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 The MDANSE Workflow.

Analysis Theory: Scattering

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 are Q vectors, which are used to perform these analyses. An in-depth discussion of this aspect is present in Appendix 2.

Theory and background

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

Double Differential Cross-Section: (S(q, ω) is closely related to the Double Differential Cross-Section[7], which is a vital measurement in neutron scattering. The double differential cross-section is defined as the number of neutrons scattered per unit time into the solid angle interval \({\left\lbrack {\Omega,{\Omega + d}\Omega} \right\rbrack}\) and into the energy interval (energy interval E). \({d^{2}{\sigma/\mathit{d\Omega dE}}}\). To make meaningful comparisons, the double differential cross-section is normalized by d, dE, and the flux of the incoming neutrons. The relationship between the double differential cross-section and the dynamic structure factor is given by:

(44)\[{{\frac{d^{2}\sigma}{d\Omega\mathit{dE}} = N}\cdot\frac{k}{k_{0}}S\left( {q,\omega} \right).}\]

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(q, ω). It includes terms related to the number of atoms (N) and wave numbers of scattered (k) and incident (k0) neutrons.

They are related to the corresponding neutron energies by

(45)\[{E = \hslash^{2}}k^{2}\text{/}2m\]

and

(46)\[{E_{0} = \hslash^{2}}k_{0}^{2}\text{/}2m\]

These equations relate the neutron energies (E and E0) to their respective wave numbers (k and k0) using the mass of the neutron (m). They are fundamental for connecting energy and momentum in neutron scattering.

Dimensionless Momentum and Energy Transfer: These equations below define the dimensionless momentum (q, dynamic structure factor) and energy (ω) transfer in units of the reduced Planck constant (ħ) based on the incident and scattered wave numbers and energies:

(47)\[{{q = \frac{k_{0} - k}{\hslash}},}\]

(48)\[{{\omega = \frac{E_{0} - E}{\hslash}}.}\]

Then, Expresses the modulus of the momentum transfer in terms of scattering angle, energy transfer, and incident neutron energy. See equations below:

(49)\[{{q = \sqrt{{2 - \frac{\mathit{\hslash\omega}}{E_{0}} - 2}\cos{\theta\sqrt{2 - \frac{\mathit{\hslash\omega}}{E_{0}}}}}}.}\]

Intermediate Scattering Function (F(q, t)):

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

(50)\[{S{\left( {q,\omega} \right) = \frac{1}{2\pi}}{\int\limits_{- \infty}^{+ \infty}\mathit{dt}}\exp\left\lbrack {{- i}\omega t} \right\rbrack F\left( {q,t} \right).}\]

F(q, t) is called the intermediate scattering function and is defined as

(51)\[{\text{F}{\left( {q,t} \right) = {\sum\limits_{\alpha,\beta}{\Gamma_{\mathit{\alpha\beta}}\left\langle {\exp\left\lbrack {{- i}q\cdot\hat{R_{\alpha}}(0)} \right\rbrack\exp\left\lbrack {iq\cdot\hat{R_{\beta}}(t)} \right\rbrack} \right\rangle}}},}\]

(52)\[{{\Gamma_{\mathit{\alpha\beta}} = \frac{1}{N}}\left\lbrack {\overline{b_{\alpha}}{\overline{b_{\beta}} + \delta_{\mathit{\alpha\beta}}}\left( {\overline{b_{\alpha}^{2}} - {\overline{b_{\alpha}}}^{2}} \right)} \right\rbrack.}\]

The operators \(\hat{R_{\alpha}}(t)\) in Eq. (51) are the position operators of the nuclei in the sample. The brackets \(\langle\ldots\rangle\) denote a quantum thermal average and the time dependence of the position operators is defined by the Heisenberg picture. The quantities \(b_{\alpha}\) 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 symbol \(\overline{...}\) appearing in \({\Gamma_{\mathit{\alpha\beta}}}\) denotes an average over isotopes and relative spin orientations of neutron and nucleus.

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. Defining

(53)\[{b_{\alpha,\mathit{coh}}\doteq\overline{b_{\alpha}},}\]

(54)\[{b_{\alpha,\mathit{inc}}\doteq\sqrt{\overline{b_{\alpha}^{2}} - {\overline{b_{\alpha}}}^{2}},}\]

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

(55)\[{\text{F}_{\text{coh}}{\left( {q,t} \right) = \frac{1}{N}}{\sum\limits_{\alpha,\beta}b_{\alpha,\mathit{coh}}}b_{\beta,\mathit{coh}}\left\langle {\exp\left\lbrack {{- i}q\cdot\hat{R_{\alpha}}(0)} \right\rbrack\exp\left\lbrack {iq\cdot\hat{R_{\beta}}(t)} \right\rbrack} \right\rangle,}\]

(56)\[{\text{F}_{\text{inc}}{\left( {q,t} \right) = \frac{1}{N}}{\sum\limits_{\alpha}{b_{\alpha,\mathit{inc}}^{2}\left\langle {\exp\left\lbrack {{- i}q\cdot\hat{R_{\alpha}}(0)} \right\rbrack\exp\left\lbrack {iq\cdot\hat{R_{\alpha}}(t)} \right\rbrack} \right\rangle}}.}\]

Rewriting these formulas, MDANSE introduces the partial terms, this consider different species (I, J) and their contributions to the scattering process.

