Emergence of glassy features in halomethane crystals

TL;DR

This study reveals glassy dynamical features in perfectly ordered halomethane crystals at low temperatures, showing that disorder and vibrational anomalies similar to glasses can occur in crystalline materials.

Contribution

It demonstrates that ordered halomethane crystals exhibit glass-like vibrational properties, supported by experimental specific heat measurements and DFT calculations, challenging the notion that disorder is necessary for such features.

Findings

Presence of a broad peak in Cp/T^3 and g(w)/w^2 indicating boson peak-like features.

Observation of a linear term in Cp, typical of glasses, in ordered crystals.

Vibrational analysis shows low-energy optical modes pile up near the boson peak energy.

Abstract

Both structural glasses and disordered crystals are known to exhibit anomalous thermal, vibrational, and acoustic properties at low temperatures or low energies, what is still a matter of lively debate. To shed light on this issue, we studied the halomethane family C-Br_n-Cl_4-n (n = 0,1,2) at low temperature where, despite being perfectly translationally ordered stable monoclinic crystals, glassy dynamical features had been reported from experiments and molecular dynamics simulations. For n = 1,2 dynamic disorder originates by the random occupancy of the same lattice sites by either Cl or Br atoms, but not for the ideal reference case of CCl4. Measurements of the low-temperature specific heat (Cp) for all these materials are here reported, which provide evidence of the presence of a broad peak in Debye-reduced Cp/T^3 and in the reduced density of states g(w)/w^2 determined by means of neutron spectroscopy, as well as a linear term in Cp usually ascribed in glasses to two-level systems in addition to the cubic term expected for a fully ordered crystal. Being CCl4 a fully ordered crystal, we also performed density functional theory (DFT) calculations, which provide unprecedented detailed information about the microscopic nature of vibrations responsible for that broad peak, much alike the boson peak of glasses, finding it to essentially arise from a piling up (at around 3 - 4 meV) of low-energy optical modes together with acoustic modes near the Brillouin-zone limits.

Tables

Table 1. Table 1: C T ​ L ​ S subscript 𝐶 𝑇 𝐿 𝑆 C_{TLS} , C D subscript 𝐶 𝐷 C_{D} and C s ​ m subscript 𝐶 𝑠 𝑚 C_{sm} are the linear, cubic and fifth-power coefficients, respectively, from a quadratic fit of C p / T subscript 𝐶 𝑝 𝑇 C_{p}/T vs T 2 superscript 𝑇 2 T^{2} at low enough temperatures (see text for details). T m ​ a ​ x subscript 𝑇 𝑚 𝑎 𝑥 T_{max} is the temperature at which C p / T 3 subscript 𝐶 𝑝 superscript 𝑇 3 C_{p}/T^{3} exhibits a broad maximum and Θ D subscript Θ 𝐷 \Theta_{D} is the Debye temperature directly obtained from the cubic coefficient C D subscript 𝐶 𝐷 C_{D} of Eqn. 1 .

	$C_{TLS}$	$C_D$	$C_{sm}$	$T_{max}$	${\Theta }_D$
	(mJ/mol$\cdot$ K2)	(mJ/mol$\cdot$ K4)	(mJ/mol$\cdot$K6)	(K)	(K)
CCl4	4.5 $\pm$ 0.4	1.81 $\pm$ 0.09	0.084 $\pm$ 0.004	9.2	175
CBrCl3	5.4 $\pm$ 1.2	2.3 $\pm$ 0.3	0.21 $\pm$ 0.02	7.7	162
CBr2Cl2	7.8 $\pm$ 1.6	5.0 $\pm$ 0.4	0.37 $\pm$ 0.03	7.5	125

Equations

$$C_p=C_{TLS}T+C_D{T}^3+C_{sm}{T}^5$$

Full text

††preprint: APS/123-QED

I Introduction

Structural (conventional) glasses (SG) are amorphous solids Elliott (1990) lacking both translational and orientational long-range order, usually obtained by quenching or freezing the supercooled liquid Debenedetti and Stillinger (2001); Cavagna (2009); Ediger and Harrowell (2012). Orientational glasses (OG) consist of crystals with positional order and orientational disorder and exhibit similar glassy properties that conventional glassesSherwood (1978); Binder and Kob (2005). Both kinds of glasses share a complex atomic dynamics that cannot be described by the standard wave-like oscillations (phonons) observed in crystalline solids. Such complexity modifies the low-frequency excitations (the low-frequency vibrational density of states, VDoS, $g(\omega)$) and thus results on thermodynamic and transport anomalies with respect to their well-known crystalline counterparts.

