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Register a new userVibrations and Relaxations in a Soft Sphere Glass: Boson Peak and Structure Factors
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Authors
H. R. Schober
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The dynamics of a soft sphere model glass, studied by molecular dynamics, is investigated. The vibrational density of states divided by $omega^2$ shows a pronounced boson peak. Its shape is in agreement with the universal form derived for soft oscillators interacting with sound waves. The excess vibrations forming the boson peak have mainly transverse character. From the dynamic structure factor in the Brillouin regime pseudo dispersion curves are calculated. Whereas the longitudinal phonons are well defined up to the pseudo zone boundary the transverse ones rapidly get over-damped and go through the Ioffe-Regel limit near the boson peak frequency. In the high $q$ regime constant-$omega$ scans of the dynamic structure factor for frequencies around the boson peak are clearly distinct from those for zone boundary frequencies. Above the Brillouin regime, the scans for the low frequency modes follow closely the static structure factor. This still holds after a deconvolution of the exact harmonic eigenmodes into local and extended m odes. Also the structure factor for local relaxations at finite temperatures resembles the static one. This semblance between the structure factors mirrors the collective motion of chain like structures in both low frequency vibrations and atomic hopping processes, observed in the earlier investigations.
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We show that a {em vibrational instability} of the spectrum of weakly interacting quasi-local harmonic modes creates the maximum in the inelastic scattering intensity in glasses, the Boson peak. The instability, limited by anharmonicity, causes a complete reconstruction of the vibrational density of states (DOS) below some frequency $omega_c$, proportional to the strength of interaction. The DOS of the new {em harmonic modes} is independent of the actual value of the anharmonicity. It is a universal function of frequency depending on a single parameter -- the Boson peak frequency, $omega_b$ which is a function of interaction strength. The excess of the DOS over the Debye value is $proptoomega^4$ at low frequencies and linear in $omega$ in the interval $omega_b ll omega ll omega_c$. Our results are in an excellent agreement with recent experimental studies.
Experimental results on the density of states and on the acoustic modes of glasses in the THz region are compared to the predictions of two categories of models. A recent one, solely based on an elastic instability, does not account for most observations. Good agreement without adjustable parameters is obtained with models including the existence of non-acoustic vibrational modes at THz frequency, providing in many cases a comprehensive picture for a range of glass anomalies.
High frequency sound is observed in lithium diborate glass, Li$_2$O--2B$_2$O$_3$, using Brillouin scattering of light and x-rays. The sound attenuation exhibits a non-trivial dependence on the wavevector, with a remarkably rapid increase towards a Ioffe-Regel crossover as the frequency approaches the boson peak from below. An analysis of literature results reveals the near coincidence of the boson-peak frequency with a Ioffe-Regel limit for sound in {em all} sufficiently strong glasses. We conjecture that this behavior, specific to glassy materials, must be quite universal among them.
We present a random matrix approach to study general vibrational properties of stable amorphous solids with translational invariance using the correlated Wishart ensemble. Within this approach, both analytical and numerical methods can be applied. Using the random matrix theory, we found the analytical form of the vibrational density of states and the dynamical structure factor. We demonstrate the presence of the Ioffe-Regel crossover between low-frequency propagating phonons and diffusons at higher frequencies. The reduced vibrational density of states shows the boson peak, which frequency is close to the Ioffe-Regel crossover. We also present a simple numerical random matrix model with finite interaction radius, which properties rapidly converges to the analytical results with increasing the interaction radius. For fine interaction radius, the numerical model demonstrates the presence of the quasilocalized vibrations with a power-law low-frequency density of states.
The inelastic scattering intensities of glasses and amorphous materials has a maximum at a low frequency, the so called Boson peak. Under applied hydrostatic pressure, $P$, the Boson peak frequency, $omega_{rm b}$, is shifted upwards. We have shown previously that the Boson peak is created as a result of a vibrational instability due to the interaction of harmonic quasi localized vibrations (QLV). Applying pressure one exerts forces on the QLV. These shift the low frequency part of the excess spectrum to higher frequencies. For low pressures we find a shift of the Boson peak linear in $P$, whereas for high pressures the shift is $propto P^{1/3}$. Our analytics is supported by simulation. The results are in agreement with the existing experiments.
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