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Register a new userApplication of the random matrix theory to the boson peak in glasses
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The density of vibrational states $g(omega)$ of an amorphous system is studied by using the random-matrix theory. Taking into account the most important correlations between elements of the random matrix of the system, equations for the density of vibrational states $g(omega)$ are obtained. The analysis of these equations shows that in the low-frequency region the vibrational density of states has the Debye behavior $g(omega) sim omega^2$. In the higher frequency region, there is the boson peak as an additional contribution to the density of states. The obtained equations are in a good agreement with the numerical results and allow us to find an exact shape of the boson peak.
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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.
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 boson peak in metallic glasses is modeled in terms of local structural shear rearrangements. Using Eshelbys solution of the corresponding elasticity theory problem (J. D. Eshelby, Proc. Roy. Soc. A241, 376 (1957)), one can calculate the saddle point energy of such a structural rearrangement. The neighbourhood of the saddle point gives rise to soft resonant vibrational modes. One can calculate their density, their kinetic energy, their fourth order potential term and their coupling to longitudinal and transverse sound waves.
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.
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