On the temperature dependence of the density of states of liquids at low energies

We report neutron-scattering measurements of the density of states (DOS) of water and liquid Fomblin in a wide range of temperatures. In the liquid phase, we confirm the presence of a universal low-energy linear scaling of the experimental DOS as a function of the frequency, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mi>ω</mml:mi></mml:mrow> </mml:math> , which persists at all temperatures. The low-frequency scaling of the DOS exhibits a sharp jump at the melting point of water, below which the standard Debye's law, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>g</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo></mml:mrow> <mml:mo>∝</mml:mo> <mml:msup><mml:mi>ω</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:mrow> </mml:math> , is recovered. On the contrary, in Fomblin, we observe a continuous transition between the two exponents reflecting its glassy dynamics, which is confirmed by structure measurements. More importantly, in both systems, we find that the slope a(T) grows with temperature following an exponential Arrhenius-like form, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∝</mml:mo> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>E</mml:mi> <mml:mo>⟩</mml:mo> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> . We confirm this experimental trend using molecular dynamics simulations and show that the prediction of instantaneous normal mode (INM) theory for a(T) is in qualitative agreement with the experimental data.