*2.3. 1H T1*<sup>ρ</sup> *Study*

To elucidate dynamic processes for all the studied compounds, the temperature dependencies of spin lattice relaxation times in the rotating frame, *T*1<sup>ρ</sup> were measured. Relaxation measurements are more sensitive to changes in molecule dynamics than spectroscopic ones [47,50]. For the studied systems, the NMR relaxation is issued mainly by fluctuating strengths of 1H–1H dipole coupling. The latter, being dependent on the relative position of the interacting nuclear spins, is altered by motional processes. As a result, this leads to fluctuations of the Larmor frequency. This process can be described through a correlation function, *G*(*t*):

$$\mathbf{G}(t) \, = \langle \Delta \boldsymbol{\omega}(0) \cdot \Delta \boldsymbol{\omega}(t) \rangle \, \, = \, \mathbf{G}(0) \cdot \mathbf{g}(t) \tag{4}$$

where the brackets represent the ensemble average; *g*(*t*) contains information about dynamic processes, and its exact expression depends on the spin interaction and diffusion mechanism; *G*(0) is determined by the mutual nuclear spin arrangement.

Commonly, to describe relaxation processes one uses a spectral relaxation function *j*(*ω*), which is a Fourier-transformed correlation function, *g*(*t*). In terms of *j*(*ω*), the dipole contribution to NMR spin-lattice relaxation time *T*1, a characteristic time for magnetization recovery after a perturbating pulse, can be written as follows:

$$1/T\_1 = G(0) \cdot \left[\frac{1}{3}j(\omega\_0) + \frac{4}{3}j(2\omega\_0)\right],\tag{5}$$

where *ω*<sup>0</sup> is the 1H NMR frequency. By analyzing the temperature dependence of spin lattice relaxation within an appropriate model, one can extract parameters of molecular motion, such as activation energies and correlation times. However, solids normally exhibit slower dynamics as compared to liquids. For such systems, the spin-locking technique is much more fruitful: by applying a locking field *ω*1, one can shift the minimum of the temperature dependence of the spin-lattice relaxation time towards the lower temperature in such a way that it falls within the measured temperature range [51–55]. More details can be found in Ref. [13]. At condition *ω*<sup>1</sup> *ω*0, the relaxation time can be written as

$$1/T\_{1\rho} = G(0) \cdot \left[\frac{1}{2}j(2\omega\_1) + \frac{5}{6}j(\omega\_0) + \frac{1}{3}j(2\omega\_0)\right].\tag{6}$$

For HCa2Nb3O10·*y*H2O, the upper limit of temperature is restricted by the water desorption, which according to the TG analysis (Figure 2a) occurs at *T* > 300 K. Application of the spin-locking technique helps to determine the spin motion parameters in a more accurate way.

The relaxation times *T*1<sup>ρ</sup> for the studied forms of HCa2Nb3O10·*y*H2O plotted versus inverse temperature are shown in Figure 7a–c. It should be noted that for all the studied forms within the experimental temperature range, the magnetization recovery is mainly described by a two-exponential function, with characteristic spin-lattice relaxation times *T*1ρ', *T*1ρ" differing from each other in one order of magnitude, except α-form, in which a mono-exponential behavior was observed above 200 K. Examples of the magnetization recovery curves (mono and two-exponential) are shown in Figure 8.

**Figure 7.** 1H relaxation times *T*1ρ', *T*1ρ" (**a**–**c**) and their relative contributions *A*', *A*" to the magnetization recovery (**d**–**f**) for α- (**a**,**d**), β- (**b**,**e**), and *γ*- (**c**,**f**) forms of HCa2Nb3O10·*y*H2O versus inverse temperature. The solid lines show the fitting using the KWW correlation function.

**Figure 8.** 1H magnetization recovery curves for the <sup>α</sup>-form of HCa2Nb3O10·*y*H2O at 297 and 167 K with the exponential fit (solid lines); for 167 K the line corresponds to the slow component only.

