## 5 Recent Experiments
### Relation of the Boson Heat Capacity Peak to the Defect Structure  
A peculiar universal feature of atomic dynamics in non-crystalline materials consists in the presence of excess (over the Debye contribution) low-frequency vibrational modes [65; 66]. These low-frequency modes are detected as a peak in the low temperature (5-15 K) heat capacity \(C\) plotted as \(C/T^{3}\) vs. \(T\). These features are usually referred to as the boson peak, which is known for metallic glasses as well [67]. The nature of the boson peak constitutes a matter of intensive ongoing debates [68; 69; 70; 71]. [PERSON] argued that the boson peak originates from low-frequency resonance vibration modes of interstitial-type defects frozen-in upon glass production [72]. He showed that the boson peak height should be proportional to the defect concentration. A refined equation for the boson peak height has the form [73]  
\[h_{B}=\frac{C^{d}}{T_{B}^{3}}=\frac{234\ R}{\Theta_{D}^{3}}\left[0.09f\left( \frac{\omega_{D}}{\omega_{r}}\right)^{3}+\frac{3}{2}\beta\right]c\equiv \Gamma c, \tag{14}\]  
where \(T_{B}\) is the boson peak temperature, \(C^{d}\) is the heat capacity related to the interstitial-type defect system, \(\Theta_{D}\) and \(\omega_{D}\) are the Debye temperature and Debye frequency of the maternal crystal, respectively, \(\omega_{r}\) is the characteristic frequency of interstitial resonance vibrations, \(f\) is the number of resonance modes per interstitial-type defect, \(R\) is the universal gas constant and other quantities are specified above. The defect concentration \(c\) can be monitored by measurements of the shear moduli of glass and maternal crystal as implied by Equation (2). Thus, the boson peak within the framework of the \(\Gamma\Gamma\) is considered to be a \"fingerprint\" of the defect glass structure. Figure 8: The melting enthalpy vs. the absolute value of sum of the enthalpies of structural relaxation and crystallization. The numbers correspond to different Zr-, Pd- and La-based MGs [64]. With permission from Elsevier, 2019. An experiment aimed to check the prediction given by Equation (14) was carried out on glassy \(Zr_{65}Cu_{15}Ni_{10}Al_{10}\)[73]. The main result of this experiment is shown in Figure 9, which gives the measured height of the boson peak as a function of the defect concentration \(c\) calculated with Equation (2) using room-temperature measurements of the shear modulus after different annealing treatments. The annealing protocol was designed to perform measurements on both fully amorphous and partially crystalline samples. It is seen that independent of the state of the samples (amorphous/partially crystalline), the boson peak height linearly increases with the defect concentration, in line with [PERSON]'s prediction [72]. The derivative \(dh_{B}/dc\) calculated from Figure 9 provides reasonable estimates for the resonant vibration frequencies of the defects assumed to be responsible for the boson peak [73]. On the other hand, since the interstitial-type defect structure determines the excess enthalpy \(\Delta H\) of glass, as discussed above in Section 5.3, one can expect that the boson peak height \(h_{B}\) should also be related to \(\Delta H\). The calculation gives the relation between the boson peak height and excess enthalpy of glass \(\Delta H\) as [74]  
\[h_{B}=\frac{\Gamma}{\alpha\beta}ln\frac{\mu}{\mu-\rho\beta\Delta H}, \tag{15}\]  
where the quantity \(\Gamma\) is defined by Equation (14), \(\rho\) is the density, and \(\mu\), \(\alpha\) and \(\beta\) are same as in Equation (2). The relationship in Equation (15) directly connects the boson peak height with the excess enthalpy of the glass. The latter may be considered as an independent variable, which can be changed by the annealing leading to either structural relaxation within the glassy state or partial crystallization.