Vibrational density of states of amorphous solids with long-ranged power-law-correlated disorder in elasticity
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Vibrational density of states of amorphous solids with long-ranged power-law–correlated disorder in elasticity Bingyu Cui1 and Alessio Zaccone1,2,3,a 1 2 3
Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB3 0HE Cambridge, UK Department of Physics “A. Pontremoli”, University of Milan, via Celoria 16, 20133 Milano, Italy Statistical Physics Group, Department of Chemical Engineering and Biotechnology, University of Cambridge, Philippa Fawcett Drive, CB3 0AS Cambridge, UK Received 25 September 2020 / Received in final form 18 October 2020 / Accepted 2 November 2020 Published online: 23 November 2020 c The Author(s) 2020. This article is published with open access at Springerlink.com Abstract. A theory of vibrational excitations based on power-law spatial correlations in the elastic constants (or equivalently in the internal stress) is derived, in order to determine the vibrational density of states D(ω) of disordered solids. The results provide the first prediction of a boson peak in amorphous materials where spatial correlations in the internal stresses (or elastic constants) are of power-law form, as is often the case in experimental systems, leading to a logarithmic enhancement of (Rayleigh) phonon attenuation. A logarithmic correction of the form ∼ −ω 2 ln ω is predicted to occur in the plot of the reduced excess DOS for frequencies around the boson peak in 3D. Moreover, the theory provides scaling laws of the density of states in the low-frequency region, including a ∼ ω 4 regime in 3D, and provides information about how the boson peak intensity depends on the strength of power-law decay of fluctuations in elastic constants or internal stress. Analytical expressions are also derived for the dynamic structure factor for longitudinal excitations, which include a logarithmic correction factor, and numerical calculations are presented supporting the assumptions used in the theory.
1 Introduction Understanding the physics of vibrational spectra of disordered systems is a classical topic in condensed-matter physics [1–4]. Glasses and other disordered solids exhibit anomalous features, compared with their crystalline counterparts. Concerning the thermal properties, at few tens of kelvin, the specific heat of glasses exhibits an excess over the Debye prediction, in the form of a characteristic maximum in the plot of C(T )/T 3 . The peak is ascribed to the presence of an excess of states over the Debye density of states (DOS) ∼ ω 2 , known as the boson peak since its temperature dependence conforms with that of the Bose function, and thus appears to strongly depend on the features of the vibrational modes in the THz frequency [5–8]. Thanks to neutron, X-ray and other inelastic scattering experiments [9–20], computer simulations [21–31], as well as analytical theories [32–50], the nature of these excited modes has been widely investigated. Since the boson
Contribution to the Topical Issue “Disordered, NonEquilibrium Systems: From Supercooled Liquids to
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