(57)\[{\text{F}_{\text{coh}}{\left( {q,t} \right) = \sum\limits_{I,J\geq I}^{N_{\mathit{species}}}}\sqrt{n_{I}n_{J}\omega_{I,\text{coh}}\omega_{J,\text{coh}}}F_{\mathit{IJ},\text{coh}}\left( {q,t} \right),}\]

(58)\[{\text{F}_{\text{inc}}{\left( {q,t} \right) = {\sum\limits_{I = 1}^{N_{\mathit{species}}}{n_{I}\omega_{I,\text{inc}}F_{I,\text{inc}}\left( {q,t} \right)}}}}\]

where:

(59)\[{\text{F}_{\mathit{IJ},\text{coh}}{\left( {q,t} \right) = \frac{1}{\sqrt{n_{I}n_{J}}}}{\sum\limits_{\alpha}^{n_{I}}{\sum\limits_{\beta}^{n_{J}}\left\langle {\exp\left\lbrack {{- i}q\cdot\hat{R_{\alpha}}\left( t_{0} \right)} \right\rbrack\exp\left\lbrack {iq\cdot\hat{R_{\beta}}\left( {t_{0} + t} \right)} \right\rbrack} \right\rangle_{t_{0}}}},}\]

(60)\[{\text{F}_{I,\text{inc}}{\left( {q,t} \right) = \frac{1}{n_{I}}}{\sum\limits_{\alpha = 1}^{n_{I}}\left\langle {\exp\left\lbrack {{- i}q\cdot\hat{R_{\alpha}}\left( t_{0} \right)} \right\rbrack\exp\left\lbrack {iq\cdot\hat{R_{\alpha}}\left( {t_{0} + t} \right)} \right\rbrack} \right\rangle_{t_{0}}}.}\]

where n_I, n_J, N_species, \(\omega\)_I,coh,inc and \(\omega\)_J,coh,inc are defined in Section target_CN.

The corresponding dynamic structure factors are obtained by performing the Fourier transformation defined in Eq. (50).

Static Structure Factor (S(q)): An important quantity describing structural properties of liquids is the static structure factor, which is defined above. (S(q)) as an integral involving the dynamic structure factor and the coherent intermediate scattering function at zero time delay (t = 0).

(61)\[{\text{S}(q)\doteq{\int\limits_{- \infty}^{+ \infty}{d\omega}}\text{S}_{\mathit{coh}}\left( {q,\omega} \right)\text{F}_{\mathit{coh}}\left( {q,0} \right).}\]

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,

(62)\[{\text{S}{\left( {q,\omega} \right) = \exp}\left\lbrack {\beta\hslash\omega} \right\rbrack\text{S}\left( {{- q}{, - \omega}} \right),}\]

and also to a loss of all odd moments

(63)\[{\left\langle \omega^{2{n + 1}} \right\rangle\doteq{\int\limits_{- \infty}^{+ \infty}{d\omega}}\omega^{2{n + 1}}S\left( {q,\omega} \right),{n = 1,2},\ldots.}\]

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 sort of atoms, it reads

(64)\[{{\left\langle \omega \right\rangle = \frac{\hslash q^{2}}{2M}},}\]

where M is the mass of the atoms.

Recoil Moment: Formula (64) shows that the first moment is given by the average kinetic energy (in units of \(\hslash\)) of a particle which receives a momentum transfer \(\hslash q\). Therefore, \(\langle\omega\rangle\) is called the recoil moment. A number of ‘recipes’ 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]

(65)\[{{\text{S}\left( {q,\omega} \right)\approx\exp\left\lbrack \frac{\beta\hslash\omega}{2} \right\rbrack}_{}\text{S}_{\mathit{cl}}\left( {q,\omega} \right)}\]

One can easily verify that the resulting dynamic structure factor fulfils the relation of detailed balance. Formally, the correction (65) is correct to first order in \(\hslash\). Therefore, it cannot be used for large q-values which correspond to large momentum transfers \(\hslash q\). This is actually true for all correction methods which have suggested so far. For more details we refer to Ref. [Ref18].