The leading macroscopic property characterizing those thermal anomalies in glasses is the specific heat ($C_{p}$). In the usual Debye-reduced representation in terms of $C_{p}(T)/T^{3}$ as a function of $T$, a hump appears with a maximum at low temperatures (typically between 4 and 12 K) deviating from the horizontal line predicted by the Debye theory. There is nowadays a broad consensus that this heat capacity excess comes from additional low-frequency phonon-like excitations Buchenau et al. (1986); Bagglioli and Zaccone (2018), which is evidenced in plots of $g(\omega)/\omega^{2}$ by a correlated broad peak at $\omega_{BP}\approx 2-4$ meV (ca. 1 THz), the so-called boson peak (BP) Phillips (1981). In addition, the specific heat at low temperatures also displays a characteristic property of glasses: the apparently universal existence of tunneling two-level systems (TLS). The systematic observation of a linear term at low temperature in $C_{p}$ for non-metallic glasses Zeller and Pohl (1971); Phillips (1981) was soon ascribed to the ubiquitous presence of a random distribution of TLS as proposed by the Tunneling Model Phillips (1972); Anderson et al. (1972). These and some other features, such as the strong contribution of TLS to thermal and acoustic properties around 1$-$2 K, or the universal plateau both in the thermal conductivity and the acoustic internal friction at about a few K, have been considered as inescapable properties of structural glasses (SG) first Phillips (1981) and, afterwards, also of orientational glasses (OG), either mixed crystals such as alkali-halide/alkali-cyanide crystals Knorr and Loidl (1985); De Yoreo et al. (1986); Höchli et al. (1990); Watson (1995) or orientationally-disordered crystals (called “glassy crystals”) obtained by quenching high lattice symmetry plastic crystals with rotational disorder Bonjour et al. (1981); Ramos et al. (1997); Talón et al. (2002); Vdovichenko et al. (2015); Vispa et al. (2017). Furthermore, some of these glassy anomalies have also been reported in other disordered solids, including quasicrystals Lasjaunias et al. (1997); Pohl et al. (2002), some kind of incommensurate modulated crystals with broken translational periodicity Reményi et al. (2015), and low-dimensional organic crystals with minimal amounts of disorder Gebbia et al. (2017); Romanini et al. (2012). All such disparate systems share some type of frustration, namely the presence of competing interactions of energetic or entropic nature which make it extremely difficult the attainment of a ground state, thus leading the system to get trapped into long lived metastable states. Some recent works have associated the excess low-frequency modes in glassy systems with transverse-like vibrations Shintani and Tanaka (2008); Schirmacher (2006); Zhang et al. (2017); Chumakov et al. (2011) through the argument that transverse vibrations are more sensitive to the lack of order than longitudinal ones, while other Rufflé et al. (2006) attributed the boson peak to the Ioffe-Regel limit of longitudinal phonons. Another approach is provided by the Soft-Potential Model (SPM), which postulates a coexistence of sound waves and quasilocalized modes, either tunneling TLS or soft vibrational modes Karpov et al. (1983); Buchenau et al. (1992); Parshin (1994); Ramos and Buchenau (1998).

Here we report on results pertaining a system composed by globule-like molecules which because of the weak and nearly spherical intermolecular interactions at play cannot easily Yamamuro et al. (2003); Chua et al. (2016) form structurally amorphous phases but rather form rotator-phase crystals Criado et al. (2000) which exhibit phenomena quantitatively close to those shown by fully disordered solids. The materials under study are members of the halomethanes family, namely CBrnCl4-n with $n=0,1,2$. They present orientationally disordered (OD) face-centered cubic or rhombohedral (plastic) crystalline phases near below room temperature Ohta et al. (1995); Tamarit et al. (2008); Zuriaga et al. (2009, 2012). As revealed by neutron diffraction Binbrek et al. (1999), on cooling from the OD phases, all of them transform into low-symmetry crystals (monoclinic C2/c, with $Z=32$ molecules per unit cell and an asymmetric unit with $Z^{\prime}=4$ molecules) below 200-220 K down to the lowest temperatures Tamarit et al. (2008), hence devoid a priori of any kind of translational or orientational disorder. Nevertheless, quasi-tetrahedral CBrnCl4-n molecules exhibit (for $n\neq 0$) statistical “occupational disorder” between Cl and Br atoms Ohta et al. (1995); Zuriaga et al. (2012, 2009), which persist down to $\sim 90$ K where a calorimetric transition much alike that exhibited by the canonical glass-transition signals the transition into a frozen disordered state where molecular reorientations cannot be detected with the available experimental means Ohta et al. (1995).

