Michael I. Klinger - Glassy Disordered Systems: Glass Formation and Universal Anomalous Low-Energy Properties

Appendix A

CONVOLUTION OF SOFT-MODE VIBRATIONAL DOS AND TRANSFORMATION KERNEL IN THE DOS OF BP AND HFS EXCITATIONS

In this appendix, approximate analytical expressions for the total vibrational DOS J( = 2), describing the spectrum of vibrational eigenvalues = (q) = 2(q) with the dispersion law (q) of ).

The total vibrational DOS can be calculated by applying its general expression (]):

where the transformation kernel, from soft-mode vibrational eigenvalues to eigenvalues resulting from soft-mode-acoustic interaction, is

Here, the dispersion relation (q;) is independent of the wave vector direction, and (q) is the range of allowed wave-number values, e.g., at qD/2 qqD/2 = /a1, while the integration limits in ]. The small width of the excitation eigenvalue = 2 can be approximated from the complex frequency = i with its width Then, the function can be approximated by an often used Lorentzian function D1[X], for typical ~ 0:

at X = and small By taking into account comments below ):

where

and

where (X) = 1 at X > 0 and (X) = 0 at x < 0. The function I(IRC)(;) and describe the region around and in the pseudogap, containing ill-defined excitations with large excitation widths, whereas I(ac)(; ) and characterise two regions of acoustic-like well-defined excitations, Debye excitations and HFS ones, below and above the pseudogap respectively; the results of numerical calculations of I(ac)(; ) are practically the same as to those obtained with the -function (X) substituted for

By introducing new, dimensionless, variables and parameters,

and by using, in particular, the resulting total vibrational DOS can be described by the following expression:

where and q(; ) is the appropriate solution of the equation (q; ) = , while (, ) denotes the variation range of such , and that Q(; ) 0. Moreover, J() can straightforwardly be reduced to the dimensionless function I(u):

where

It is taken into account in the derivation of that where

with

and the integration range () for the system two-valued continuous spectrum (q, ) in to the two-valent integration range (z, t).

Figure 16: (a) Initial integration range (z, t) in the double integral of , after changing the sequence of integrations.

The latter includes the range t1tt2 and is shown in , consisting of four two-dimensional strips:

, the integral I(u) can explicitly be described as follows:

where

Note that at = 0 the function K() in

GENERAL DESCRIPTION OF GLASSES AND GLASS TRANSITION

1.1 Metastability and disorder. Types of glasses

Glasses and glassy materials are macroscopic condensed systems constituting a very important variety of amorphous solids of which the formation is related to the so-called glass transition briefly described in what follows. Many materials are practically prepared as glasses and their number rapidly increases. It is important to understand how glasses differ from stable crystals in thermal equilibrium and from other types of amorphous solids and what the nature of phenomena characteristic of glasses is, including glass formation. Actually, as with other types of amorphous solids; glasses are metastable systems with a macroscopic life time