Total Structure Factors:

MDANSE computes the partial S(Q)’s as the Fourier transform of the partial g(r), corresponding to the Faber-Ziman definition:

(66)\[{S_{\alpha\beta}(Q) = 1 + \frac{4\pi\rho_0}{Q}\int\limits_{0}^{\infty}{r\left\lbrack {g_\alpha\beta}(r)-1 \right\rbrack\text{sin}(Qr)dr}}\]

The total S(Q) is computed as a weighted sum similar to the one used for the total g(r). In the case of the analysis ‘X-ray Static structure factor’, the Q-dependence of the atomic form factors is taken into account in this weighted sum.

X-ray Observable Normalization: Again, Soper has provided experimental data (table 4 in ISRN Physical Chemistry, 279463 (2013), given in file soper13_fx.dat). Here a source of confusion is that the data can be normalized in different ways (see Soper’s paper). Using the normalization II in that reference we have that:

(67)\[{D_{x}{(Q) = \frac{\sum\limits_{\mathit{\alpha\beta}\geq\alpha}{\left( {2 - \delta_{\mathit{\alpha\beta}}} \right)\times c_{\alpha}c_{\beta}f_{\alpha}{(Q)}f_{\beta}{(Q)}\left\lbrack {S_{\mathit{\alpha\beta}}{(Q) - 1}} \right\rbrack}}{\sum\limits_{\alpha}{c_{\alpha}f_{\alpha}^{2}{(Q)}}} = \left\lbrack {S{(Q) - 1}} \right\rbrack}\times\frac{\sum\limits_{\mathit{\alpha\beta}}{c_{\alpha}c_{\beta}f_{\alpha}{(Q)}f_{\beta}{(Q)}}}{\sum\limits_{\alpha}{c_{\alpha}f_{\alpha}^{2}{(Q)}}}}\]

Where S(Q) would be the static structure factor (going to 1 at large Q) computed by MDANSE. Therefore, even after using MDANSE we should recalculate the x-ray observable using the atomic factors.

Current Correlation Function

The correlation function is a fundamental concept in the study of dynamical processes in various physical systems, including disordered materials. It provides insights into how fluctuations or excitations propagate through a system over time. In the context of disordered systems, understanding the correlation function can help reveal the behavior of particles or components in a disordered environment, such as a disordered solid or a supercooled liquid.

In the context of MDANSE, researchers calculate two essential components of the correlation function:

• Longitudinal Component: This component is associated with density fluctuations, offering insights into how particle or atom densities change at specific locations within the disordered system over time.

• Transverse Component: The transverse component is linked to propagating shear modes, helping researchers comprehend the relative displacements of neighboring particles or atoms and the propagation of these shear modes throughout the disordered material.

Dynamic Coherent Structure Factor

In materials science and condensed matter physics, dynamic coherent structure factors are crucial. They enable a comprehensive understanding of complex particle or atom movements and interactions over time. These factors provide invaluable insights into the dynamic behavior of materials, aiding researchers in deciphering particle evolution and characterizing properties such as diffusion rates, elasticity, and phase transitions. They play a pivotal role in enhancing our understanding of system dynamics and significantly benefit research in these fields.

In this analysis, MDANSE proceeds in two steps. First, it computes the partial and total intermediate coherent scattering function using equation (57). Then, the partial and total dynamic coherent structure factors are obtained by performing the Fourier Transformation, defined in Eq. (50), respectively on the total and partial intermediate coherent scattering functions.

Coherent Intermediate Scattering Function Calculation MDANSE computes the coherent intermediate scattering function on a rectangular grid of equidistantly spaced points along the time-and the q-axis, respectively:

(68)\[\begin{split}{{F}_{\text{coh}}\left( {q_{m},k\cdot\Delta t} \right)\doteq{\sum\limits_{{I = 1},J\geq I}^{N_{\mathit{species}}}\sqrt{n_{I}n_{J}\omega_{I,\text{com}}\omega_{I,\text{com}}}}{\overline{\left\langle {\rho_{I}\left( {{-q},0} \right)\rho_{J}\left( {q,k\cdot\Delta t} \right)} \right\rangle}}^{q},} \\ {{k = 0}\ldots{N_{t} - 1},{m = 0}\ldots{N_{q} - 1.}}\end{split}\]

Equation defines the computation of the coherent intermediate scattering function in terms of particle densities, species, and time steps.

Fourier-Transformed Particle Density where N_t is the number of time steps in the coordinate time series, N_q is a user-defined number of q-shells, N_species is the number of selected species, n_I the number of atoms of species I, \(\omega\)_I the weight for species I (see Section target_CN for more details) and \({\rho_{I}\left( {q,k\cdot\Delta t} \right)}\) Below defines the Fourier-transformed particle density for species I.