The dynamics of such disorder in halomethane crystals has been studied by means of dielectric spectroscopy (for $n\neq 0$) and nuclear quadrupole resonance (NQR) spectroscopy (for $n=0$) Zuriaga et al. (2009, 2012). Both techniques and the help of extensive molecular dynamics simulations have demonstrated the existence of large-angle rotations of tetrahedra about their higher molecular symmetry axes (CBrCl3, and CBr2Cl2 with C3v and C2v point-group symmetries, respectively) due to the statistical occupancy of 75% for Cl and 25% for Br atoms in the case of CBrCl3 Parat et al. (2005); Caballero et al. (2017), and 50% for Cl and 50% for Br atoms in CBr2Cl2 Barrio et al. (2008). As for CCl4 (Td point-group symmetry), results have revealed identical dynamics of the monoclinic phase as a function of temperature Zuriaga et al. (2009, 2011) and, consequently, the four nonequivalent molecules of the asymmetric unit ($Z^{\prime}=4$) perform reorientational jumps each one with slightly different time scales due to their different crystalline (and well-defined) environment. Moreover, it was found that the molecules sitting on nonequivalent crystal sites exhibit dynamics characterized by different timescales. Even more surprising, the results from dielectric spectroscopy revealed relaxation patterns for the Bromine-Chlorine-Methane crystals astonishingly close to those exhibited by glass-forming materials near the glass transition. Therefore, these structurally-ordered crystals seem to present dynamical heterogeneity Richert (2002); Berthier et al. (2011); Caballero et al. (2017); Zuriaga et al. (2012); Caballero et al. (2016); Zuriaga et al. (2011).

Although the occupational disorder should be absent in the reference case of tetrachloromethane, CCl4 Hicks et al. (1944); Atake and Chihara (1971); Bagastkii and Manzhelii (1971), the same dynamics has been strikingly revealed by NQR measurements Zuriaga et al. (2009) and molecular dynamics simulation Caballero et al. (2016). On the other hand, thermal conductivity measurements Krivchikov et al. (2015) in the same three halomethane crystals have shown an absence of the characteristic glassy anomalies at moderately low temperatures. The thermal conductivity of these molecular crystals was found to display two distinct temperature regions, the boundary of which roughly coincided with the abovementioned glass-like transitions. The absence of the ubiquitous plateau in the thermal conductivity of glassy systems was interpreted as a consequence of a large mismatch between frequencies characteristic of heat-carrying phonons and those expected for reorientational jumps. The aim of this work is thus to investigate whether a thermodynamic correlate of the dynamic anomalies already reported on could be found in the form of universal glassy anomalies at low temperatures/frequencies in the halomethane compounds CBrnCl4-n ($n=0,1,2$) and, in particular, for the reference case $n=0$, which shows the same dynamical properties but without exhibiting crystallographic occupational disorder. Then, measurements of the specific heat $C_{p}$ at low temperatures have been carried out as well as measurements of the frequency distribution $g(\omega)$ or vibrational density of states (DoS) amenable to experiment. In addition, and to provide us with a reference state onto which the experimental data can be referred to, fully atomistic calculations for a fully ordered monoclinic CCl4 crystal structure have been carried out by means of Density Functional Theory (DFT). Such results will enable us not only to compare the experimental $g(\omega)$ to that resulting from calculation but also to assign the most salient features of the observed distribution to well defined physical entities present in the ordered ground state. In fact, the crystalline nature of the reference monoclinic phase of CCl4 enables us to monitor the behavior of the crystal lattice phonons, which, contrary to the case of amorphous materials are here well defined entities all along the Brillouin zone Zuriaga et al. (2011). We will show that glass-like features such as a broad peak at around 7$-$9 K in $C_{p}/T^{3}$ and around 3$-$4 meV in $g(\omega)/\omega^{2}$ (similar to the BP of glasses), as well as a linear contribution to the specific heat at lower temperatures associated with tunneling two-level systems, are in fact observed, all of them larger in CBr2Cl2 than in CBrCl3. Most strikingly, we have also observed such similar features in the specific heat and in the density of states of crystalline CCl4. As we will demonstrate, this unexpected and provocative experimental finding cannot be attributed to a secondary source of occupational disorder brought about by the random natural isotopic distribution of 75% 35Cl atoms and 25% 37Cl atoms. The dynamic distortion introduced by this kind of isotopic disorder is not enough to make appear the universal set of low-energy excitations typical of the glass state. By using DFT calculations for the CCl4 case, we will show that the abovementioned broad peak, arises from a piling up (at around 3$-$4 meV) of low-energy optical modes together with acoustic modes near the Brillouin-zone boundaries.

