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Quantum Electric Dipole Lattice

Water Molecules Confined to Nanocavities in Beryl
  • Martin Dressel
  • Elena S. Zhukova
  • Victor G. Thomas
  • Boris P. Gorshunov
Article
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Abstract

Water is subject to intense investigations due to its importance in biological matter but keeps many of its secrets. Here, we unveil an even other aspect by confining H2O molecules to nanosize cages. Our THz and infrared spectra of water in the gemstone beryl evidence quantum tunneling of H2O molecules in the crystal lattice. The water molecules are spread out when confined in a nanocage. In combination with low-frequency dielectric measurements, we were also able to show that dipolar coupling among the H2O molecules leads towards a ferroelectric state at low temperatures. Upon cooling, a ferroelectric soft mode shifts through the THz range. Only quantum fluctuations prevent perfect macroscopic order to be fully achieved. Beside the significance to life science and possible application, nanoconfined water may become the prime example of a quantum electric dipolar lattice.

Keywords

Water Quantum tunneling Dipolar interaction Ferroelectricity Dielectric spectroscopy THz spectroscopy Fourier transform infrared spectroscopy 

1 Introduction

Geometric frustration has attracted considerable attention in classical magnetic systems; the antiferromagnetically interacting spins in a triangular arrangement illustrated in Fig. 1a are the most prominent example. In addition also hexagonal, kagome and hyper-kagome lattices, tetragonal structures, etc. have been subjected to intense studies, mainly theoretical, but also experimental. The issue becomes more intriguing in the case of quantum spin systems when geometrical frustration and quantum fluctuations may prohibit the formation of long-range ordering even at the lowest temperatures. In these cases, liquid-like ground states are expected and the numerous investigations of quantum spin liquids and spin ice reflect these efforts [1, 2, 3].
Fig. 1

a Antiferromagnetically interacting Ising spins in a triangular arrangement lead to frustration. b Classical Heisenberg spins can arrange in an ordered fashion when rotated 120 with respect to each other. c The hydrogen-bonded water molecules in ice are arranged in a tetrahedral arrangement; adopted from (http://www.denniskalma.com/river/waterassolid.html)

Pioneering works on geometrical frustration date back to the 1920s, when Pauling [4] realized that the hydrogen bonds between H2O molecules in ice can be allocated in multiple ways, as sketched in Fig. 1c. A given oxygen atom in water ice is situated at a vertex of a diamond lattice and has four nearest-neighbor oxygen atoms, each connected via an intermediate proton, shown in the annexed Fig. 2. According to the ice rule, the lowest energy state has two protons positioned close to the oxygen and two protons positioned farther away, forming a so-called two-in two-out state. Although these considerations used electric dipoles, Anderson [5] mapped them to a spin model possessing an extensive degeneracy of states. Two-dimensional arrangements and many generalizations have been studied subsequently.
Fig. 2

Schematic view of water molecules in nano-sized cages of the beryl crystal lattice. a H2O molecules confined in the channels within the beryl crystal lattice. Three-dimensional and top views with the crystal plotted dark gray and the water molecules colored (oxygen green and hydrogen blue). The one-dimensional channels are arranged in a hexagonal fashion with 9.2-Å distance and contain cages in a distance of 4.6 Å. b Water molecules located within structural voids formed by lattice ions. The cages (diameter 5.1 Å) are separated by narrower bottlenecks (2.8 Å). Molecules of type I have their dipole moments (red arrows) perpendicular to the crystallographic c-axis with the plane of H2O molecules parallel to c; they can perform hindered rotations around the c-axis experiencing a six-well potential (depth A) due to the hexagonal crystal symmetry. Type-II water molecules are turned by 90 relative to those of type I due to Coulomb interactions with alkali ions (Li and Na, shown in yellow) blocking the bottleneck; their dipole moments are directed along the c-axis. c Dipole moments of type-I molecules. The moments can rotate within the planes perpendicular to the c-axis. The dipole–dipole interactions (magenta wavy lines) act between the molecular dipoles within the channels where molecular doublets, triplets, and so on are formed; the interactions between dipoles in adjacent channels are much weaker owing to their greater mutual distances. d Photograph of the typical studied beryl crystal

