[Paper Review] Universality in the Vibrational Spectra of Amorphous Systems

Gurpreet S. Matharoo|ArXiv.org|Dec 25, 2008

Theoretical and Computational Physics2 references3 citations

TL;DR

This thesis investigates universality in the vibrational spectra of amorphous systems using numerical simulations of atomic clusters and bulk-like disordered structures. By analyzing diverse pair potentials—Morse and generalized Lennard-Jones (GLJ)—it demonstrates that the normalized density of states converges to an identical, universal form in the fully disordered limit, independent of potential details, suggesting a fundamental underlying mechanism for the observed trans-boson peak universality in molecular glasses.

ABSTRACT

We have studied both clusters and bulk systems while investigating amorphous states. We have varied the nature of interaction amongst the particles of the system under consideration in order to reveal the possible presence of universality (i.e. independence of the potential). For clusters, the number of particles is varied to investigate the effect of finite size on various properties. To study bulk amorphous systems, we actually use periodic crystals with as large a primitive cell as possible. For any finite disorder, the number of particles in the primitive cell should ideally be made arbitrarily large. However, in practice, this is limited by the available computational resources. We should note here that due to the periodic nature of our approximation to the bulk disordered system, the analysis of spectral fluctuations becomes a lot more subtle and has not been reported in this thesis. We report results only on: (1) Universality in the density of states for clusters, (2) Universality in spectral fluctuations in the case of clusters, and (3) Universality in the density of states for bulk amorphous systems.

Motivation & Objective

• To investigate whether the vibrational spectra of amorphous systems exhibit universality independent of the specific interatomic potential.
• To examine how finite-size effects and disorder parameters influence spectral fluctuations and density of states in amorphous clusters.
• To determine whether the asymptotic vibrational density of states becomes independent of the potential form as disorder increases.
• To provide a numerical explanation for the experimentally observed universality in the trans-boson peak region of molecular glasses.
• To explore the convergence behavior of vibrational spectra toward a universal limit across different families of pair potentials.

Proposed method

• Numerical computation of inherent structures (local minima of potential energy) for single-component and binary amorphous clusters using standard minimization techniques.
• Solution of the Hessian eigenvalue problem for each inherent structure to obtain the vibrational spectrum and density of states.
• Application of random matrix theory concepts to analyze statistical fluctuations in the spectra, particularly focusing on level spacing distributions.
• Systematic variation of the potential parameters (e.g., m, n in GLJ and Morse potentials) to approach the asymptotic limit of infinite disorder (ϕ → ∞).
• Normalization of the density of states across different potentials and system sizes to compare spectral shapes in the limit of high disorder.
• Semi-logarithmic plotting of G(ω)/ω² to compare with experimental data and assess functional form similarity.

Experimental results

Research questions

• RQ1 Does the vibrational density of states of amorphous systems converge to a universal form in the limit of infinite disorder, regardless of the specific interatomic potential?
• RQ2 How do spectral fluctuations and the density of states evolve with increasing disorder and system size in amorphous clusters?
• RQ3 To what extent do different pair potentials (Morse vs. GLJ) yield similar vibrational spectra in the high-disorder regime?
• RQ4 Can the observed experimental universality in the trans-boson peak region of molecular glasses be explained by the convergence of normalized density of states to a common functional form?
• RQ5 What is the role of both attractive and repulsive parts of the pair potential in enabling asymptotic universality?

Key findings

• The normalized density of states converges to an identical functional form across both the Morse and GLJ potential families in the limit of infinite disorder (ϕ → ∞), indicating asymptotic universality.
• Within the error bars of the numerical calculations, the asymptotic normalized density of states is indistinguishable between the two potential families, even though their functional forms differ significantly.
• The semi-logarithmic plot of G(ω)/ω² shows a substantial linear segment in the trans-boson peak region, matching experimental observations of a straight-line behavior in this frequency range.
• The functional form g(ω) = αω²exp(−ω/ω₀) is empirically supported by the data, suggesting a universal scaling behavior in the high-disorder limit.
• Spectral convergence to universality occurs well before the strict asymptotic limit is reached, implying that many real amorphous materials may already exhibit near-universal vibrational spectra.
• The observed universality is robust only when both the attractive and repulsive parts of the pair potential contribute to the approach to infinite disorder; otherwise, universality breaks down.