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Theory and implementation

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

| (4.174)

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

| (4.175)

where decays to zero for infinite time. Taking now the Fourier transform it follows immediately that

| (4.176)

The EISF appears as the amplitude of the elastic line in the neutron scattering spectrum. Elastic scattering is only present for sytems 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 [7],

| (4.177)

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

| (4.178)

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 approaches for large times. This is the case for molecules in liquids and gases.

For computational purposes it is convenient to use the following representation of the EISF [14]:

| (4.179)

where is the number of selected species, the number of atoms of species I, the weight for specie I (see Section 4.2.1 for more details) and for each specie the following expression for the elastic incoherent scattering function is

| (4.180)

This expression is derived from definition (4.174) of the EISF and expression (4.142) for the intermediate scattering function, using that for infinite time the relation

| (4.181)

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.

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

| (4.182)

where is a user-defined number of q-shells, the values for are defined as , and for each specie the following expression for the elastic incoherent scattering function is:

| (4.183)

Here the symbol denotes an average over the q-vectors having the same modulus . The program corrects the atomic input trajectories for jumps due to periodic boundary conditions.

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