**3. Elastic properties of alkali borate glasses**

Since the low-energy excitations in glass is closely related to the acoustic modes, the sound velocity was measured by the ultrasonic pulse-echo overlap method at a frequency of 10 MHz and at 298 K [15, 16]. The sound velocities of longitudinal acoustic (LA) and transverse acoustic (TA) modes of the binary alkali metal borate glasses, *x*M2O(1�*x*)B2O3 (M = Li, Na, K, Rb, Cs) are shown in **Figure 6a** and **b**,

**Figure 6.** *Alkali content dependences of velocity of (a) longitudinal, and (b) transverse acoustic modes.*

**Figure 7.** *Alkali content dependences of (a) Young's modulus, E, and (b) the ratio between bulk and shear moduli, B/G.*

respectively [17].These dependences have a similarity with those of the reciprocal plots of *κV*m(B) and (b) *κV*m(O) in **Figure 5a** and **b** reflecting the variation of structural units by alkali ions.

Using the LA and TA velocities, the following elastic moduli were calculated.

$$\text{Shear modulus}: G = \rho V\_T^2 \tag{10}$$

$$\text{Longitudinal modulus}: L = \rho V\_L^2 \tag{11}$$

$$\text{Poisson's ratio}: \sigma = \frac{1}{2} \frac{V\_L^2 - 2V\_T^2}{V\_L^2 - V\_T^2} \tag{12}$$

$$\text{Young's modulus}: E = 2G(1+\sigma) \tag{13}$$

$$\text{Bulk modulus} : B = L - \frac{4}{3}G \tag{14}$$

$$\mathbf{\color{red}{Compressibility}} : \kappa = \mathbf{1}/\mathcal{B} \tag{15}$$

**Figure 7a** shows the alkali content dependences of Young's modulus. In the Young's modulus of LiB and KB glasses, the monotonic increase was observed. However, In that of KB, RbB, and CsB glasses concave and convex shapes were observed. Such alkali dependence of Young's modulus is similar to that of *κV*m(B) in **Figure 5a**.

The plot of bulk versus shear moduli is helpful in distinguishing ductile from brittle behavior beyond the elastic limit. When B/G>>1 (σ=0.5), materials are extremely incompressible. For ceramics, B/G≈1. 7(σ≈0.25). For polymers, B/G≈2.7(σ≈0.33). When B/G< <1 (σ = �1), materials are extremely compressible. The B/G is also related to the boson peak intensity and the fragility index [18]. The ratio between bulk and shear modulus is plotted in **Figure 7b**. The B/G is between 1.8 and 2.9. The *B* /*G* shows the concave and convex shapes were observed in all the alkali borate glasses. These dependences are explained by the division of four alkali content ranges. (a) 0≤*x*≤0.07: The BØ3 changes into BØ4 � only and does not change into BOØ2 �. (b) 0.07≤*x*≤0.20: The BØ3 changes into both BØ4 � and BOØ2 �. The effect of BOØ2 �is predominant over that of BØ4 �. (c) 0.20≤*x*≤0.30: The BØ3 changes into both BØ4 � and BOØ2 �. The effect of BØ4 �is predominant over that of BOØ2 �. (d) 0.30≤*x*≤0.40: The BØ3 changes only into BOØ2 � without the further formation of BØ4 �.