(69)\[{\rho_{I}{\left( {q,k\cdot\Delta t} \right) = \sum\limits_{\alpha}^{n_{I}}}\exp\left\lbrack {\mathit{iq}\cdot R_{\alpha}\left( {k\cdot\Delta t} \right)} \right\rbrack.}\]

The symbol \({\overline{...}}^{q}\) in Eq. (68) denotes an average over q-vectors having approximately the same modulus

**q-Vectors on a Reciprocal Lattice ** Below describes the selection of q-vectors on a lattice reciprocal to the MD box lattice.

(70)\[{{q_{m} = {q_{\mathit{\min}} + m}}\cdot\Delta q}\]

The particle density must not change if jumps in the particle trajectories due to periodic boundary conditions occur. In addition, the average particle density, \(N/V\) , must not change.

Position Vector in the MD Cell This can be achieved by choosing q-vectors on a lattice which is reciprocal to the lattice defined by the MD box. Let b₁, b₂, b₃ be the basis vectors which span the MD cell. Any position vector in the MD cell can be written as

(71)\[{{R = x^{'}}{b_{1} + y^{'}}{b_{2} + z^{'}}b_{3},}\]

Eq defines the position vector in the MD cell.

** Dual Basis Vectors** with x’, y’, z’ having values between 0 and 1. The primes indicate that the coordinates are box coordinates. A jump due to periodic boundary conditions causes x’, y’, z’ to jump by \(\pm1\). The set of dual basis vectors b¹, b², b³ is defined by the relation

(72)\[{b_{i}{b^{j} = \delta_{i}^{j}}.}\]

Eq defines the dual basis vectors and their relation to the basis vectors.

**Selection of q-Vectors with Phase Changes ** If the q-vectors are now chosen as

(73)\[{{q = 2}\pi\left( {k{b^{1} + l}{b^{2} + m}b^{3}} \right),}\]

Describes the selection of q-vectors with phase changes for handling jumps in particle trajectories

where k,l,m are integer numbers, jumps in the particle trajectories produce phase changes of multiples of \(2\pi\) in the Fourier transformed particle density, i.e. leave it unchanged. One can define a grid of q-shells or a grid of q-vectors along a given direction or on a given plane, giving in addition a tolerance for q. MDANSE looks then for q-vectors of the form given in Eq. 61 whose moduli deviate within the prescribed tolerance from the equidistant q-grid. From these q-vectors only a maximum number per grid-point (called generically q-shell also in the anisotropic case) is kept.

Negative Coherent Scattering Lengths The q-vectors can be generated isotropically, anisotropically or along user-defined directions. The \(\sqrt{\omega_{I}}\) may be negative if they represent normalized coherent scattering lengths, i.e.

(74)\[{{\sqrt{\omega_{I}} = \frac{b_{I,\text{coh}}}{\sqrt{\sum\limits_{I = 1}^{N_{\mathit{species}}}{n_{I}b_{I,\text{coh}}^{2}}}}}.}\]

Defines the use of negative coherent scattering lengths for hydrogenous materials.

Negative coherent scattering lengths occur in hydrogenous materials since \(b_{\mathit{coh},H}\) is negative [Ref20]. The density-density correlation is computed via the FCA technique described in the section The FCA algorithm.

When the default value of weights (\(b_{coherent}\)) is chosen for this analysis, the result will correspond to that of the equation (106) from the analysis-ndtsf.

Dynamic Incoherent Structure Factor

In this analysis, MDANSE proceeds in two steps. First, it computes the partial and total intermediate incoherent scattering function F_inc(q, t) using equation (57). Then, the partial and total dynamic incoherent structure factors are obtained by performing the Fourier Transformation, defined in Eq. (50), respectively on the total and partial intermediate incoherent scattering function.

Computation of Incoherent Intermediate Scattering Function

MDANSE computes the incoherent intermediate scattering function on a rectangular grid of equidistantly spaced points along the time-and the q-axis, respectively:

(75)\[\begin{split}{\text{F}_{\text{inc}}\left( {q_{m},k\cdot\Delta t} \right)\doteq{\sum\limits_{I = 1}^{N_{\mathit{species}}}{n_{I}\omega_{I,\text{inc}}}}\text{F}_{I,\text{inc}}\left( {q_{m},k\cdot\Delta t} \right),\\ {k = 0}\ldots{N_{t} - 1},{m = 0}\ldots{N_{q} - 1.}}\end{split}\]

Eq: Incoherent Intermediate Scattering Function Calculation

where N_t is the number of time steps in the coordinate time series, N_q is a user-defined number of q-shells, N_species is the number of selected species, n_I the number of atoms of species I, \(\omega\)_I the weight for species I (see Section target_CN for more details) and \({F_{I,\text{inc}}\left( {q_{m},k\cdot\Delta t} \right)}\) is defined as:

