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Theory and implementation

Please refer to Section 4.2.5.1 for more details about the theoretical background related to the dynamic incoherent structure factor. In this analysis, nMOLDYN proceeds in two steps. First, it computes the partial and total intermediate incoherent scattering function ${\cal F}_{\mathrm{inc}}({\bf q},t)$ using equation 4.142. Then, the partial and total dynamic incoherent structure factors are obtained by performing the Fourier Transformation, defined in Eq.4.134, respectively on the total and partial intermediate incoherent scattering function.

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

\begin{displaymath}
{\cal F}_{\mathrm{inc}}(q_m,k\cdot\Delta t) \doteq \sum_{I =...
...\Delta t),
\qquad k = 0\ldots N_t - 1,\; m = 0\ldots N_q - 1.
\end{displaymath} (4.159)

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, $\omega_{I,\mathrm{inc}}$ the weight for specie I (see Section 4.2.1 for more details) and $F_{I,\mathrm{inc}}(q_m,k\cdot\Delta t)$ is defined as:
\begin{displaymath}
F_{I, inc,\alpha}(q_m,k\cdot\Delta t) = \sum_{\alpha=1}^{n_I...
...f R}_\alpha(0)]\exp[i{\bf q}\cdot{\bf R}_\alpha(t)]\rangle}^q.
\end{displaymath} (4.160)

The symbol $\overline{\rule{0pt}{5pt}\ldots}^{q}$ in (4.160) denotes an average over q-vectors having approximately the same modulus $q_m = q_{min} + m\cdot\Delta q$. The particle density must not change if jumps in the particle trajectories due to periodic boundary conditions occcur. 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 ${\bf b}_1,{\bf b}_2,{\bf b}_3$ be the basis vectors which span the MD cell. Any position vector in the MD cell can be written as
\begin{displaymath}
{\bf R} = x'{\bf b}_1 + y'{\bf b}_2 + z'{\bf b}_3,
\end{displaymath} (4.161)

with $x',y',z'$ having values between $0$ and $1$. The primes indicate that the coordinates are box coordinates. A jump due to periodic bounday conditions causes $x',y',z'$ to jump by $\pm 1$. The set of dual basis vectors ${\bf b}^1,{\bf b}^2,{\bf b}^3$ is defined by the relation
\begin{displaymath}
{\bf b}_i{\bf b}^j = \delta_i^j.
\end{displaymath} (4.162)

If the q-vectors are now chosen as
\begin{displaymath}
{\bf q} = 2\pi\left(k{\bf b}^1 + l{\bf b}^2 + m{\bf b}^3\right),
\end{displaymath} (4.163)

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. 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 correlation functions defined in 4.160 are computed via the FCA technique described in Section A. 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 nMOLDYN.


next up previous contents
Next: Parameters Up: Dynamic Incoherent Structure Factor Previous: Dynamic Incoherent Structure Factor   Contents
pellegrini eric 2009-10-06