continuous characteristic of the viscoelastic crossover from the liquid state into the solid one when the transition is not accompanied by crystallization—ergo the supercooled viscous liquid. Thus we see the intimate correlation between transverse acoustic phonons (or shear waves) and the onset of rigidity upon vitrification, as described by Bartenev in his mechanical description of the vitrification process. This concept leads to defining the glass transition in terms of the vanishing or significant lowering of the low-frequency shear modulus, as shown quantitatively in the work of Zaccone and Terentjev on the example of polymer glass. In fact, the shoving model stipulates that the activation energy of the relaxation time is proportional to the high-frequency plateau shear modulus, a quantity that increases upon cooling thus explaining the ubiquitous non-Arrhenius temperature dependence of the relaxation time in glass-forming liquids. The velocities of longitudinal acoustic phonons in condensed matter are directly responsible for the thermal conductivity that levels out temperature differentials between compressed and expanded volume elements. Kittel proposed that the behavior of glasses is interpreted in terms of an approximately constant "mean free path" for lattice phonons, and that the value of the mean free path is of the order of magnitude of the scale of disorder in the molecular structure of a liquid or solid. The thermal phonon mean free paths or relaxation lengths of a number of glass formers have been plotted versus the glass transition temperature, indicating a linear relationship between the two. This has suggested a new criterion for glass formation based on the value of the phonon mean free path. It has often been suggested that heat transport in dielectric solids occurs through elastic vibrations of the lattice, and that this transport is limited by elastic scattering of acoustic phonons by lattice defects (e.g. randomly spaced vacancies). These predictions were confirmed by experiments on commercial glasses and glass ceramics, where mean free paths were apparently limited by "internal boundary scattering" to length scales of . The relationship between these transverse waves and the mechanism of vitrification has been described by several authors who proposed that the onset of correlations between such phonons results in an orientational ordering or "freezing" of local shear stresses in glass-forming liquids, thus yielding the glass transition. Electronic structure The influence of thermal phonons and their interaction with electronic structure is a topic that was appropriately introduced in a discussion of the resistance