(76)\[{\text{F}_{I,\mathit{inc},\alpha}{\left( {q_{m},k\cdot\Delta t} \right) = \sum\limits_{\alpha = 1}^{n_{I}}}{\overline{\left\langle {\exp\left\lbrack {{-i}q\cdot R_{\alpha}(0)} \right\rbrack\exp\left\lbrack {iq\cdot R_{\alpha}(t)} \right\rbrack} \right\rangle}}^{q}.}\]

Eq: Definition of F_{I,inc,alpha}(q_m, k * Δt)

The symbol \({\overline{...}}^{q}\) in Eq. (76) denotes an average over q-vectors having approximately the same modulus \({{q_{m} = {q_{\mathit{\min}} + m}}\cdot\Delta q}\). The particle density must not change if jumps in the particle trajectories due to periodic boundary conditions occur.

Selection of q-Vectors on a Reciprocal Lattice In addition, the average particle density, N/V, must not change. This can be achieved by choosing q-vectors on a lattice which is reciprocal to the lattice defined by the MD box. Let b₁, b₂, b₃ be the basis vectors which span the MD cell. Any position vector in the MD cell can be written as

(77)\[{{R = x^{'}}{b_{1} + y^{'}}{b_{2} + z^{'}}b_{3},}\]

Eq: Position Vector in the MD Cell

with x’, y’, z’ having values between 0 and 1. The primes indicate that the coordinates are box coordinates. A jump due to periodic boundary conditions causes x’, y’, z’ to jump by \(\pm 1\). The set of dual basis vectors b¹, b², b³ is defined by the relation

(78)\[{b_{i}{b^{j} = \delta_{i}^{j}}.}\]

Eq: Dual Basis Vectors

If the q-vectors are now chosen as

(79)\[{{q = 2}\pi\left( {k{b^{1} + l}{b^{2} + m}b^{3}} \right),}\]

Eq: Selection of q-Vectors with Phase Changes

where k,l,m are integer numbers, jumps in the particle trajectories produce phase changes of multiples of 2π in the Fourier transformed particle density, i.e. leave it unchanged. One can define a grid of q-shells or a grid of q-vectors along a given direction or on a given plane, giving in addition a tolerance for q. MDANSE looks then for q-vectors of the form given in Eq. (79) whose moduli deviate within the prescribed tolerance from the equidistant q-grid. From these q-vectors only a maximum number per grid-point (called generically q-shell also in the anisotropic case) is kept.

The q-vectors can be generated isotropically, anisotropically or along user-defined directions.

Computation of Correlation Functions The correlation functions defined in (76) are computed via the FCA technique described in the section The FCA algorithm. Although the efficient FCA technique is used to compute the atomic time correlation functions, the program may consume a considerable amount of CPU-time since the number of time correlation functions to be computed equals the number of atoms times the total number of q-vectors. This analysis is actually one of the most time-consuming among all the analysis available in MDANSE.

When the default value of weights (\({b^{2}}_{incoherent}\)) is chosen for this analysis, the result will correspond to that of the equation (107) from the analysis-ndtsf.

Elastic Incoherent Structure Factor

The 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 biological systems. Incoherent scattering from hydrogen dominates by far all other contributions.

Van Hove Self-correlation Function

The Elastic Incoherent Structure Factor (EISF) is defined as the limit of the incoherent intermediate scattering function for infinite time,

(80)\[{\mathit{EISF}(q)\doteq\lim\limits_{t\rightarrow\infty}\text{F}_{\mathit{inc}}\left( {q,t} \right).}\]

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

(81)\[{\text{F}_{\text{inc}}{\left( {q,t} \right) = \mathit{EISF}}{(q) + \text{F}_{\text{inc}}^{'}}\left( {q,t} \right),}\]

where F_inc‘(q,t) decays to zero for infinite time. Taking now the Fourier transform it follows immediately that

(82)\[{\text{S}_{\text{inc}}{\left( {q,\omega} \right) = \mathit{EISF}}(q)\delta{(\omega) + \text{S}_{\text{inc}}^{'}}\left( {q,\omega} \right).}\]

The 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 biological systems. Incoherent scattering from hydrogen dominates by far all other contributions. Using the definition of the van Hove self-correlation function G_s(r, t) [Ref20],

(83)\[{b_{\text{inc}}^{2}G_{s}\left( {r,t} \right)\doteq\frac{1}{2\pi^{3}}{\int d^{3}}q\exp\left\lbrack {{- i}q\cdot r} \right\rbrack\text{F}_{\mathit{inc}}\left( {q,t} \right),}\]

which can be interpreted as the conditional probability to find a tagged particle at the position r at time t, given it started at r = 0, one can write:

