\omega} ).}\]

Using the definition of the self-part of the van Hove function

(109)\[\rho G_{\mathrm{s}} (\mathbf{r},t) = \frac{1}{2\pi^{3}} \int \mathrm{d} \mathbf{q} \, F_{\mathrm{inc}}(\mathbf{q},t) e^{-i\mathbf{q}\cdot\mathbf{r}}\]

which can be interpreted as the conditional probability to find a tagged particle at the position \(\mathbf{r}\) at time \(t\), given it started at the origin, one can write

(110)\[{\mathrm{EISF}(\mathbf{q}) = {\int \mathrm{d}}\mathbf{r} \, G_{\mathrm{s}}( {\mathbf{r},{t = \infty}} )e^{-i\mathbf{q}\cdot\mathbf{r}}.}\]

The EISF gives the sampling distribution of the points in space in the limit of infinite time. In a real experiment this means times longer than the time which is observable with a given instrument. The EISF vanishes for all systems in which the particles can access an infinite volume since \(G_{\mathrm{s}}(r, t)\) approaches \(N^{-1}\) for large times, this is the case for molecules in liquids and gases.

EISF Computation: For computational purposes it is convenient to use the following representation of the EISF

(111)\[\mathrm{EISF}(\mathbf{q}) = \sum\limits_{\alpha} W_{\alpha} \mathrm{EISF}_{\alpha}(\mathbf{q}), \qquad \mathrm{EISF}_{\alpha}(\mathbf{q}) = \frac{1}{Nc_{\alpha}} \sum\limits_{j \in \alpha} \left\vert \left\langle \exp[i \mathbf{q} \cdot \mathbf{r}_{j}] \right\rangle \right\vert^{2}\]

where \(\mathrm{EISF}_{\alpha}(\mathbf{q})\) is the EISF of atom-types \(\alpha\). This expression is derived from definition Eq. (106) of the EISF and expression Eq. (104) for the intermediate scattering function, using that for infinite time the relation

(112)\[\lim\limits_{t\rightarrow\infty}{\left\langle {\exp\left\lbrack {{- i\mathbf{q}}\cdot \mathbf{r}_{j}(0)} \right\rbrack\exp\left\lbrack { i\mathbf{q}\cdot \mathbf{r}_{j}(t)} \right\rbrack} \right\rangle = \left\langle \exp[-i \mathbf{q} \cdot \mathbf{r}_{j}(0)] \right\rangle \left\langle \exp[i \mathbf{q} \cdot \mathbf{r}_{j}(t)] \right\rangle}\]

holds. In this way the computation of the EISF is reduced to the computation of a static thermal average. We remark at this point that the length of the MD trajectory from which the EISF is computed should be long enough to allow for a representative sampling of the conformational space.

Gaussian Dynamic Incoherent Structure Factor

Cumulant Expansion: The MSD can be related to the incoherent intermediate scattering function via the cumulant expansion [Ref11], [Ref22]

(113)\[F_{\text{inc}}(\mathbf{q},t ) = \sum\limits_{\alpha} W_{\alpha} F_{\text{inc},\alpha}( \mathbf{q},t),\]
(114)\[F_{\text{inc},\alpha}(\mathbf{q},t) = \frac{1}{N c_{\alpha} } \sum\limits_{j \in \alpha} \exp\left[ -q^2 p_{1,j}(t) + q^4 p_{2,j}(t) + \cdots \right].\]

The cumulants \(p_{n,j}(t)\) are identified as

(115)\[p_{1,j}(\hat{\mathbf{q}},t){ = \frac{1}{2!}\left\langle d_{j}^{2}(\hat{\mathbf{q}}, t) \right\rangle},\]
(116)\[p_{2,j}(\hat{\mathbf{q}},t){ = \frac{1}{4!}}\left\lbrack {{\left\langle {d_{j}^{4}(\hat{\mathbf{q}}, t)} \right\rangle - 3}\left\langle {d_{j}^{2}(\hat{\mathbf{q}}, t)} \right\rangle^{2}} \right\rbrack\]

where

(117)\[ \Delta_{j}^{2}(\hat{\mathbf{q}},t) = \langle d_{j}^{2}(\hat{\mathbf{q}}, t) \rangle = \langle \left\vert \hat{\mathbf{q}} \cdot [\mathbf{r}_{j}(t) - \mathbf{r}_{j}(0)] \right\vert^2 \rangle\]

is the means squared displacement of atom \(j\) along the axis \(\hat{\mathbf{q}}\).

