Detection of boson peak and fractal dynamics of disordered systems using terahertz spectroscopy

The boson peak is a largely unexplained excitation found universally in the terahertz vibrational spectra of disordered systems; the so-called fracton is a vibrational excitation associated with the self-similar structure of monomers in polymeric glasses. We demonstrate that such excitations can be detected using terahertz spectroscopy. In the case of fractal structures, we determine the infrared light-vibration coupling coefficient for the fracton region and show that information concerning the fractal and fracton dimensions appears in the exponent of the absorption coefficient. Finally, using terahertz time-domain spectroscopy and low-frequency Raman scattering, we experimentally observe these universal excitations in a protein (lysozyme) system that has an intrinsically disordered and fractal structure and argue that the system should be considered a single supramolecule. These findings are applicable to amorphous and fractal objects in general and will be valuable for understanding universal dynamics of disordered systems via terahertz light.

Figure 1

THz and Raman spectra of lysozyme. (a) Real ${\varepsilon }^{\prime }\left(\nu \right)$ and (b) imaginary ${\varepsilon }^{\prime \prime }\left(\nu \right)$ parts of complex dielectric constant, and (c) BP plot $\alpha \left(\nu \right)/{\nu }^2$ of lysozyme determined by THz-TDS measurements at room temperature. Insets of (a), (b), and (c) show temperature dependence of IR spectra, as measured in a heating process with a temperature range of 13 K to 295 K and plotted every 10 K above 20 K. (d) Measured Raman intensity $I\left(\nu \right)$ of lysozyme at room temperature. (e) Imaginary part of Raman susceptibility ${\chi }^{\prime \prime }\left(\nu \right)$. (f) BP plot of Raman spectra ${\chi }^{\prime \prime }\left(\nu \right)/\nu$. For convenience, 1 THz = 33.3 ${\mathrm{cm}}^{-1}$ = 48 K = 4.14 meV.

Figure 2

Comparison of THz, Raman, and v-DOS spectra. (a) Imaginary parts of complex dielectric constant ${\varepsilon }^{\prime \prime }\left(\nu \right)$ (orange) and Raman susceptibility ${\chi }^{\prime \prime }\left(\nu \right)$ (blue) of lysozyme. (b) BP plots of IR $[\alpha \left(\nu \right)/{\nu }^2$ (red)] and Raman $[{\chi }^{\prime \prime }\left(\nu \right)/\nu$ (green)] spectra. (c) BP plot of v-DOS $g\left(\nu \right)/{\nu }^2$. The data of $g\left(\nu \right)$ are quoted from previous studies based on INS [42, 47]. The log-normal fit used to obtain the BP frequency is shown by the pink curve.

Figure 3

Adjustment of $g\left(\nu \right)$ by inelastic neutron scattering with the absolute value determined by performing MEM analysis on the low-temperature specific heat ($C_p$) data of the lysozyme. We depicted and connected $g\left(\nu \right)$ from inelastic neutron scattering measurements by Perticaroli et al. [47] and Lushnikov et al. [42] for achieving higher accuracy of $g\left(\nu \right)$ [47] in the BP region and $g\left(\nu \right)$ [42] in the fracton region. Based on data depicted from the results of low-temperature specific heat measurements reported in the work by Crupi et al. [25], the specific heat v-DOS as $g_{\mathrm{heat}}\left(\nu \right)$ was extracted by MEM analysis. We determined the absolute value of the $g_{\mathrm{INS}}\left(\nu \right)$ used in this work through conformation with the absolute value of $g_{\mathrm{heat}}\left(\nu \right)$ at the BP frequency.

Figure 4

Log-log plot of THz, Raman, and v-DOS spectra and coupling coefficients. (a) Log-log plot of imaginary parts of complex dielectric constant ${\varepsilon }^{\prime \prime }\left(\nu \right)$ (orange), Raman susceptibility ${\chi }^{\prime \prime }\left(\nu \right)$ (arb. units) (blue), and v-DOS $g\left(\nu \right)$ (cyan) of lysozyme. $g\left(\nu \right)$ data are quoted from previous studies based on INS [42, 47]. (b) IR light-vibration (red) and Raman (green) coupling coefficients $[C_{\mathrm{IR}}\left(\nu \right)$ and $C_{\mathrm{Raman}}\left(\nu \right)]$ of lysozyme in the log-log representation. The inset shows the linear representation of $C_{\mathrm{IR}}\left(\nu \right)$ and $C_{\mathrm{Raman}}\left(\nu \right)$ in the vicinity of the BP frequency.

Figure 5

Calculation of mass fractal dimension of lysozyme molecule. (a) Atoms (red balls) contained in a sphere of radius $R$ around the center of gravity of the lysozyme molecule. For the lysozyme atomic coordinate arrangement, information on 1LYZ was taken from the PDB. (b) Log-log plot of the $R$ dependence of molecular weight contained in a sphere of radius $R$ around the center of gravity of lysozyme. When the diameter of the sphere exceeds the diameter of the lysozyme molecules, the molecular weight levels off. From fitting to the linear region, the mass fractal dimension of the lysozyme molecule was found to be $D_{f-\mathrm{calc}}=2.75$.

Figure 6

Dispersion relation in fracton region and THz and Raman spectra. (a) Dispersion relation of TA and fracton modes of lysozyme with the hypothesis of continuous connection at the BP frequency (green solid line). The LA (red dashed line) and TA (gray dashed line) modes are also shown, with sound velocity as measured by Perticaroli et al. [47] The dotted lines represent the structure correlation length (wine red dashed line) and the characteristic wavelengths (blue (Raman) and orange (IR) dashed lines). The values of these lengths are indicated by bars on the figure of the lysozyme molecule. (b) Imaginary parts of complex dielectric constant ${\varepsilon }^{\prime \prime }\left(\nu \right)$ (orange) and Raman susceptibility ${\chi }^{\prime \prime }\left(\nu \right)$ (blue) of lysozyme. (c) BP plots of IR $[\alpha \left(\nu \right)/{\nu }^2$ (red)] and Raman $[{\chi }^{\prime \prime }\left(\nu \right)/\nu$ (green)] spectra.