In magnetically ordered crystals, the ground state of the spin system is determined by exchange or Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions and the influence of the magneto-dipole interaction is only of minor importance [6, 7]. Experimental breakthroughs in trapping, cooling, and controlling ultracold gases of polar molecules and magnetic and Rydberg atoms have paved the way to investigate tunable quantum systems, where anisotropic, long-range dipolar interactions play a prominent role at the many-body level [8, 9, 10]. Very recently, it was suggested that in a quantum paraelectric hexaferrite with geometric frustration, an electric dipole liquid might evolve [11]. Frustration in dielectrics has been investigated previously [12] and also the importance of quantum fluctuations in SrTiO3, for instance, was studied intensively [13, 14, 15, 16]. The experimental realization in these perovskites, however, is far from an ideal situation.

In the present study, we place separate H2O molecules in the structural channels of a beryl single crystal so that they are located far enough to prevent hydrogen bonding, but close enough to keep the dipole-dipole interaction, resulting in incipient ferroelectricity in the water molecular subsystem. This is well-defined system that can be investigated spectroscopically and theoretically analyzed in full depth. We observe a ferroelectric soft mode that causes Curie–Weiss behavior of the static permittivity, which saturates at very low temperatures due to quantum fluctuations.

2 Confined Water Molecules

There have been attempts to place single H2O molecules inside a C60 sphere [17, 18] and study their properties. The fullerene cavity, however, is rather large and thus the water molecules still contain significant degrees of freedom. In the case of carbon nanotubes filled with water [19], the situation is even worse; ice-like layers are formed at the inner walls while additional H2O molecules behave water-like in the center. Recently, Alabrase et al. [20] placed H2O in the hydrophilic nanopores of AlPO4-54 and investigated the behavior as temperature decreases. The interaction with the pore surface causes orientation of water molecules and suppresses crystallization down to T = 173 K. It was observed that a single water layer does not freeze, but has a glassy or liquid-like structure with orientational order.

We have taken a completely different approach by isolating single H2O molecules in lattice pores of single crystals. The gemstone beryl (Be3Al2Si6O18) contains channels of 5.1 Å in diameter with constrictions of approximately 2.8 Å, as sketched in Fig. 2, leaving a cavity just large enough for a single water molecule. By placing alkali metal ions in the bottleneck, the H2O dipole moment points along c-direction (Water II). In all other cases, the strong H2O electric dipole of 1.85 Debye is oriented within the ab-plane [Water I, Fig. 2a] and rotates more or less freely at elevated temperatures due to the absence of hydrogen bonds or noticeable interaction with surrounding crystalline matrix. As the temperature is reduced, the sixfold symmetry of the surrounding channel increasingly constrains this motion. At low temperatures, however, the protons become coherently delocalized due to quantum tunneling through the energy barrier as revealed by THz [21, 22, 23, 24] and neutron scattering experiments [25]. The hydrogens are somewhat circularly spread, implying that we do not have a well-defined and directed electric dipole moment in the classical sense in undulated rings. First-principle calculations of the partial vibrational density of states reproduce these findings [26, 27].

At first glance, the situation seems similar to water or ice, where the network of hydrogen bonds prevents any long-range dielectric order: in the present case, the symmetric distribution of charge density should prohibit dielectric order. Hence, it was rather surprising to find evidence for an electrically ordered state when we conducted dielectric and optical measurements on water-containing beryl in a broad frequency and temperature range [28]. Dipolar coupling between the H2O molecules within the c-axis column but also in adjacent channels leads to collective effects.

3 Experimental Details

Beryl single crystals were grown in stainless steel autoclaves according to the regular hydrothermal growth method [29] at the temperature of 600 C and under pressure of 1.5 kbar by a recrystallization of natural beryl to a seed {5.5–10.6}. The chemical composition of the crystals amounted in (mass %): SiO2—65.79, Al2O3—17.32, BeO—13.75, Fe2O3—1.30, MnO—0.09, Li2O—0.15, and H2O—1.93, with some traces of Na2O, K2O, CuO, and MgO. Recalculation of these values to the crystallographic formula gives (Be2.988Li0.012)(Al1.865Fe\(^{3+}_{0.090}\)Mn\(^{3+}_{0.007}\)Si0.038)(Si5.971Be0.029)((H2O)0.532Li0.043) according to recipe given in Ref. [30]. The procedure can be repeated using heavy water D2O as well as DHO [31]. In order to identify water-related features in the optical spectra, we have performed experiments on dehydrated samples. To that end, we extract the crystal water from the beryl crystals by heating to 1000 C in vacuum for 24 h.