II Materials and Methods

II.1 Sample preparation

Samples of CCl4 and CBrCl3 were obtained from Across with purity better than 99% and used without further purification. CBr2Cl2 was purchased from Aldrich with a purity of 95% and was twice fractionally distilled.

II.2 Specific-heat measurements

The heat capacity of the studied samples was measured in a versatile calorimetric system Pérez-Enciso and Ramos (2007), especially designed for glass-forming liquids. Firstly using liquid nitrogen as thermal sink, we were able to concurrently measure the absolute heat capacity and to characterize the phase transitions in the range 77$-$300 K (data not shown here) employing a self-developed quasiadiabatic continuous method Pérez-Enciso and Ramos (2007); Pérez-Castañeda et al. (2013). Once the respective monoclinic crystals were obtained at approximately 221 K ($n=0$), 234 K ($n=1$), and 254 K ($n=2$), and cooled down to 77 K, liquid nitrogen was replaced by liquid helium as thermal bath in the experimental cryostat, and accurate measurements of the heat capacity were conducted between 1.8 K and 25 K by the thermal relaxation method. Many more details about the cryogenic system employed, electronic control, thermal sensors and heating elements can be found in Pérez-Castañeda et al. (2013). The calorimetric cell is essentially a thin-walled vacuum-tight closed copper can where the liquid sample has been previously inserted and carefully weighed. The calorimeter set-up is formed by a copper ring, which acts as thermal sink, and a sapphire disk suspended by nylon wires from the ring. An additional copper wire (chosen as to provide thermal relaxation times of tens of seconds) acts as thermal link between the sapphire substrate and the copper-ring reservoir. The latter has a germanium thermometer and a 50 $\Omega$ resistor (heater) that allow temperature control and increasing its temperature. In the present work, the sample cell was put on a sapphire disk and attached to it with a tiny amount of Apiezon vacuum grease to ensure good thermal contact. On the sample cell, a carbon ceramic sensor thermometer (CCS A2) is installed, also attached by Apiezon vacuum grease, and another heater (a resistor chip of 1 k$\Omega$), is put on the sapphire substrate for heating the calorimetric cell at a given fixed thermal reservoir temperature. This calorimeter set-up is firmly attached to an insert, which is sunk in the cryogenic liquid within a cryostat. With both a rotary pump and a diffusion pump a vacuum of $10^{-8}$ mbar in the internal chamber is reached. Thermometers and heaters are controlled by employing a LakeShore 336 Temperature Controller, a Keithley 224 Current Source and a Keithley microvoltmeter. A home-made program is used to automatically run the measurements. The addenda of the calorimeter using an empty copper cell was independently measured and thus subtracted from the obtained heat-capacity data. In each case, small corrections due to slight differences in the carefully measured masses of Apiezon grease, low-temperature varnish or copper in the cell were made, since their specific-heat curves are known.

