**5. Excess heat capacity at low temperatures of alkali borate glasses**

The excess heat capacity has been observed as the deviation from the Debye *T*<sup>3</sup> law at low temperatures. The broad peak in a *Cp/T*<sup>3</sup> vs. *T* plot is the thermal boson peak, where *C*<sup>p</sup> is the heat capacity at a constant pressure. It is related to the non-Debye excess heat capacity. The alkali content dependence of *Cp/T*<sup>3</sup> in lithium borate glasses is shown in **Figure 12** [35]. In the pure borate glass, the peak of *Cp/T*<sup>3</sup> was observed at the temperature,*T*<sup>m</sup> = 5.8 K [39]. As the lithium content increases, the peak value decreases and the peak temperature increases. Such a behavior of the peak value and the peak temperature of thermal boson peaks is similar to the peak intensity and peak frequency of boson peaks observed by Raman scattering and neutron inelastic scattering, respectively.

**Figure 12.** *Temperature dependence of* C*p/*T*<sup>3</sup> of lithium borate glasses at low temperatures.*

#### **Figure 13.**

*(a) Temperature dependence of* C*p/*T*<sup>3</sup> of alkali borate glasses normalized by Debye contribution at low temperatures, and (b) Scaled thermal boson peaks of alkali borate glasses.*

**Figure 14.** *Correlation between the peak temperature of* C*p/*T*<sup>3</sup> , boson peak frequency, and shear modulus.*

The temperature dependence of *C*p/*T*<sup>3</sup> of alkali borate glasses with *x*=0.22 below 50 K is shown in **Figure 13** [35]. The peak temperature of CsB glass is close to that of pure borate glass. However, as the ionic radius of alkali metal ions decreases, the peak temperature markedly increases. The peak temperature of LiB glass is *T*m=14.6 K, which is about three times of a pure borate glass.

For all the alkali borate glasses, the universal nature of the master plot in *Cp/T*<sup>3</sup> vs. *T* is also observed as shown in **Figure 13b**. This universal nature is the same as that of boson peaks observed by Raman scattering and neutron inelastic scattering. This fact indicates that the distribution of the low-energy excess VDoS remains the same for all the alkali modification.

On the discussion on the origin of a boson peak, the correlation between the transverse acoustic mode and a boson peak is very interesting. According to the numerical simulation, the equality of the boson peak frequency to the Ioffe–Regel limit for "transverse" phonons was reported. The boson peak energy is proportional to the shear modulus [40]. Since the peak temperature of a thermal boson peak is proportional to the boson peak frequency, the correlation is examined between the peak temperature of *C*p/*T*<sup>3</sup> , boson peak energy, and shear modulus. The good correlation is found between these quantities as shown in **Figure 14**. This fact indicates that the origin of a boson peak is closely related to the Ioffe-Regal limit for transverse acoustic waves.