For optical measurements, the beryl crystal of about a cubic centimeter size was oriented by X-ray diffraction and cut in slices containing the crystallographic c-axis within their planes. This geometry allowed to measure the optical response in two principal polarizations, when the electrical vector of the probing radiation was parallel and perpendicular to the c-axis; referred to as “parallel polarization” (Ec) and “perpendicular polarization” (Ec) throughout the text. Different kinds of spectrometers were employed for the optical experiments: for infrared measurements, a standard Fourier transform spectrometer Bruker IFS 113v was used to measure the spectra of reflection and transmission coefficients, R(ν) and Tr(ν), respectively. The reflectivity spectra were recorded using samples of about 1-mm thickness. For transmission measurements, thin (about 100 μ m) samples were prepared. The same samples were used for the measurements at lower frequencies where a quasi-optical THz spectrometer was utilized based on monochromatic and continuously frequency-tunable radiation generators—backward-wave oscillators (BWOs) described in details in Refs. [32, 33]. Complementary experiments were conducted utilizing a pulsed THz time-domain TeraView spectrometer. It allows us to directly (without using the Kramers-Kronig analysis—a necessary procedure for Fourier transform reflection spectroscopy) determine the spectra of complex conductivity \(\hat \sigma =\sigma _{1}+\text {i}\sigma _{2}\) (or dielectric permittivity \(\hat \epsilon =\epsilon ^{\prime }+\mathrm {i} \epsilon ^{\prime \prime }\)) at frequencies 1 to 70–80 cm− 1, in the temperature interval from 5 to 300 K.

4 Results and Analysis

4.1 Single-Particle Response

Figure 3a, b displays typical broadband spectra of transmission and reflection coefficients of a beryl crystal measured for perpendicular polarization (Ec) at liquid helium temperature. The spectra contain rich sets of absorption lines identified as minima in the Tr(ν) spectra and as characteristic dispersions in the R(ν) spectra. The lowest-frequency resonance absorption is observed at around 20–30 cm− 1—the corresponding minimum is seen in the transmission coefficient of Fig. 3a [Note, for both polarizations (Ec, Ec), no additional resonances were detected at even lower frequencies down to approximately 2 cm− 1 (quantum energy 0.25 meV)]. Narrow lines at the high-frequency side, above 1000 cm− 1, correspond to the well-known intramolecular modes ν1 = 3656.65 cm− 1, ν2 = 1594.59 cm− 1, and ν3 = 3755.79 cm− 1 of the water molecule. They are indicated by arrows in Fig. 3a. In accordance with earlier measurements [34, 35], these modes are accompanied by satellite resonances due to their coupling to lower-energy H2O vibrations of caged H2O molecules. The rich structures between 100 and 1000 cm− 1 are composed by the mixed response of the crystal lattice (phonons) and of water molecules in the pores of beryl.
Fig. 3

a Spectra of transmission coefficient (sample thickness 102 μ m) and b reflection coefficient (sample thickness 1.34 mm in the infrared and 0.237 mm in the THz range) of beryl measured for the polarization Ec at T = 5 K. The arrows at the upper panel indicate frequencies of intramolecular stretching and bending modes of the water molecule. The dots in panel a correspond to measurements on the THz spectrometer and the lines in panels a and b are recorded by an infrared Fourier spectrometer. The dots in panel b indicate the reflectivity values calculated from directly measured 𝜖 and 𝜖. c Transmission coefficient spectra of beryl (sample thickness 102 μ m) measured for polarization Ec at T = 5 K before and after dehydration (annealing in vacuum at 1000 C for 24 h). The black solid line at low frequencies in panel c gives an example of model fits of the spectra (see text)