(84)\[{\mathit{EISF}{(q) = b_{\text{inc}}^{2{\int d^{3}}}}r\exp\left\lbrack {\mathit{iq}\cdot r} \right\rbrack G_{s}\left( {r,{t = \infty}} \right).}\]

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_s(r, t) approaches 1/V 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 [Ref21]:

(85)\[{\mathit{EISF}{(q) = {\sum\limits_{I = 1}^{N_{\mathit{species}}}{n_{I}\omega_{I,\text{inc}}\mathit{EIS}F_{I}(q)}}}}\]

where N_species is the number of selected species, n_I the number of atoms of species I, \(\omega\)_I,inc the weight for species I (see Section target_CN for more details) and for each species the following expression for the elastic incoherent scattering function is

(86)\[{\mathit{EIS}F_{I}{(q) = \frac{1}{n_{I}}}{\sum\limits_{\alpha}^{n_{I}}\left\langle {|{\exp\left\lbrack {\mathit{iq}\cdot R_{\alpha}} \right\rbrack\left. {} \right|^{2}}} \right\rangle}.}\]

This expression is derived from definition (80) of the EISF and expression (58) for the intermediate scattering function, using that for infinite time the relation

(87)\[{\left\langle {\mathit{ex}p\left\lbrack {{- \mathit{iq}}\cdot R_{\alpha}(0)} \right\rbrack\mathit{ex}p\left\lbrack {\mathit{iq}\cdot R_{\alpha}(t)} \right\rbrack} \right\rangle = \left\langle {|{\mathit{ex}p\left\lbrack {\mathit{iq}\cdot R_{\alpha}} \right\rbrack\left. {} \right|^{2}}} \right\rangle}\]

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.

Grid Computation

MDANSE allows one to compute the elastic incoherent structure factor on a grid of equidistantly spaced points along the q-axis:

(88)\[{\mathit{EISF}\left( q_{m} \right)\doteq{\sum\limits_{I = 1}^{N_{\mathit{species}}}{n_{I}\omega_{I}\mathit{EIS}F_{I}\left( q_{m} \right)}},{m = 0}\ldots{N_{q} - 1.}}\]

where N_q is a user-defined number of q-shells, the values for q_m are defined as

(89)\[{{q_{m} = {q_{\mathit{\min}} + m}}\cdot\Delta q}\]

, and for each species the following expression for the elastic incoherent scattering function is:

(90)\[{\mathit{EIS}F_{I}{\left( q_{m} \right) = \frac{1}{n_{I}}}{\sum\limits_{\alpha}^{n_{I}}{\overline{\left\langle {|{\exp\left\lbrack {\mathit{iq}\cdot R_{\alpha}} \right\rbrack\left. {} \right|^{2}}} \right\rangle}}^{q}}.}\]

Here the symbol \({\overline{...}}^{q}\) denotes an average over the q-vectors having the same modulus q_m. The program corrects the atomic input trajectories for jumps due to periodic boundary conditions.

Gaussian Dynamic Incoherent Structure Factor

The Gaussian Dynamic Incoherent Structure Factor is a concept used to study how particles or atoms move independently within materials over time, with a focus on their distribution. It’s valuable in materials science and condensed matter physics for understanding dynamic behavior at the atomic level.

MSD Calculation

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

(91)\[{\text{F}_{\text{inc}}^{g}{\left( {q,t} \right) = {\sum\limits_{I = 1}^{N_{\mathit{species}}}{n_{I}\omega_{I,\text{inc}}}}}\text{F}_{I,\text{inc}}^{g}\left( {q,t} \right)}\]

where N_species is the number of selected species, n_I the number of atoms of species I, \(\omega\)_I,inc the weight for species I (see Section target_CN for more details) and

(92)\[{\text{F}_{I,\text{inc}}^{g}{\left( {q,t} \right) = \frac{1}{n_{I}}}\sum\limits_{\alpha}^{n_{I}}\exp\left\lbrack {{- q^{2}}\rho_{\alpha,1}{(t) + q^{4}}\rho_{\alpha,2}(t)\mp\ldots} \right\rbrack.}\]

The cumulants

(93)\[{\rho_{\alpha,k}(t)}\]

are identified as

(94)\[{\rho_{\alpha,1}{(t) = \left\langle {d_{\alpha}^{2}\left( {t;n_{q}} \right)} \right\rangle}}\]

(95)\[{\rho_{\alpha,2}{(t) = \frac{1}{4!}}\left\lbrack {{\left\langle {d_{\alpha}^{4}\left( {t;n_{q}} \right)} \right\rangle - 3}\left\langle {d_{\alpha}^{2}\left( {t;n_{q}} \right)} \right\rangle^{2}} \right\rbrack}\]

\[{\vdots}\]

