**4. Boson peaks of alkali borate glasses**

In glasses the universal features of their thermal properties at low temperatures have been observed. The heat capacity shows an excess part as the deviation from the Debye *T*<sup>3</sup> law and the thermal conductivity has a plateau at around 10 K [19]. These universal behaviors are caused by the anomalous phonon dispersion in the terahertz frequency (THz) range. In the inelastic scattering spectra, the peak of g(*E*)/*E*<sup>2</sup> has been observed, where g(*E*) and *E*=*hν* are the vibrational density of states (VDoS) and energy, respectively. The origin of a peak is the low-energy excess part of VDoS over the Debye model defined by g(*E*)/*E*<sup>2</sup> . This THz peak is called the boson peak [20]. In the measurement of heat capacity *Cp* at low temperatures, a bump in *Cp/T*<sup>3</sup> at around 10 K is called a thermal boson peak. The plateau of the thermal conductivity indicates the strongly scattered of phonons above the boson peak frequency. It indicates that phonons meet the transverse Ioffe–Regel (IR) limit [21] around the boson peak.

The microscopic origin of a boson peak has been discussed by various theoretical models, such as (1) the structure and elastic constants heterogeneity [22–24]; (2) soft potential model [24–26]; (3) the resonant vibration of medium range order [27]; (4) mode-coupling theory on density fluctuations of arrested glass structures [28]; (5) broadening of the lowest van Hove singularity of the transverse phonon branch [29]; (6) the phonon-saddle transition in the energy landscape [30]; (7) the random firstorder transition theory (RFOT) [31]; (8) anharmonic effects [32], and (9) recent numerical calculations reported that the boson peak originates from quasi-localized vibrations of string-like dynamical defects [33]. However, this situation has remained quite controversial because of a lack of distinct evidence.

The Stokes-component of Raman scattering intensity *I*(ν) is related to the imaginary part of Raman susceptibility <sup>χ</sup>″(ν).

$$I(\nu) = I\_0 \chi''(\nu) \{ n(\nu) + 1 \}, \ n(\nu) = \frac{1}{\exp\left(\frac{h\nu}{k\_B T}\right) - 1} \tag{16}$$

where *n*ð Þ*ν* is the Bose-Einstein factor and *I*<sup>0</sup> is a constant which depends on the experimental condition. For the discussion of the boson peak in a Raman spectrum, the following quantity is plotted. Here, *<sup>C</sup>*ð Þ¼ *<sup>ν</sup> <sup>ν</sup><sup>α</sup>* (α=0�2) is the lightvibration coupling constant and its frequency dependence shows a monotonic increase [20].

$$\frac{\chi''(\nu)}{\nu} = \frac{I(\nu)}{\nu \{ n(\nu) + 1 \}} \propto \mathcal{C}(\nu) \frac{\mathbf{g}(\nu)}{\nu^2} \tag{17}$$

**Figure 8a** shows the boson peak spectra of lithium borate glasses observed by Raman scattering using a triple-grating spectrometer with the additive dispersion. The boson peak frequency νBP = 26 cm�<sup>1</sup> at *x* = 0.02 increases up to 72 cm�<sup>1</sup> at *x* = 0.26 as the lithium content increases and the increase is related to the increase of transverse sound velocity shown in **Figure 5b** [34]. In LiB glasses, the fragility index *m* increases from 30 at *x*=0.00 to 62 at *x*=0.28. In strong glass, the boson peak intensity is high, while the intensity of the fast β-relaxation is weak. As the fragility index *m* increases, the boson peak intensity becomes weak and that of the fast β-relaxation increases. As shown in **Figure 8a** the boson peak intensity decreases as the Li content increases.

#### **Figure 8.**

*(a) Reduced Raman spectra of boson peaks of lithium borate glasses observed by Raman scattering, and (b) Scaled boson peak spectra.*

**Figure 9.** *Reduced Raman spectra of boson peaks of alkali borate glasses (*x*=0.14).*

It is interesting to check that the boson peak spectra of inelastic neutron scattering, and Raman scattering are scaled by their peak positions and scattering intensity. In LiB glasses, it is found that the scaled boson peak spectra have a universal shape as shown in **Figure 8b**. The universal scaling of boson peaks indicates that the way of the distribution of VDoS basically remains the same, even though the glass structures drastically change by the alkali metal modification.

The alkali dependence of boson peak spectra at *x*=0.14 is shown in **Figure 9**. The origin of the boson peak in a pure borate glass was attributed to the coherent libration of several boroxol rings based on the study of the hyper-Raman scattering. As the alkali content increases, these boroxol rings change into other boron-oxygen structural units. As the ionic radius of alkali ions increases, the boson peak frequency decreases reflecting the difference in the modification of glass structure by alkali ions. Since the large Cs ions with the low charge density only slightly changes the boronoxygen network structure. However, the small Li ions with the high charge density cause shrinking of the boron-oxygen network structure [35, 36]. By application of high pressure, it was reported that the boson peak frequency significantly increases up to 68 cm<sup>1</sup> at 4 GPa by shrinking of the boron-oxygen network structure [37]. The alkali content dependence of boson peaks of LiB glasses has the similarity with the densified borate glasses.

The boson peak frequency in a Raman spectrum includes the influence of the lightvibration coupling constant *C*(ν) as shown in Eq. (17). However, the boson peak in a neutron inelastic spectrum enables the direct observation of a boson peak frequency or energy even in the S/N ratio of scattering intensity is much lower than that of Raman scattering. **Figure 10** shows the alkali dependence of boson peak spectra of alkali borate glasses at *x*=0.22 observed by cold neutron inelastic scattering. Neutron inelastic scattering measurements of all the alkali borate glasses were carried out at 25° C (far below the *T*g) using a direct geometry chopper-type ToF spectrometer AGNES belonging to the Institute for Solid State Physics, University of Tokyo [38]. The neutron boson peak energy also decreases as the ionic radius of alkali ions increases. As the charge density decreases with the increase in ionic radius, the contraction of the boron–oxygen structural units become weaker and the boson peak energy may decrease.

If we assume that the boson peak is connected to the nano-heterogeneity of the shear modulus [27]. In this approach, a dynamic length scale, *L*BP, is given by

$$L\_{\rm BP} = V\_{\rm t} / 2\pi\nu\_{\rm BP},\tag{18}$$

where *V*<sup>t</sup> is the transverse sound velocity, and νBP is the boson peak frequency. The *L*BP corresponds to a medium-range scale important for characterization of structure

**Figure 10.** *Boson peaks of alkali borate glasses at* x*=0.22 observed by cold neutron inelastic scattering [37].*

**Figure 11.** *The correlation between* L*BP and ionic radius of alkali ions in alkali borate glasses.*

correlations in glasses. The good correlation between *L*BP and ionic radius of alkali ions is found as shown in **Figure 11**. It is found that *L*BP is proportional to the ionic radius of alkali ions. It indicates that the dynamic length of a boson peak may relate to the size of alkali ion in the void of boron-oxygen network structure.