II.3 Inelastic neutron scattering measurements

Inelastic neutron scattering measurements were performed using TOSCA and MARI spectrometers at the ISIS Pulsed Neutron and Muon Source of the Rutherford Appleton Laboratory (Oxfordshire, UK). TOSCA indirect geometry time-of-flight spectrometer Pinna et al. (2018a, b, 2017); Pinna, Roberto S. et al. (2015); Parker et al. (2014) is characterized with high spectral resolution ($\Delta E/E\sim 1.25\%$) and broad spectral range ($-24:4000$ cm*-1*). The samples ($2–3$ g) were placed in thin walled and flat aluminum cans. To reduce the impact of the Debye$–$Waller factor on the observed spectral intensity, the sample chamber was cooled to approximately 10 K by a closed cycle refrigerator (CCR) and the spectra were recorded for 6 to 12 hours. The direct geometry spectrometer MARI has continuous detector coverage from $3.5-135$ deg. The samples were loaded into an aluminum sample holder with an annular geometry. Using a top loading CCR system with temperature range of $5-600$ K the samples were cooled down to 5 K. The inelastic neutron spectra of monoclinic phases were taken at this temperature with an incident energy of 18 meV, selected using a Fermi chopper system with a Gd foil chopper pack rotating at 200 Hz. The chosen configuration of the instrument ensured an elastic line resolution of $3\%$ $\delta E/E_{i}$ and a $|Q|$ range $\approx 0.4-6$ $\AA^{-1}$.

II.4 DFT calculations

Calculations of the generalized frequency spectrum $g(\omega)$, also known as vibrational density of states (VDoS), have been carried out for the reference monoclinic phase of CCl4 using the CASTEP DFT code Clark et al. (2005) for a crystal lattice structure comprising one half of the 32 molecules in the unit cell. The contribution of isotopic effects on the chlorine atoms was evaluated and found to be of scant relevance (see Supplemental Material at [URL will be inserted by publisher]). Results here reported on correspond to an isotopic mixture of 35Cl and 37Cl corresponding to the natural abundance of chlorine (75% and 25%, respectively). Different exchange-correlation functionals Kohn and Sham (1965); Grimme (2006) have been tested by calculation of the frequency spectrum at the crystal $\Gamma$-point and compared with spectra measured by neutron techniques as described below. A GGA functional supplemented with a dispersion correction PBE-G06 was found to provide the best results. The calculation of the full set of dispersion branches has been carried out spanning the full Brillouin zone (see Supplemental Material at [URL will be inserted by publisher] for DFT phonon calculation details)Monkhorst and Pack (1976); Payne et al. (1992); Refson et al. (2006); Xavier Gonze and Teter (1994).

III Results and Discussion

The heat capacity of the different studied substances was measured below room temperature employing a calorimetric system especially designed for liquid samples at room temperature that solidify in situ whereas being cooled and measured. The expected liquid $\rightarrow$ plastic crystal(s) $\rightarrow$ monoclinic crystal transitions in CBrnCl4-n ($n=0,1,2$) Tamarit et al. (2008); Zuriaga et al. (2012); Ohta et al. (1995); Hicks et al. (1944) and reverse when increasing temperature were observed and monitored by means of quasiadiabatic continuous calorimetry. Then, measurements of the heat capacity were performed between 1.8 K and 25 K by the thermal relaxation method.

Low-temperature specific-heat data obtained for the three substances CBrnCl4-n ($n=0,1,2$) in their monoclinic phases are shown in Fig. 1(A). They are presented as $C_{p}/T^{3}$ vs $T$ to emphasize the deviation from the expected Debye behavior at low temperature $C_{p}\propto T^{3}$ valid for insulating stable crystals. The corresponding Debye coefficients $C_{D}$ (Table 1) are also indicated there by horizontal dashed lines. Earlier published data for CCl4 at not so low temperatures Hicks et al. (1944); Atake and Chihara (1971); Bagastkii and Manzhelii (1971) are also shown for comparison. CBr2Cl2 and CBrCl3 exhibit clear maxima in $C_{p}/T^{3}$ at about 7.5 and 7.7 K, respectively, that show a clear resemblance with the ubiquitous BP observed in most glasses, including above-mentioned OG or glassy crystals. Unexpectedly, CCl4 also exhibits a noticeable and similar peak, though clearly smaller and shifted to 9.2 K. Nonetheless, its size, shape and position are very similar to those found in many glasses and different from the $C_{p}/T^{3}$ narrower maxima observed at about 12$-$30 K in many crystals Chumakov et al. (2011) due to van Hove singularities (see Figs. 3.3 and 3.10 in Ref. Phillips (1981) and Refs. Ramos et al. (1997); Talón et al. (2002, 2001); Hassaine et al. (2012), as well as Fig. 2 below).