In order to extract parameters of each resonance and to trace their temperature evolution, we have analyzed the measured Tr(ν) and R(ν) spectra using the well-known Fresnel expressions for complex transmission \(\hat {\text {Tr}}\) and reflection \(\hat {\mathrm {R}}\) coefficients of a plane-parallel layer [36, 37]:
$$\begin{array}{@{}rcl@{}} \hat{\text{Tr}} = \frac{T_{12}T_{21}\exp\{\mathrm{i}\delta\}} {1+T_{12}T_{21}\exp\{2i\delta\}} \hat{\mathrm{R}} = \frac{R_{12}+R_{21}\exp\{2\mathrm{i}\delta\}} {1+R_{12}R_{21}\exp\{2i\delta\}} . \end{array} $$
(1)
Here,
$$\begin{array}{@{}rcl@{}} T_{pq}= t_{pq}\exp\{\mathrm{i}\phi^{T}_{pq}\} R_{pq}= r_{pq}\exp\{\mathrm{i}\phi^{R}_{pq}\} ; \end{array} $$
(2)
$$\begin{array}{@{}rcl@{}} t^{2}_{pq}=\frac{4({n^{2}_{p}}+{k^{2}_{p}})}{(k_{p}+k_{q})^{2}+(n_{p}+n_{q})^{2}} r^{2}_{pq}=\frac{(n_{p}-n_{q})^{2}+(k_{p}-k_{q})^{2}}{(k_{p}+k_{q})^{2}+(n_{p}+n_{q})^{2}} ; \end{array} $$
(3)
$$\begin{array}{@{}rcl@{}} \phi^{T}_{pq}&=&\arctan\left\{ \frac{k_{p} n_{q}-k_{q} n_{p}}{{n^{2}_{p}}+{k^{2}_{p}}+ n_{p}n_{q} +k_{p} k_{q}} \right\} , \end{array} $$
(4)
$$\begin{array}{@{}rcl@{}} \phi^{R}_{pq}&=&\arctan\left\{ \frac{2(k_{p} n_{q}-k_{q} n_{p})}{{n^{2}_{p}}+{k^{2}_{p}}- {n^{2}_{q}} -{k^{2}_{q}}} \right\} \end{array} $$
(5)
are the Fresnel coefficients for the interfaces “air-sample;” indices p,q = 1,2 correspond to the following: “1” to air (refractive index n1 = 1, extinction coefficient k1 = 0) and “2” to the material of the sample (n2, k2); and \(\delta =\frac {2\pi d}{\lambda }(n_{2}+\mathrm {i}k_{2})\), where d is the sample thickness and λ is the radiation wavelength. Most of the observed absorption resonances could be described using regular Lorentzian expressions for the complex dielectric permittivity
$$ \hat\epsilon(\nu)=\epsilon^{\prime}(\nu)+\mathrm{i} \epsilon^{\prime\prime}(\nu)={\sum}_{j}\frac{f_{j}}{\nu_{j}\gamma_{j} +\mathrm{i}({\nu_{j}^{2}} -\nu^{2})} , $$
(6)
where \(\epsilon ^{\prime }(\nu )={n^{2}_{2}}-{k^{2}_{2}}\) and 𝜖(ν) = 2n2k2 are real and imaginary parts of \(\hat \epsilon (\nu )\), \(f_{j}={\Delta } \epsilon _{j}{\nu _{j}^{2}}\) is the oscillator strength of the j th resonance, Δ𝜖j is its dielectric contribution, νj is the resonance frequency, and γj is the damping. For some resonance absorptions, however, we were not able to describe the shape of the curves in Tr(ν) and R(ν) spectra with the Eq. 6. Specifically, these are the absorption lines located at 120–160 cm− 1. Satisfactory descriptions could be reached by using the expression for coupled Lorentzians when the complex dielectric permittivity is written as follows [38]:
$$ \hat\epsilon(\nu)= \frac{f_{1}({\nu_{2}^{2}} - \nu^{2} + \mathrm{i}\nu \gamma_{2}) + f_{2}({\nu_{1}^{2}} - \nu^{2} + \mathrm{i}\nu \gamma_{1}) - 2\sqrt{f_{1}f_{2}}(\alpha + \mathrm{i}\nu\delta)}{({\nu_{1}^{2}} - \nu^{2} + \mathrm{i}\nu \gamma_{1})({\nu_{2}^{2}} - \nu^{2} + \mathrm{i}\nu \gamma_{2}) - (\alpha + \mathrm{i}\nu \delta)^{2}}, $$
(7)
where j = 1,2; \(f_{j} = {\Delta }\epsilon _{j} {\nu ^{2}_{j}}\) is the oscillator strength of the j th Lorentzian with νj being the eigenfrequency, and α is the real and δ is the imaginary coupling constants. Successful results in applying Eq. 7 indicate that corresponding absorptions are not independent, as, for example, was observed for some dielectric compounds [38].

