Liquid and solid aluminum

In this first example aluminium will be studied. An EAM potential was used [Phys. Rev. B 59, 3393 (1999)] to describe both the solid and liquid phases. The melting point was found to be about 930 K. The lattice parameters used for the NVT MD simulations were found running NPT and averaging the box length.

Liquid

First we look at alumnium above the melting point at 1400 K, i.e., as a liquid using a cell comprising 2048 atom. For liquids a spherically average over \(\boldsymbol{q}\) is desirable. To compute static properties such as \(S(q)\) no time correlation needs to be computed which reduces the tomputation time a lot. Therefore there are two different bash scripts setupt that calls dynasor, run_dynasor_static.sh and run_dynasor_dynamical.sh. Note that these scripts may take a while to run so starting with fewer \(\mathrm{MAX\_FRAMES}\) is probably good idea. Two plotting scripts are also included in order to reproduce the figures seen here below if running the dynasor scripts as are.

In the figure the structure factor \(S(q)\) is shown.

../_images/Al_liquid_static.png

Structure factor \(S(q)\) computed by running run_dynasor_static.sh. The \(q=0\) point has been skipped for the structure factor due to \(S(0) = N\), see theory.

The structure factor is in good agreement with other MD simulations as well as with experimental data, see Figure 2 in [AVM06].

A map of the dynamical structure factor and the two current correlations is seen below as a function of \(q\) and \(\omega\). The figure is cut at \(q=2.5nm^{-1}\) because the resolution in \(q\)-space is very poor below this point.

The intensity in \(C_l(q,\omega)\) is more pronounced than \(S(q,\omega)\), which means longitudinal vibrations are more easily observed in the current correlation rather than the dynamical structure factor.

These heatmap seem to agree very well with spectral intensity plots in Figure 3 of [AVM06]. The clear dispersion in \(C_l(q,\omega)\) agrees with the dispersion relation in Figure 4 of [AVM06].

../_images/Al_liquid_heatmap.png

Dynamical structure factor (a), longitudinal current (b) and transverse current (c).

Crystalline

Todo

Update these old figures.

In this example we look at alumnium below the melting point at \(T=300K\). The crystal structure is face-centered-cubic (fcc) and \(q\)-space sampling was done along three paths

\[L=\frac{2\pi}{a} \Big(\frac{1}{2},\frac{1}{2},\frac{1}{2}\Big) \quad , \quad K=\frac{2\pi}{a} \Big(\frac{3}{4},\frac{3}{4},0\Big)\]

Since the sampling along a path is much faster than doing a spherical average the number of atoms used here is 6912 (12x12x12 fcc).

The transverse and longitudinal current correlation is shown both in time and frequency between \(\Gamma\) and K. Here the \(q\)-value is between 0 and 1 and represent how far along the path you are.

../_images/C_ktw_GK.png

Current correlations between \(\Gamma\) and K in both time (top) and frequency .

Looking at the current correlations in time we see that the longitudinal is oscillating with one frequency and the transverse with many. This agrees with what is seen in the frequency domain.

The sum of the longitudinal and transverse current correlation is seen below together with the phonon dispersion calculated, using the same potential, by phonopy.

../_images/dispersion_T300.png

Sum of the longitudinal and transverse current show along the path L to \(\Gamma\) to K. The black dots indicate the results from phonopy calculations.

The agreement with the phonon dispersion calculated with phonopy is very good. The fact that there is discrete \(q\)-values which give rise to high intensitiy is due to the finite supercell size not supporting all kinds of oscillations.