The deviation from the expected Debye behavior of $C_{p}/T^{3}$ at low temperature of these insulating crystals is emphasized in Fig. 1(B) which plots $C_{p}/T$ vs. $T^{2}$. A glance for the three materials immediately reveals a significant departure, without any possible resemblance with a crystalline counterpart, which dominates at temperatures below 1$-$2 K Zeller and Pohl (1971); Phillips (1981). As said above, the thermal excitations producing this typical glassy feature (and other ones in thermal conductivity or in acoustic and dielectric properties) Phillips (1981) are long thought to be tunneling TLS Phillips (1972); Anderson et al. (1972), although some criticisms about the precise interpretation of these universal low-energy excitations have been posited long ago Yu and Leggett (1988). It has been shown Ramos (2004) that the best way to assess quantitatively the TLS contribution is to conduct a quadratic fit in the appropriate temperature range below the $C_{p}/T^{3}$ maximum by using:

[TABLE]

where $C_{TLS}$, $C_{D}$ and $C_{sm}$ coefficients account for the contributions from the TLS, Debye acoustic modes and additional “soft modes”, respectively, following the phenomenological Soft-Potential Model (SPM) Ramos and Buchenau (1998); Ramos (2004).

The results from these fits are shown in Table 1, also including the correspondingly obtained Debye temperatures $\theta_{D}$. It should be stressed that the definition of the Debye temperature $\theta_{D}$ depends on the number of vibrating particles in the Debye lattice. Although only acoustic phonons should be taken into account to evaluate the genuine Debye temperature or frequency of a solid, often in the literature other more loosely ways of counting active phonons are considered, thus providing very different, usually much higher values. A direct comparison between $\theta_{D}$ values reported by different authors is therefore often misleading. Table 1 provides the usual atomic Debye temperatures (see Supplemental Material at [URL will be inserted by publisher] for discussion of different approaches of Debye temperature)Ashcroft and Mermin (1978); Kittel (1996); Hassaine et al. (2012); Ohta et al. (1995); Krivchikov et al. (2015).

The data listed in Table 1 show that all the three compounds display relatively large values for the $C_{TLS}$ coefficient if compared with the Debye term. The ratio $C_{D}/C_{TLS}$ increases from 0.4 to 0.64 as the molecular mass is increased and the molecule point group symmetry is lowered. In contrast $C_{D}/C_{sm}$ shows a decreasing value with increasing molecular mass.

Data displayed in Fig. 2 emphasizes the deviation from harmonic, Debye behavior as manifested by an upturn of the measured quantity for temperatures below the minimum (corresponding to temperatures below 3 K), an upturn without any possible resemblance with a crystalline counterpart (as shown for ethanol and glycerol crystals), as well as the similar shape for the BP of different glasses of different nature.

Inelastic neutron scattering (INS) measurements were undertaken by means of the TOSCA and MARI spectrometers at ISIS (details can be found above).

Fig. 3(A) displays a comparison of the $g(\omega)$ frequency distribution derived from the DFT calculation with the $S(Q,\omega)$ spectrum measured using the TOSCA spectrometer at 10 K for CCl4 Dymkowski et al. (2018). The spectrum extends up to 124 meV although all features above some 12 meV contain contributions from internal molecular high-frequency modes which are beyond the scope of the present study. The spectral region of interest thus comprises frequencies below some 10 meV. The calculated spectrum comprises purely harmonic, one-phonon, single scattering events only, and is not affected by sizeable resolution or attenuation (i.e. Debye-Waller) effects as it is the case for the experimental data. Such effects account for most of the significant differences between experiment and calculation. An improved measurement of the low-frequency part of the distribution is provided in Fig. 3B which compares the results with measurements carried out using a chopper spectrometer (MARI). The region below 10 meV should therefore comprise all the relevant features as far as glassy dynamics is concerned.

The Debye-reduced $g(\omega)/\omega^{2}$ in Fig. 3B shows peaks at frequencies $\omega_{max}$ around 4.3, 3.3 and 3.1 meV for $n=0,1,2$, respectively. These peaks in the VDoS are in very good agreement with the corresponding peaks at $T_{max}$ in $C_{p}/T^{3}$, since they follow the expected numerical factor around 4$-$5 between them, i.e., $\hbar\omega_{max}=(4-5)k_{B}T_{max}$, as occurs for the BPs in glasses Chumakov et al. (2009); Carini Jr et al. (2016).