Gaussian Approximation: In the Gaussian approximation the above expansion is truncated after the \(q^2\)-term

(118)\[F_{\text{inc},\alpha}(\mathbf{q},t) \approx F_{\text{inc},\alpha}^{\text{G}}(\mathbf{q},t) = \frac{1}{N c_{\alpha} } \sum\limits_{j \in \alpha} \exp\left[ -q^2 p_{1,j}(\hat{\mathbf{q}}, t) \right].\]

For certain model systems like the ideal gas, the harmonic oscillator, and a particle undergoing Einstein diffusion, this is exact. For these systems the incoherent intermediate scattering function is completely determined by the MSD.

Similar to the Dynamic Incoherent Structure Factor described above, b_incoherent weight setting is used and the weights are generated using the incoherent scattering lengths.

Neutron Dynamic Total Structure Factor

This is a combines the coherent and incoherent intermediate scattering functions and corresponding dynamic structure factors. It is a fully neutron-specific analysis, so that coherent neutron scattering lengths b_coherent and the atomic incoherent neutron scattering lengths b_incoherent are used to weight the corresponding coherent and incoherent signals.

In this analysis the total incoherent, total coherent and total (coherent + incoherent) signals are calculated as

(119)\[F_{\mathrm{inc}}(\mathbf{q},t) = \sum_{\alpha} c_{\alpha}b_{\text{inc},\alpha}^{2} F_{\mathrm{inc},\alpha}(\mathbf{q},t),\]
(120)\[F_{\mathrm{coh}}(\mathbf{q},t) = \sum_{\alpha\beta} \sqrt{c_{\alpha}c_{\beta}} b_{\text{coh},\alpha}b_{\text{coh},\beta} F_{\mathrm{coh},\alpha\beta}(\mathbf{q},t),\]
(121)\[F(q,t) = F_{\mathrm{inc}}(\mathbf{q},t) + F_{\mathrm{coh}}(\mathbf{q},t).\]

These expressions correspond to the formalism and equations given in [Ref47] - Chapter 1: β€œAn introduction to neutron scattering” .

Scattering Length Density Profile

Calculates the time-averaged scattering length density profile of the system along one of the simulation box axes. This results is typically used for further calculations in other software packages for neutron reflectometry.

The result is the time-averaged coherent scattering length density profile given in units of \(10^{-6} \mathrm{Γ…}^{-2}\), as used in neutron reflectometry calculations.

Static Structure Factor

MDANSE computes the partial partial static structure factor (SSF) as the Fourier transform of the partial pair distribution function following the Faber-Ziman definition

(122)\[S_{\alpha\beta}(q) = 1 + \frac{4\pi\rho}{q}\int\limits_{0}^{\infty}{\mathrm{d}r \, \left\lbrack {g_{\alpha\beta}}(r)-1 \right\rbrack} r \sin(qr).\]

The SSF here is related to the coherent intermediate scattering function calculated in MDANSE via the following expression

(123)\[S_{\alpha\beta}(q) = \delta_{\alpha\beta} + \sqrt{c_{\alpha}c_{\beta}} \left[ F_{\mathrm{coh},\alpha\beta}(q, 0) - 1 \right].\]

Structure Factor From Scattering Function

Calculates the static structure factor from the coherent intermediate scattering function via the following expression

(124)\[S_{\alpha\beta}(q) = \delta_{\alpha\beta} + \sqrt{c_{\alpha}c_{\beta}} \left[ F_{\mathrm{coh},\alpha\beta}(q, 0) - 1 \right].\]

To obtain the total results MDANSE will use the same weights that were used in the DCSF calculation.

X-ray Static Structure Factor

MDANSE’s xray static structure factor analysis is tailored for neutron and X-ray scattering experiments in material science. It systematically investigates material structural properties by analyzing particle distribution and ordering. Researchers gain precise insights into fundamental aspects like atomic spacing and ordered patterns. MDANSE provides fine-grained control over β€œ\(r\)-values” and β€œ\(q\)-values,” enabling customization for probing specific material structural characteristics. This tool is invaluable for advancing scientific and industrial research, especially in neutron scattering experiments.