Next: Parameters Up: Dynamic Coherent Structure Factor Previous: Dynamic Coherent Structure Factor Contents

####
Theory and implementation

Please refer to Section 4.2.5.1 for more details about the theoretical background related to the dynamic coherent structure factor. In this analysis, nMOLDYN proceeds in two steps. First, it computes the partial and total intermediate coherent scattering function using equation 4.141. Then, the partial and total dynamic coherent structure factors are obtained by performing the Fourier Transformation, defined in Eq. 4.134, respectively on the total and partial intermediate coherent scattering functions.

nMOLDYN computes the coherent intermediate scattering function on a rectangular grid of equidistantly spaced points along the time-and the q-axis, repectively:

| (4.150)

where is the number of time steps in the coordinate time series, is a user-defined number of q-shells, 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 is the Fourier transformed particle density for specie I defined as,

| (4.151)

The symbol in (4.150) denotes an average over q-vectors having approximately the same modulus . The particle density must not change if jumps in the particle trajectories due to periodic boundary conditions occcur. In addition the average particle density, , 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 be the basis vectors which span the MD cell. Any position vector in the MD cell can be written as

| (4.152)

with having values between and . The primes indicate that the coordinates are box coordinates. A jump due to periodic bounday conditions causes to jump by . The set of dual basis vectors is defined by the relation

| (4.153)

If the q-vectors are now chosen as

| (4.154)

where k,l,m are integer numbers, jumps in the particle trajectories produce phase changes of multiples of 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. nMOLDYN looks then for q-vectors of the form (4.163) 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.

The may be negative if they represent normalized coherent scattering lenghts, i.e.

| (4.155)

Negative coherent scattering lengths occur in hydrogenous materials since is negative [7]. The density-density correlation is computed via the FCA technique described in Section A.

Next: Parameters Up: Dynamic Coherent Structure Factor Previous: Dynamic Coherent Structure Factor Contents