The ab-initio lattice dynamics calculations using DFT for the reference case of CCl4 are shown in Fig. 4. In the first Brillouin zone, in addition to the ubiquitous acoustic phonon branches, there are 477 optical phonons (see Figs. 4A and 4C). The most striking feature concerning Fig. 4 regards the large number of branches lying between $\approx 2$ meV and $\approx 9$ meV as well as the manifold of excitations which appear in the calculation as a broad band centred at about 89 meV (not shown) which corresponds to a resolved multiplet in the experimental spectra centered at some 96 meV. It arises from the triply-degenerated $\nu_{3}$ band shown by the free molecule, comprising three out of the nine internal molecular modes which represent motions where the molecular center of mass gets displaced from its equilibrium position. The number of branches within 3$-$9 meV contrasts with the expectancy of a far more reduced number of motions of librational or librational-translational character expected for known, fully ordered molecular crystals Decius and Hexter (1977), although it seems to be a common feature for materials of lower dimensionality Mayer et al. (2004). The natural compound C35Cl${}_{3}{}^{37}$Cl, with C3v molecular point symmetry and with an exchange between sites occupied by 35Cl and 37Cl, that makes it similar to the CBrCl3 case, qualitatively displays the same dispersion relations that pure C35Cl4, giving rise only to an overall lowering of the energy (softening) of acoustic and optical modes (see Supplemental Material at [URL will be inserted by publisher]), but with a non-essential effect in the VDoS behavior. It can also be noticed (see Fig. 4A) that the low-energy (soft) modes at about 3$-$4 meV, responsible for the BP-like, mainly arise from the lowest optical modes that merged with Brillouin-zone-boundary acoustic phonons. The observed “piling up” of optical modes makes it impossible to rationalize the BP-like feature in terms of a modification of the van Hove singularity as some precedent works proposed Chumakov et al. (2011).

The characteristic frequencies at $Q=0$ for the set of the thirteen lowest energy branches are bounded below 3.55 meV, and comprise a group of ten optical branches with energies down to 2.33 meV. To deepen into the character of such low-frequency modes, the mode eigenvectors resulting from the solution of the eigenvalue problem have been analyzed in some detail. The result of such an exercise is best gauged by means of direct visualization of the time dependence of molecular motions within the crystal lattice as given by a set of motion pictures (see Supplemental Material at [URL will be inserted by publisher] for movies). A glance to such graphical representations reveals the following. First, as it could be expected, all such modes show a mixed rotation-translation character also including minor molecular deformation contributions. Furthermore, the reorientational motions do involve significant displacements of the molecular centers of mass, a finding which is glaringly exemplified by the complete lifting of the degeneracy of the higher-frequency $\nu_{3}$ band of individual molecules. Such results vividly exemplify how strong translational-rotational coupling dominates molecular motions with a crystal lattice even for considerably large frequencies. A result which merits to delve attention at, concerns the three lowest frequency motions, although such features are also found at higher frequencies. The analysis of the mode eigenvectors for such motions has revealed that the angular excursions of single molecules are coupled to highly cooperative out-of-phase displacements of pairs of crystal planes. Finally, the highly cooperative character of the set of optical modes here considered is progressively lost as the mode energy increases while molecular deformational components increase in importance. The result may appear somewhat surprising since such modes have energies some 22 meV below that of the lowest-lying normal mode of an isolated molecule.

In Fig. 5, the Debye-reduced specific-heat data of crystalline C35Cl${}_{3}{}^{37}$Cl, $C_{p}/T^{3}$, scaled to its height value, are compared to the calculated values by DFT, where the contribution of acoustic phonons and optical phonons to the specific heat is depicted separately, besides showing the total.

