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Calculation of the BP frequency
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Ultrafast extreme rejuvenation of metallic glasses by shock compression

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The total specific heat capacity Cp of the glass is expressed as$Cp=Cpele+nDCpDebye+nqCpquasi$(1)where nD and nq represent the Debye and quasi-localized contributions per mole, respectively. The Debye term $CpDebye$ is$CpDebye=9R(TθD)3∫0θD/Tξ4eξ(eE−1)2dξ$(2)where R is the gas constant and θD is the Debye temperature that can be obtained by fitting the Cp data below 8 K (see fig. S6). The quasi-localized term $Cpquasi$ can be written as$Cpquasi=3R∫0ω*(ħωkBT)2eħω/kBT(eħω/kBT−1)2gq(ω)dω$(3)where ħ, kB, and ω are the Planck constant, the Boltzmann constant, and the frequency, respectively. Considering the hybrid nature of quasi-localized and Debye modes (55), the upper limit ω* of integral was reasonably chosen as the Debye frequency ωD = kBθD/ħ. The gq(ω) is the vibrational density of states or the frequency distribution of the quasi-localized modes, which is the key to determine the $Cpquasi$ and resultant BP. It is well accepted that the spatial heterogeneity of elastic modulus incurs the emergence of quasi-localized modes (48, 49, 56). Shear modulus and BP intensity are strongly correlated (46, 57). We therefore examined the distribution of elastic modulus in metallic glasses (58, 59) and found that the data can be well fitted by a formula analogous to the Planck’s law of black-body radiation. Therefore, we assumed that the quasi-localized modes gq(ω)/ω2 normalized by the Debye law obeys a Planck-like distribution$gq(ω)/ω2=φω3eℓω−1$(4)where the parameter φ = 5.572 × 10−15 is determined by the high-temperature limit (3R) of Cp and the ℓ can be determined by the position TBP of the Cp-BP. It is noticed that $limω→0gq(ω)∼ω4$, which is consistent with the statistic law of low-frequency quasi-localized modes revealed recently (60). Uniting Eqs. 1 to 4, both the Cp and (Cp − γT)/T3 data were satisfactorily fitted, and the contributions from the electronic $Cpele$, the Debye $CpDebye$, and the quasi-localized vibrational $Cpquasi$ can be calculated, respectively. The results are shown in fig. S7, and all relevant parameters are listed in table S3. According to the measured TBP, the frequency ωBP of BP can be determined by ∂[gq(ω)/ω2]/∂ω∣ωBP = 0.

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