Vibrational spectra of amorphous clusters: Universal aspects

We have performed extensive numerical computations on the vibrational spectra of isolated amorphous clusters of medium to large size and containing one or two types of atoms. The interaction potential is also varied to study possible universality. For all the potentials and cluster sizes we find that the cumulative density of states can be described very accurately by the same functional form over a large central region of the spectrum. This functional form contains only one scale of frequency. We also find that the statistical fluctuations of the spectra are described by the Gaussian orthogonal ensemble of random matrices. For the largest clusters this is tested to a very high degree of precision in the central region and to a somewhat lower degree for most of the rest of the spectrum. We put forward a conjecture regarding the domain of the space of local minima of the potential energy function where universality with respect to the density of states function may be expected.

Figure 1

(a) Filled circles: Eigenvalue $\left(\lambda \right)$ vs eigenvalue number plot of a complete spectrum (Morse potential with $N=2000$). Some data points have been removed for clarity. Continuous line: Best-fit curve obtained after removing bottom 10% and top 20% of the spectrum. (b) Misfit function in the region of best fit for the same spectrum.

Figure 2

Probability density $\left[p\left(s\right)\right]$ for normalized nearest-neighbor spacing $\left(s\right)$ in the region II of the spectrum. Filled circles: our data. Crosses: Wigner’s surmise for GOE. Continuous line: Exact prediction for the GOE. (a) Morse $(N=2000)$, (b) Sutton-Chen $(N=400)$, and (c) Gupta potential for nickel $(N=400)$.

Figure 3

Variance of the number of levels in intervals of length $r$ plotted as a function of $r$ for the region II of the spectrum. Continuous line: Prediction for the GOE. Filled circles: Our data. Number of particles $\left(N\right)$ increases from top to bottom in each figure. (a) Morse with $N=200$, 500, 1000, and 2000; (b) Sutton-Chen with $N=100$, 200, and 400; and (c) Gupta potential for nickel with $N=100$, 200, and 400.

Figure 4

Skewness and excess parameters of the distribution of $n\left(r\right)$, the number of levels in interval of length $r$, plotted as a function of $r$ for the region II of the spectrum. Filled circles: Our data. Continuous lines: Predictions for GOE. (a) Morse with $N=2000$; (b) Sutton-Chen with $N=400$; and (c) Gupta potential for nickel with $N=400$.

Figure 5

Probability density $\left[p\left(s\right)\right]$ for normalized nearest-neighbor spacing $\left(s\right)$, obtained with Morse potential $(N=2000)$, in the regions I and III of the spectrum. Filled circles: Our data. Crosses: Wigner’s surmise for GOE. Continuous line: Exact prediction for the GOE. (a) Region I and (b) region III.

Figure 6

Variance of the number of levels in intervals of length $r$ plotted as a function of $r$ for the regions I and III of the spectrum obtained with Morse potential $(N=2000)$. Continuous line: Prediction for the GOE. Filled circles: Our data. (a) Region I and (b) region III.

Figure 7

Probability density $\left[p\left(s\right)\right]$ of normalized nearest-neighbor spacing $\left(s\right)$ for the binary LJ mixture $(N=2000)$ in the region II of the spectrum. Filled circles: Our data. Crosses: Wigner’s surmise for GOE. Continuous line: Exact prediction for the GOE. (a) Case I and (b) case II.

Figure 8

Variance of the number of levels in intervals of length $r$ plotted as a function of $r$ for the binary LJ mixture $(N=2000)$ in the region II of the spectrum. Continuous line: Prediction for the GOE. Filled circles: Our data. (a) Case I and (b) case II.

Figure 9

Skewness and excess parameters of the distribution of $n\left(r\right)$, the number of levels in interval of length $r$, plotted as a function of $r$ in the region II of the spectrum of the binary LJ mixture $(N=2000)$. Filled circles: Our data. Continuous lines: Predictions for GOE. (a) Case I and (b) case II.

Figure 10

Normalized density of states $\left[n\left(\nu \right)\right]$ for rescaled frequency $\left(\nu \right)$ with rescaling done by the average frequency of the corresponding spectrum. Vertical bars denote approximately the limits of region II. (a) Filled circles: LJ (500). Open circles: Morse (500). Open triangles: Sutton-Chen (400). Stars: Gupta for nickel (400). Filled squares: Gupta for vanadium (400). Open squares: Binary LJ (500). (b) Filled circles: LJ (2000). Open squares: Morse (2000). Crosses: LJ mixture, Case I (2000). Open inverted triangles: LJ mixture, case II (2000).

Figure 11

Normalized density of states $\left[n\left(\nu \right)\right]$ for rescaled frequency $\left(\nu \right)$ with rescaling done by the best-fit frequency parameter of the region II of the corresponding spectrum. Vertical bars denote approximately the limits of region II. (a) Filled circles: LJ (500). Open circles: Morse (500). Open triangles: Sutton-Chen (400). Stars: Gupta for nickel (400). Filled squares: Gupta for vanadium (400). Open squares: Binary LJ (500). (b) Filled circles: LJ (2000). Open squares: Morse (2000). Crosses: LJ mixture, case I (2000). Open inverted triangles: LJ mixture, case II (2000).