A Noncommutative Glass Model and the Boson Peak

A new theoretical model for the structure of glasses is presented and used to study the boson peak found in glasses. The model is based on a simple lattice model familiar from crystals, which is disordered using techniques from noncommutative fluid models. First classical crystal models and concepts of lattice vibrations are reviewed, focusing on acoustic and optical waves, the density of vibrational states, heat capacity and the Debye model. Then noncommutative fluid theory and noncommutative geometry are shortly introduced to show the connection to fluids in our model. After these introductions, the glass model is formulated and used to calculate the dispersion relations, the density of vibrational states and the heat capacity. The density of states has a Van Hove singularity at low frequencies, which generates the boson peak seen in experiments. The glass is found to have both acoustic and optical waves, and the acoustic waves are located very close to the frequency of the Van Hove singularity, which hints that the boson peak should be related to acoustic waves.Työssä esitellään uusi teoreettinen malli lasien rakenteelle, ja sitä hyödynnetään lasien niin sanotun bosonipiikin tutkinnassa. Malli perustuu yksinkertaiseen kiteistä tuttuun hilaan, joka muutetaan epäjärjestyneeksi epäkommutoivista fluidimalleista tutuilla menetelmillä. Aluksi tutkielmassa käydään läpi kiteiden rakenteen kannalta tärkeitä käsitteitä, keskittyen erityisesti akustisiin ja optisiin aaltoihin, hilavärähtelyjen tilatiheyteen, lämpökapasiteettiin sekä Debyen värähtelymalliin. Sen jälkeen perehdytään lyhyesti epäkommutoiviin fluidimalleihin sekä epäkommutoivaan geometriaan, jotta lasimallia esitellessä nähtäisiin sen yhteys fluideihin. Lopuksi esitellään itse malli lasien rakenteelle, ja siitä lasketaan lasien dispersiorelaatiot, tilatiheys sekä lämpökapasiteetti. Tilatiheydessä havaitaan Van Hoven singulariteetti matalilla taajuuksilla, joka vastaa lasien kokeissa löydettyä bosonipiikkiä. Lasilla on sekä akustisia että optisia aaltoja, joista akustiset aallot sijaitsevat hyvin lähellä Van Hoven singulariteetin taajuutta. Löydön perusteella lasien bosonipiikki johtuu akustisista aalloista

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