Using the least-square fitting procedure to process the Tr(ν) and R(ν) spectra with the expressions (1) through (7) in combination with the ac conductivity and permittivity spectra measured directly at the terahertz range allowed us to obtain broadband spectra of real and imaginary parts of the dielectric permittivity of the beryl crystal. Since the phonon absorption of beryl has been studied earlier (see, for example, Ref. [39, 40, 41]), we will concentrate here on the response caused by presence of water molecules. To distinguish water-related absorptions from phonon resonances, we have performed comparative measurements of transmission coefficient of a thin (100 μ m) sample before and after the dehydration, the dehydration process described above. Two characteristic transmission coefficient spectra are presented in Fig. 3c. Importantly, the absorption structures disappear in the dehydrated crystal; this fact clearly points to their connection to the H2O molecular response. (Note that all phonon resonances, like the minimum at 250 cm− 1 in Fig. 3c, stayed unchanged in the dehydrated crystals). This kind of inspection of the broadband spectra of beryl crystals allows us to identify the water-related absorption lines for both principle polarizations at all temperatures. After that, the phonon resonances have been subtracted from the spectra by setting corresponding oscillator strengths in Eqs. 6 and 7 to zero.

Finally, we obtained the temperature-dependent spectra of 𝜖(ν,T) and 𝜖(ν,T) and of the real part of the dynamical conductivity \(\sigma _{1}(\nu ,T)=\sigma (\nu ,T)=\frac {\nu }{2}\epsilon ^{\prime \prime }(\nu ,T)\)—related purely to the water molecules inside beryl. Only these spectra will be discussed for the rest of the paper. For the presentation of the results, we have chosen the spectra of the real parts of the dielectric permittivity and of the ac conductivity. We use σ(ν,T) instead of 𝜖(ν,T) because this quantity is proportional to the absorptivity and because the area under the resonance peaks seen in the σ(ν) spectra is directly connected to the intensity of the resonance, that is with the oscillator strength of the resonance
$$ f=\frac{4}{\pi}\int\sigma(\nu)\,\mathrm{d}\nu . $$
(8)
Broadband conductivity spectra of water in beryl measured for two polarizations at the lowest temperature of T= 5 K are presented in Fig. 4. For comparison, we put in the same figure spectra of the ac conductivity of the liquid water and of the hexagonal ice taken from the data of Ref. [42, 43, 44, 45]. The hatched area denotes the spectral range where the phonon resonances are observed and subtracted (as described above); this procedure introduces some uncertainty to the water-related modes parameters in the range 300–1000 cm− 1. The spectra presented in Fig. 4 are highly anisotropic, especially below 1000 cm− 1: while for the parallel polarization, there are only two absorption lines at 90 and 170 cm− 1, much richer structures are observed for the E⊥ c polarization. Here, three regions can be distinguished: a broad peak at the lowest frequency of 25 cm− 1 with two narrower resonances at its high-frequency shoulder, and two bands above 100 cm− 1 centered at approximately 150 and 400 cm− 1, each composed of several narrower peaks. Higher frequencies, ν > 1000 cm− 1, are occupied by resonances related to the internal modes of the H2O molecule. This high-frequency range is shown in more detail in Fig. 5 that has a linear frequency scale, as opposite to the graph in Fig. 4. Note that positions of the intramolecular resonances are shifted relative to their positions in free H2O molecule, in agreement with earlier measurements [34, 35]. Additionally, these resonances are accompanied by absorption peaks related to combinations of (internal) high-frequency and (external) low-frequency vibrations of H2O [34, 35]. Finally, for both polarizations, Ec and Ec, peaks at approximately 5300 cm− 1 have to be treated as combinations of the intramolecular ν1 and ν2 modes, since they are located near the frequency ν1 + ν2.
Fig. 4

a Conductivity spectra (proportional to optical absorptivity) of water and ice. Doted and dashed lines show the spectra of liquid water at T = 27 C, obtained from Refs. [42, 43, 44] and ice (T = 100 K, calculations based on Ref. [45]). b Dynamic conductivity caused by water molecules in beryl measured at T = 5 K for two principle polarizations, Ec (red) and Ec (blue). The sharp peaks above 1000 cm− 1 correspond to the free H2O intramolecular stretching and bending modes. The dielectric constant for the polarization Ec is shown by the magenta dashed line corresponding to the right axis

