Structure factors of harmonic and anharmonic Fibonacci chains by molecular dynamics simulations

The dynamics of quasicrystals is characterized by the existence of phason excitations in addition to the usual phonon modes. In order to investigate their interplay on an elementary level we resort to various one-dimensional model systems. The main observables are the static, the incoherent, and the coherent structure factor, which are extracted from molecular dynamics simulations. For the validation of the algorithms, results for the harmonic periodic chain are presented. We then study the Fibonacci chain with harmonic and anharmonic interaction potentials. In the dynamic Fibonacci chain neighboring atoms interact by double-well potentials allowing for phason flips. The difference between the structure factors of the dynamic and the harmonic Fibonacci chain lies in the temperature dependence of the phonon line width. If a bias is introduced in the well depth, dispersionless optic phonon bands split off.

Figure 1

Double-well potential $V\left(x\right)=x^4-2x^2$ of the dynamic Fibonacci chain. The equilibrium distances are $S$ and $L$.

Figure 2

Density-density correlation function $G(x,t)$ of the HPC for $k_{B}T=0.5$. The Gaussians are centered at integer multiples of $x=a\approx 4.47$.

Figure 3

Incoherent structure factor $S_i(q,\omega )$ of for HPC for $q=\pi ∕a$, and different temperatures. The symbols mark the data from MD simulations with $N=1000$ particles, and the lines result from the analytical formula (7b). The peaks and edges are at integer multiples of $\omega =2{\omega }_0=2\sqrt{8}$.

Figure 4

(Color online) Coherent structure factor $S(q,\omega )$ of the HPC with $N=6500$ particles from MD simulation. The temperatures are $k_{B}T=0.01$ (a) and $k_{B}T=0.1$ (b). One-, two-, and three-phonon branches are observed. They start at the reciprocal lattice points $2\pi n∕a$. In (b) the output from the MD simulation (left side) is compared to the output of the analytical formula Eq. (7a) (right side).

Figure 5

Static structure factor $S\left(q\right)$ of the HPC for $k_{B}T=0.02$. The symbols mark the data from a MD simulation with $N=1000$ particles, and the line is the result from the analytical formula (9).

Figure 6

(Color online) Coherent structure factor $S(q,\omega )$ of the HFC with $N=13000$ particles for $k_{B}T=0.02$. It consists of a dense set of phonon branches starting from the reciprocal lattice points.

Figure 7

Static structure factor $S\left(q\right)$ of the HFC from a MD simulation with 2000 particles at the temperature $k_{B}T=0.02$.

Figure 8

Snapshot of the particle trajectories of the DFC at the temperature $k_{B}T=0.6$. Changes in the particle distances from $L$ to $S$ and $S$ to $L$ are marked with a cross (×) and a plus (+). A $L\to S$ change and a $S\to L$ change combined form a phason flip.

Figure 9

Average flip frequency as a function of the temperature $k_{B}T$. In the temperature range of the figure: ${\omega }_{\text{flip}}<{\omega }_0\approx 2.83$.

Figure 10

Incoherent structure factor $S_i(q,\omega )$ of the DFC (solid) and the HFC/HPC (dashed) for $q=\pi ∕a$ at different temperatures. The data was calculated using MD simulations with $N=1000$ particles.

Figure 11

Cuts through $S(q,\omega )$ of the DFC for a fixed $\omega =1.0$ including a one-phonon peak. The solid curves are fits with Lorentzians.

Figure 12

Width of the Lorentzian peak of Fig. 11 as a function of temperature for the HFC (inset) and the DFC.

Figure 13

(Color online) Coherent structure factor $S(q,\omega )$ of a randomized Fibonacci chain with 2000 particles at the temperature $k_{B}T=0.02$. The same interaction potentials as in Fig. 6 have been used.

Figure 14

Interaction potential $V_{\mathrm{AFC}}\left(x\right)$ of the AFC for different values of the parameter $\chi$ and $ϵ=1$. $V_{\mathrm{DFC}}$ is shown with dashed lines.

Figure 15

(Color online) Coherent structure factor $S(q,\omega )$ of the AFC for different values of $\chi$ and $ϵ$ from MD simulations with 6500 particles at $k_{B}T=0.01$. Band gaps are observed. The band gaps also appear in the DOS $D\left(\omega \right)$.