Gaussian Approximation

The vector nq is the unit vector in the direction of q. In the Gaussian approximation the above expansion is truncated after the q²-term. 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. MDANSE allows one to compute the total and partial incoherent intermediate scattering function in the Gaussian approximation by discretizing equation (91):

(96)\[{\text{F}_{\text{inc}}^{g}\left( {q_{m},k\cdot\Delta t} \right)\doteq{\sum\limits_{I = 1}^{N_{\mathit{species}}}{n_{I}\omega_{I,\text{inc}}\text{F}_{I,\text{inc}}^{g}\left( {q_{m},k\cdot\Delta t} \right)}},{k = 0}\ldots{N_{t} - 1},{m = 0}\ldots{N_{q} - 1.}}\]

Intermediate Scattering Function

with for each species the following expression for the intermediate scattering function:

(97)\[{\text{F}_{I,\alpha,\text{inc}}^{g}{\left( {q_{m},k\cdot\Delta t} \right) = \frac{1}{n_{I}}}\sum\limits_{\alpha}^{n_{I}}\exp\left\lbrack {\frac{- \left( q_{m} \right)^{2}}{6}\Delta_{\alpha}^{2}\left( {k\cdot\Delta t} \right)} \right\rbrack\mathit{isotropic}\mathit{system}}\]

(98)\[{\text{F}_{I,\alpha,\text{inc}}^{g}{\left( {q_{m},k\cdot\Delta t} \right) = \frac{1}{n_{I}}}\sum\limits_{\alpha}^{n_{I}}\exp\left\lbrack {\frac{- \left( q_{m} \right)^{2}}{2}\Delta_{\alpha}^{2}\left( {k\cdot\Delta t;n} \right)} \right\rbrack\mathit{isotropic}\mathit{system}}\]

N_t is the total number of time steps in the coordinate time series and N_q is a user-defined number of q-shells. The (q, t)-grid is the same as for the calculation of the intermediate incoherent scattering function (see Dynamic Incoherent Structure Factor). The quantities

(99)\[{\Delta_{\alpha}^{2}(t)}\]

and

(100)\[{\Delta_{\alpha}^{2}\left( {t;n} \right)}\]

are the mean-square displacements, defined in Equations (6) and (7), respectively. They are computed by using the algorithm described in the Mean Square Displacement section. MDANSE corrects the atomic input trajectories for jumps due to periodic boundary conditions. It should be noted that the computation of the intermediate scattering function in the Gaussian approximation is much ‘cheaper’ than the computation of the full intermediate scattering function, F_inc(q, t), since no averaging over different q-vectors needs to be performed. It is sufficient to compute a single mean-square displacement per atom.

Neutron Dynamic Total Structure Factor

The Neutron Dynamic Total Structure Factor is a term used in scientific research, especially in neutron scattering experiments, to investigate how particles or atoms within a material contribute to its overall structure and dynamics. This factor provides valuable insights into how these components move and interact over time.

Calculation of Partial Coherent Intermediate Scattering Functions and Dynamic Structure Factors

This is a combination of the Dynamic Coherent and the Dynamic Incoherent Structure Factors. It is a fully neutron-specific analysis, where the coherent part of the intermediate scattering function is calculated using the atomic coherent neutron scattering lengths \(b_{coherent}\) and the incoherent one is calculated using the square of the atomic incoherent neutron scattering lengths \({b^{2}}_{incoherent}\). Therefore, in this analysis the weights option is not available.

The partial coherent intermediate scattering functions \(I_{\alpha\beta}^{coh}(Q,t)\) (and their corresponding Fourier transforms giving the partial coherent dynamic structure factors, \(S_{\alpha\beta}^{coh}(Q,\omega)\)) are calculated exactly in the same way as in the DCSF analysis, i.e.:

(101)\[I_{\alpha\beta}^{coh}(Q,t) = \left| \frac{1}{\sqrt{N_{\alpha}N_{\beta}}}\sum_{i \in \alpha,j \in \beta}^{N_{\alpha},N_{\beta}}\left\langle e^{- i\mathbf{Q}\mathbf{r}_{i}(t_{0})}e^{i\mathbf{Q}\mathbf{r}_{j}(t_{0} + t)} \right\rangle \right|_{\mathbf{Q}}\]

where \(\alpha\) and \(\beta\) refer to the chemical elements, \(N_{\alpha}\) and \(N_{\beta}\) are the respective number of atoms of each type, \(i\) and \(j\) are two specific atoms of type \(\alpha\) and \(\beta\), respectively, and \(\mathbf{r}_{i}(0)\) and \(\mathbf{r}_{j}(t)\) are their positions at the time origin and at the time \(t\), respectively. The notation \(\left\langle \ldots \right\rangle\) indicates an average over all possible time origins \(t_{0}\) and \(|\ldots|_{\mathbf{Q}}\) represents an average over all the \(\mathbf{Q}\) vectors contributing to the corresponding \(Q\)-bin.