Fig. 5

Details of the optical conductivity data for two principal polarizations with the frequency axis in linear scale to shows H2O intramolecular modes and satellite peaks. Magenta dashed lines indicate the frequencies to show free H2O intramolecular stretching and bending modes. (Data from [21])

The presented spectra have been discussed in full detail in Ref. [21, 22, 23, 24]. Complementary investigations considered heavy water HDO and D2O as well as doping with different ions; comprehensive calculations of the vibrational spectra have been performed using density functional theory [26, 27, 31]. In addition, inelastic neutron scattering experiments combined with numerical studies [25] confirmed our findings. For these reasons, we want to move right to the main point and conclusions drawn from there. It becomes clear that the H2O molecules actually rotate within the nanocavities of the beryl crystal lattice at high temperatures. During cooling down, they become increasingly confined by the six-well potential of the surrounding beryl lattice. However, due to quantum tunneling, the positions of the protons are not well defined but actually spread out into corrugated rings. Thus, from quantum mechanical point of view, the macroscopic dipole moment of the confined water should disappear.

By looking at Fig. 4, besides the narrow vibrational peaks and side-band, for Ec, there is a pronounced low-frequency band growing in the THz range as the temperature is reduced that deserves further inspection. At these energies, no single-particle excitations are expected, but modes due to the collective response. Note, that also in the spectra of ice and water, broad low-frequency features can be identified. While these systems are dominated by the network of hydrogen bonds, in the present case of isolated water molecules, dipolar coupling is the dominant part. As there is a coupling of the water molecules, we can conclude that tunneling does not completely wipe out the H2O dipoles. With approximately 10 meV, this coupling in fact is rather strong and could eventually lead to ferroelectric order [28].

4.2 Collective Response

Broad-band dielectric spectroscopy was utilized to look for characteristic fingerprints of the ferroelectric phase due to ordering of the type-I molecules. In particular, we compared the dielectric response corresponding to the electric field vector E of the probing radiation within the plane of molecular rotation, i.e., Ec, with that for the Ec polarization. Figure 6 displays the temperature dependence of the low-frequency dielectric permittivity measured for the polarization Ec, compared with analogous data from a water-free sample and with the Ec response. In the geometry of electric field within the plane of H2O molecular rotation, the permittivity can be well fitted by a Curie–Weiss law [46]
$$ \epsilon^{\prime} = \epsilon^{\prime}(T) + \epsilon_{\infty} = C(T-T_{C})^{-1} + \epsilon_{\infty} , $$
(9)
where C is the Curie constant, TC is the Curie–Weiss temperature, and 𝜖 is the temperature-independent contribution to the permittivity from higher-frequency excitations. For Ec and for the water-free sample, 𝜖(T) is almost temperature-independent.
Fig. 6

Temperature dependence of the dielectric permittivity of a hydrated beryl crystal at low frequency of 1 Hz for Ec (blue open dots) and Ec (brown dots). For comparison, the data for a dehydrated crystal in the polarization Ec (green symbols) are also plotted. The inset enlarges the low-temperature range in a logarithmic temperature scale. The solid and dashed lines correspond to fits by Curie–Weiss (Eq. 9) and Barrett equations (Eq. 12), respectively. The Curie–Weiss parameters are as follows: TC = − 20 K and C = 255 K, and the Barrett fit parameters are as follows: TC = − 20 K, T1 = 20 K, and C = 255 K; the temperature-independent contribution to the permittivity from higher-frequency excitations is \(\epsilon _{\infty } = 7\). (data from [28])