It is important to stress that the excellent agreement with experimental data of our DFT calculations for the $C_{p}/T^{3}$ BP-like shape and position (Fig. 5) is not the result of any fit nor any temperature scaling. Moreover, the consistency of both experimental INS data and DFT calculations is also confirmed here by their mutual agreement in the resulting VDoS, $g(\omega)$, as shown in Fig. 4B. In particular, DFT results enables to separate the contribution of the lowest frequency modes (three acoustic and the three lowest lying optical branches) from the manifold of modes above 3 meV to the specific heat, revealing that the dynamics of a perfectly ordered, harmonic crystal accounts for the measured specific heat down to some 6 K. Finally, it could be remarked that we are finding essentially the same set of glassy features ubiquitously observed in non-crystalline solids of disputed origin. Nonetheless, in our case the low-frequency vibrational modes producing the BP-like feature are identified as genuine low-lying optical phonons within a genuine Brillouin zone merged with near-boundary acoustic phonons, producing equivalent effects. The current results are thus of relevance for current discussions on the dynamics of glassy matter. The comparison of the experimental data for $C_{p}$ with those calculated under the harmonic approximation using the VDoS derived from the lattice dynamics results show that features appearing in plots of $C_{p}/T^{3}$ such as well defined maxima, sometimes dubbed as ”boson peaks”, arise in the current case, from the dense manifold of optic-like branches above some 2$-$3 meV for CCl4 as well as for the occupationally disordered monoclinic phases of CBrCl3, and CBr2Cl2. The access to detailed microscopic information provided by the DFT calculations concerning rather low-frequency optic modes revealed a strikingly strong coupling of molecular reorientations with out-of-phase displacements of crystal planes. We would expect that some remnants of such motions will be present in the real, dynamically disordered crystal and thus provide a strong coupling mechanism of molecular reorientations to low-frequency acoustic phonons. Such finding may thus be of relevance for ascribing a microscopic origin to motions giving rise to the observed anomalies in the specific heat.

Finally, it should be mentioned that the (harmonic) DFT lattice dynamics calculations cannot describe the upturn of the $C_{p}/T^{3}$ at low-temperature (below 2 K) corresponding to tunneling TLS. This upturn, experimentally evidenced for the three title compounds, is also a glassy fingerprint of the here presented crystals.

IV Conclusion

In summary, the major finding of our work is that the three studied CBrnCl4-n monoclinic crystals exhibit the characteristic glassy behavior at low temperature, namely a broad (“boson”) peak in both $g(\omega)/\omega^{2}$ and $C_{p}/T^{3}$, and a non-zero linear term $C_{TLS}$ well beyond experimental error, which increase with the number $n$ of Br atoms per molecule, $n=0,1,2$. For CBrCl3 and CBr2Cl2, a random occupancy of the halogen atoms appears to distort the dynamical network as to produce these low-frequency excitations. The freezing-in of this atomic disorder appears to be the origin of the glassy signatures as those emerging in canonical, structural glasses. Even more surprisingly, the subtle dynamical disorder involving the chlorine atoms in the “well-ordered” low-temperature phase of CCl4 is able to produce the same glassy phenomenology, though to a lesser extent. Being CCl4 a fully-ordered crystal, we have been able to determine and study the dispersion relations for acoustic and optical phonons over the whole Brillouin zone. The picture that emerges portrays molecular motions within this material as able to couple to the acoustic field up to surprisingly large frequencies. Such coupling leads to a lift of the degeneracy of individual crystal and molecular modes giving rise to a dense mesh of optic branches at low and even relatively high frequencies. The former are shown to give rise to the peak in $C_{p}/T^{3}$, as well to a peak in the frequency distribution $g(\omega)/\omega^{2}$. Although this conclusion is applicable stricto sensu only to the studied CCl4 crystal, the universality found in the glassy anomalies makes it very likely that some lessons or hints can be drawn from these findings to the much debated issue of the boson peak and other glassy anomalies in structural glasses and disordered crystals. Moreover, our results are at variance with some of the recent proposals Chumakov et al. (2011) which suggest the attribution of glassy features to van Hove-like singularities for acoustic phonons.

Acknowledgments

The authors are grateful for the financial support received within the projects FIS2017-82625-P and FIS2017-84330-R from MINECO. We also acknowledge the Generalitat de Catalunya under Project 2017SGR-042 and the Autonomous Community of Madrid through program NANOFRONTMAG-CM (S2013/MIT-2850). One of us (M.A.R.) also acknowledges the “María de Maeztu” Program for Units of Excellence in R&D (MDM-2014-0377).

The authors gratefully acknowledge the Science & Technology Facilities Council for financial support for this project, including personnel training and visits to the ISIS Molecular Spectroscopy Group, access to beam time at the ISIS Facility (RB910454, RB1320105), as well as computing resources from the STFC Scientific Computing Department’s SCARF cluster.