Tunneling centers

A. Quantum Mixing of a Tunneling Center and the Black-Halperin Paradox... 

Appendix B. Frequency Cutoff in the Interaction Between the Tunneling Centers and the Linear Strain References... 

Figure 9. A schematic of a tunneling center is shown. is its typical size, is a typical displacement on the order of the Lindemann distance. The doubled circles symbolize the atomic positions corresponding to the alternative internal states. The internal contour, encompassing N beads, illustrates a transition state size, to be explained later in the text.

As we will see later, the tunneling barriers, and hence the relaxation times of the tunneling centers, are distributed. This would lead to a time-dependent heat capacity. Ignoring this complication for now, the classical, long-time heat capacity is easy to estimate aheady (assuming it exists). Since our degrees of... 

With the knowledge of g, we can estimate the inverse mean free path of a phonon with frequency co. As done originally within the TLS model, the quantum dynamics of the two lowest energies of each tunneling center are described by the Hamiltonian //tls = gcTz/2 + Aa /2. This expression, together with Eqs. (15) and (17), is a complete (approximate) Hamiltonian of... 

We hope to have convinced the reader by now that the tunneling centers in glasses are complicated objects that would have to be described using an enormously big Hilbert space, currently beyond our computational capacity. This multilevel character can be anticipated coming from the low-temperature perspective in Lubchenko and Wolynes [4]. Indeed, if a defect has at least two alternative states between which it can tunnel, this system is at least as complex as a double-well potential—clearly a multilevel system, reducing to a TLS at the lowest temperatures. Deviations from a simple two-level behavior have been seen directly in single-molecule experiments [105]. In order to predict the energies at which this multilevel behavior would be exhibited, we first estimate the domain wall mass. Obviously, the total mass of all the atoms in the droplet... 

The existence of the domain wall vibrations explains and allows us to visualize, at least in part, the multilevel character of the tunneling centers as exhibited at temperatures above the TLS regime. Curiously, the existence of TLSs, even though displayed at the lowest T, is basically of classical origin due to the... 

Figure 21. A low-energy portion of the energy level structure of a tunneling center is shown. Here e < 0, which means that the reference, liquid, state structure is higher in energy than the alternative configuration available to this local region. A transition to the latter configuration may be accompanied by a distortion of the domain wall, as reflected by the band of higher energy states, denoted as ripplon states.

As the starting point in the discusion, we consider a simplified version of the diagram of a tunneling center s energy states from Fig. 14 with e < 0, as shown on the left hand side of Fig. 21. We remind the reader that the e < 0 situation, explicitly depicted in Fig. 21, implies lower transition energies than when the semiclassical energy difference e > 0 and thus dominates the low-temperature onset of the boson peak and the plateau. 

Figure 23. This caricature demonstrates the predicted phenomena of energy level crossing in domains whose energy bias is comparable or larger than the vibronic frequency of the domain wall distortions. The vertical axis is the energy measured from the bottom state the horizontal axis denotes temperature. The diagonal da ed line denotes roughly the thermal energies. A tunneling center that would become thermally active at some temperature Tq will not possess ripplons whose frequency is less than To.

The ripplon-TLS term, as estimated here, therefore seems somewhat larger relative to the ripplon-ripplon term than seen in experiment, consistent with our earlier notion that it is somewhat overestimated. StiU, qualitatively our estimates are consistent with the observed tendency of y to increase in magnitude, when the temperature is lowered. We point out that the results obtained above disregard possible effects of a specific distribution of A that will influence the precise value of the coupling between phonons and tunneling centers. 

To complete the discussion of the second-order interaction between tunneling centers, we note that the corresponding contribution to the heat capacity in the leading low T term comes from the ripplon-TLS term and scales as 7 +2 where a is the anomalous exponent of the specific law. Within the approximation adopted in this section, a = 0. However, it is easily seen that the magnitude of the interaction-induced specific heat is down from the two-level system value by a factor of 10(a/ ) ([Pg.188]

We have carrried out an analysis of the multilevel structure of the tunneling centers that goes beyond a semiclassical picture of the formation of those centers at the glass transition, which was primarily employed in this chapter. These effects exhibit themselves in a deviation of the heat capacity and conductivity from the nearly linear and quadratic laws, respectively, that are predicted by the semiclassical theory. 

Since the spatial locations r,- of active droplets are not correlated, we can replace the summation over the droplets by a continuous integral, assuming at the same time that the ripplon frequency corresponding to co/ varies from droplet to droplet within a (normalized) distribution Vi (oi) centered around co/ and having a characteristic width 8co/, whose value will be discussed shortly. There is no reason to believe that the frequency and location of the tunneling centers are correlated therefore one obtains... 

APPENDIX B FREQUENCY CUTOFF IN THE INTERACTION BETWEEN THE TUNNELING CENTERS AND THE LINEAR STRAIN... 

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