The measurements of the THz electrodynamic response presented in Fig. 7 reveal a broad excitation with a peak frequency ν0 that gradually decreases as the temperature is reduced. This softening of the excitation together with the corresponding enhancement of the dielectric contribution
$$ {\Delta} \epsilon = f\,\nu_{0}^{-2} $$
(10)
leads to the Curie–Weiss increase (Eq. 9) of the dielectric contribution; here, f is the oscillator strength of the excitation defined in Eq. 8. For details on a second contribution, we refer to Ref. [28]. No traces of the THz soft mode and of its narrower satellites are present for the polarization direction Ec. Since the soft THz mode is absent in water-free beryl crystals and for the Ec polarization, our experiments undoubtedly probe the dynamics of interacting type-I water dipoles as sketched in Fig. 2. The Curie–Weiss dependence of the radio-frequency dielectric permittivity 𝜖(T) and the soft mode in the dielectric spectra are unambiguous signs of a paraelectric behavior expressing the ability of elementary dipole moments to become polarized by the external electric field; this is a typical precursor of a possible phase transition into a state where the dipole moments are macroscopically aligned, leading to ferroelectricity [46].
Fig. 7

Terahertz spectra of the real (a) and imaginary (b) parts of the dielectric permittivity (dots) of a hydrated beryl crystal measured for Ec at different temperatures T as indicated. The gray solid lines are fits to the data by a sum of Eq. 7 (coupled oscillators) for the broad soft mode and Eq. 6 (damped Lorentzians) for the narrow resonances above 40 cm− 1, shown by arrows in panel b. The dashed line illustrates a fit with a single Lorentzian term of the spectra at T = 300 K. Black and blue solid lines show the fits according to the six-well librator-rotator model. (adopted from [28])

More information on the collective dynamics of the confined water dipoles can be reached by fits of the spectra at different temperatures as demonstrated in Fig. 7. The narrow resonances that develop at low temperatures can be described by a sum of Lorentz oscillators as given in Eq. 6. The soft excitation, however, has a pronounced asymmetric line shape that cannot be satisfactorily described neither by Lorentzian (dashed line in Fig. 7b) nor by Gaussian profiles. Nevertheless, at all temperatures, its spectral shape can be reproduced by the model of two coupled damped harmonic oscillators given in Eq. 7, as demonstrated by the gray solid lines in Fig. 7, where the bilinear coupling is active between modes of the same symmetry. The lower-frequency component exhibits the behavior typical of a ferroelectric soft mode [46] for its dielectric contribution follows the Curie–Weiss behavior, given in Eq. 9, and its frequency fulfills the Cochran law characteristic of displacive ferroelectrics
$$ \nu_{0} \propto (T-T_{\mathrm{C}})^{1/2} . $$
(11)
The temperature dependence is plotted in Fig. 8
Fig. 8

a The temperature dependence of the dielectric contribution Δ𝜖 follows the Curie–Weiss law (9) indicated by the red line. b Over a large range in temperature, the resonance frequency ν0 of the soft mode decreases according to Cochran’s law given in Eq. 11. (modified from [47])

5 Discussion

Figure 8 summarizes the findings on the collective behavior of the H2O dipoles in the beryl crystal lattice due to their dipolar coupling: a soft mode develops and shifts towards lower frequencies as the temperature is reduced. As the spectral weight shifts to lower frequencies, its contribution to the dielectric constant increases significantly according to Eq. 10. It can be well described by Cochran’s law given in Eq. 11. The corresponding dielectric constant 𝜖(T) in static limit increases strongly. This behavior is well described by the Curie–Weiss law (Eq. 9) down to rather low temperatures. However, in the inset of Fig. 6, it was already indicated that deviations from the Curie–Weiss behavior occur for very small temperatures. The system does not become a completely ordered ferroelectrics, but quantum fluctuations lead to a saturation of the dielectric constant below T ≈ 10 K. In Fig. 9a, the quasi-static permittivity 𝜖(T) is plotted down to 30 mK, indicating the saturation below approximately 20 K. The observed behavior can be fitted by the Barrett formula that describes incipient ferroelectricity in regular lattice ferroelectrics
$$\begin{array}{@{}rcl@{}} \epsilon^{\prime}&=& \epsilon_{\infty} + C\left[\frac{T_{1}}{2}\coth\left\{\frac{T_{1}}{2T}\right\}-T_{\mathrm{C}}\right]^{-1} \\ {\nu_{0}^{2}}&=& B \left[\frac{T_{1}}{2}\coth\left\{\frac{T_{1}}{2T}\right\} - T_{\mathrm{C}}\right] .\end{array} $$
(12)
Here, T1 = 20 K sets the temperature scale where quantum effects prevent further ordering.
Fig. 9