Similarly, the partial incoherent intermediate scattering functions \(I_{\alpha}^{inc}(Q,t)\) and the partial incoherent dynamic structure factors \(S_{\alpha}^{inc}(Q,\omega)\) are obtained as in the DISF analysis:

(102)\[I_{\alpha}^{inc}(Q,t) = \left| \frac{1}{N_{\alpha}}\sum_{i \in \alpha}^{N_{\alpha}}\left\langle e^{- i\mathbf{Q}\mathbf{r}_{i}(t_{0})}e^{i\mathbf{Q}\mathbf{r}_{i}(t_{0} + t)} \right\rangle \right|_{\mathbf{Q}}\]

Combination of Partial Contributions

The main difference between this analysis and the DCSF and DISF analyses, apart from the fact that the coherent and incoherent contributions are calculated simultaneously, is the way the different partial contributions are combined. In this analysis the total incoherent, total coherent and total (coherent + incoherent) signals are calculated as:

(103)\[I^{inc}(Q,t) = \sum_{\alpha}^{N_{\alpha}}{c_{\alpha}b_{\alpha,\text{inc}}^{2}}I_{\alpha}^{inc}(Q,t)\]

(104)\[I^{coh}(Q,t) = \sum_{\alpha,\beta}^{N_{\alpha},N_{\beta}}{\sqrt{c_{\alpha}c_{\beta}}b_{\alpha,\text{coh}}b_{\beta,\text{coh}}I_{\alpha\beta}^{coh}(Q,t)}\]

(105)\[I^{tot}(Q,t) = I^{inc}(Q,t) + I^{coh}(Q,t) = \sum_{\alpha}^{N_{\alpha}}{c_{\alpha}b_{\alpha,\text{inc}}^{2}}I_{\alpha}^{inc}(Q,t) + \sum_{\alpha,\beta}^{N_{\alpha},N_{\beta}}{\sqrt{c_{\alpha}c_{\beta}}b_{\alpha,\text{coh}}b_{\beta,\text{coh}}I_{\alpha\beta}^{coh}(Q,t)}\]

where \(c_{\alpha} = \frac{N_{\alpha}}{N}\) and \(c_{\beta} = \frac{N_{\beta}}{N}\) are the concentration numbers for elements \(\alpha\) and \(\beta\), respectively.

These expressions correspond to the formalism and equations given in [Ref47], chapter 1: “An introduction to neutron scattering” .

Units Conversion

As in the MDANSE database the coherent and incoherent neutron scattering lengths are given in Å, the total intermediate scattering functions above will be given in Ų/sterad/atom. Therefore, multiplying the output from MDANSE by a factor 10⁸ we can obtain these neutron observables in barn/sterad/atom and compare them directly to the experimental results (assuming the later have been properly normalized and presented in absolute units).

On the other hand, the DISF and DCSF analyses use the standard weight normalization procedure implemented in MDANSE (see Normalize). Therefore the total coherent intermediate scattering function returned by the DCSF analysis is (assuming that the chosen weights are b_coherent):

(106)\[I^{coh}(Q,t) = \frac{\sum_{\alpha\beta}^{n}{c_{\alpha}c_{\beta}b_{\alpha,coh}b_{\beta,coh}I_{\alpha\beta}^{coh}(Q,t)}}{\sum_{\alpha\beta}^{n}{c_{\alpha}c_{\beta}b_{\alpha,coh}b_{\beta,coh}}}\]

And the incoherent intermedicate scattering function given by the DISF analysis is (assuming that the chosen weights are b_incoherent2):

(107)\[I^{inc}(Q,t) = \frac{\sum_{\alpha}^{n}{c_{\alpha}b_{\alpha,inc}^{2}I_{\alpha}^{inc}(Q,t)}}{\sum_{\alpha}^{n}{c_{\alpha}b_{\alpha,inc}^{2}}}\]

Naturally, similar expressions apply to the dynamic structure factors, \(S_{\alpha\beta}^{coh}(Q,\omega)\) and \(S_{\alpha}^{inc}(Q,\omega)\).

Structure Factor From Scattering Function

• available for analysis results only

The “Structure Factor From Scattering Function” is a concept used in scientific research, particularly in the field of neutron scattering experiments. It relates to how particles or atoms within a material contribute to its overall structural properties based on their scattering behavior. This concept provides valuable insights into the material’s internal structure, dynamics, and interactions. Researchers use the Structure Factor From Scattering Function to better understand the atomic-level details of materials, which has applications in diverse areas, including materials science and condensed matter physics