a Temperature-dependent dielectric permittivity 𝜖(T) of a hydrated beryl crystal measured for Ec at ν = 1 kHz (dots) and its description by the Barrett formula (Eq. 12) with the parameters: T1 = 20 K, TC = − 20 K, C = 211 K, \(\epsilon _{\infty } = 7.9\). Inset: Lowest temperature (down to 30 mK) permittivity in an expanded scale. The dependence demonstrates a weak maximum around T = 2 K associated with the onset of short-range spatial ferroelectric (antiferroelectric) correlations between the dipole moments. b Temperature dependences of the dielectric contribution Δ𝜖 and frequency ν0 of the bare soft mode. The lines are fits by the Barrett formulas (Eq. 12) with the parameters T1 = 20 K, TC = − 20 K, C = 75 K, B = 22. (adapted from [28])

6 Outlook

Although the system of isolated H2O molecules in the beryl lattice remains in an incipient ferroelectric state and complete order does not develop due to quantum fluctuations, we discovered a versatile laboratory to study quantum electric dipole lattices. A few comment are in order here:
  • The speculation of Kolesnikov et al. that the macroscopic dipole moment of the H2O molecule might disappear due to the spreading by rotational tunneling inside the nanocavities [25] seems to be challenged as we do macroscopically observe an even enhanced dielectric response as the temperature decreases. Except one assumes a collective and coherent tunneling within the six-well potential. This would lead to a novel and massive ground state that extends over a rather large distance. This possibility has not been explored, not even theoretically.

  • In the particular case of the beryl crystal, the channels arrange in a triangular pattern. As inferred from Fig. 1a, frustration could be an issue; however, for purely dipolar coupling, the interaction is long range and not isotropic [6, 48]. The type-I H2O dipole crystals have the dipole moment oriented within the ab-plane and most likely arrange aniferroelectrically along the c-direction of the one-dimensional channels. The channels are arranged in a triangular pattern and ferroelectric coupling is preferential along the lines while the orientation is antiparallel in neighboring lines. Of course, at the beginning, there is no preference between the three respectively six distinct directions, leading to domains and with boundaries and stackfaults.

  • It would be interesting to explore the development of the ferroelectric order when the coupling is increased by external pressure. Is it possible to achieve complete ferroelectric order of the H2O dipoles at finite temperatures? This would be the first time that a complete ferroelectric state of water is actually realized, despite more than 15 phases of solid ice identified in the pressure-temperature phase diagram.

  • The gemstone beryl is just one of numerous natural crystals that contain cavities to host H2O molecules. Comparable investigations on cordierite (Mg,Fe)2Al4Si5O18 and water-chain bikitaite system are on the way. Going away from the triangular pattern will mainly change the domain structure.

Notes

Acknowledgements

We would like to thank all our collaborator who participated to the project over the years, and colleagues we had intense discussion with M.A. Belyanchikov, H.-P. Büchler, D.A. Fursenko, M. Fyta, V.S. Gorelik, U. Kaatze, C. Kadlec, F. Kadlec, L.S. Kadyrov, R.K. Kremer, V.V. Lebedev, T. Ostapchuk, E.V. Pestrjakov, J. Petzelt, A.S. Prokhorov, J. Prokleska, M. Savinov, G.S. Shakurov, J. Smiatek, P.V. Tomas, V.I. Torgashev, S. Tretiak, F. Uhlig, V.V. Uskov, and A. Zhugayevych.

Funding Information

The work was supported by the Russian Ministry of Education and Science (Program 5top100), MIPT grant for visiting professors and Project N3.9896.2017/BY.

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Copyright information

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Authors and Affiliations

  1. 1.1. Physikalisches InstitutUniversität StuttgartStuttgartGermany
  2. 2.Sobolev Institute of Geology and Mineralogy, SB RASNovosibirskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia
  4. 4.Moscow Institute of Physics and Technology (State University)DolgoprudnyRussia

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