As it is clearly seen from Figure 7, depending on the hydration level, HCa2Nb3O10· *y*H2O demonstrates rather different *T*1ρ(1/*T*) behaviors. Let us first discuss the *γ*-form. The temperature dependence of *T*1<sup>ρ</sup> for the least hydrated form of HCa2Nb3O10·*y*H2O exhibits features similar to H1.83K0.17La2Ti3O10·0.17H2O [13]. However, it should be noted that the applied locking field was not sufficient to displace the minimum in the middle of the studied temperature range. This complicates the analysis of the experimental data, but the higher pulse power would heat the system excessively.

To determine the proton motion parameters, we used the Kohlrausch-Williams-Watts (KWW) model [56–58] successfully applied to H1.83K0.17La2Ti3O10·0.17H2O [13]. Commonly, the relaxation in isotropic systems like liquids is described by the well-known Bloembergen-Purcell-Pound (BPP) model [59], which supposes that the exponential function *g*(*t*) is as follows:

$$g(t) := e^{-|t| / \tau\_c} \tag{7}$$

and that the correlation time *<sup>τ</sup>*<sup>c</sup> obeys the Arrhenius law: *<sup>τ</sup>*<sup>c</sup> <sup>=</sup> *<sup>τ</sup>*<sup>0</sup> exp *<sup>E</sup>*<sup>a</sup> *k*B*T* . Here *E*<sup>a</sup> is the activation energy of hydrogen motion, *k*<sup>B</sup> is the Boltzmann constant, and *τ*<sup>0</sup> is a pre-exponential factor. The function in Equation (7) results in the following form of the spectral density:

$$j(\omega) = \frac{2\tau\_{\rm c}}{1 + \left(\omega \tau\_{\rm c}\right)^2}.\tag{8}$$

This model can be applied to solids, e.g., to describe the translational motion of hydrogen in metallic lattice, but requires some corrections to account for activation energy distribution [47,48], contribution of conduction electrons [60–62], and an exchange between different fractions [48,61,63].

In anisotropic systems, such as ionic conductors, the *T*1(1/*T*) dependence is asymmetric, and a stretched exponential KWW correlation function *g*(*t*) = *e*−(|*t*|/*τs*) *β* is more appropriate for its description. This means that the motion is correlated. These cooperative effects, similar to conduction electrons, contribute mainly at low temperatures, and the corresponding slope is reduced by β [55–58,64]. The spectral density function in this case can be written as

$$j(\omega) = \frac{2\tau\_{\rm c}}{1 + (\omega \tau\_{\rm c})^{1 + \beta}} \,\prime \tag{9}$$

with the stretching exponent β ranging from 0 to 1.

As was mentioned above, due to the system limitations for the *γ*-form of HCa2Nb3O10· *y*H2O, one cannot observe the high temperature slope of the *T*1ρ(1/*T*) (Figure 7c), and hence one cannot determine correctly the stretching exponent. That is why to estimate the activation energy we used the parameter β = 0.28, as determined forH1.83K0.17La2Ti3O10·0.17H2O [13].

As seen from Figure 7c, the fast (*T*1ρ ) and slow (*T*1ρ) components of *T*1<sup>ρ</sup> exhibit very similar temperature dependencies. Moreover, within the studied temperature range, their contributions are almost tantamount. The fitting within the KWW model results in the very close values of *E*<sup>a</sup> and *τ*0: {*E*<sup>a</sup> = 0.223(2) eV, *τ*<sup>0</sup> = 8.8(5) × <sup>10</sup>−<sup>10</sup> s} and {*E*a = 0.213(4) eV, *τ*<sup>0</sup> = 7.8(3) × <sup>10</sup>−<sup>10</sup> s} for the slow and fast components, respectively. Accounting for the 1H MAS NMR data, one can suggest that these lines correspond to the isolated interlayer H<sup>+</sup> ions or those in the vicinity of the water molecules.

For the most hydrated α-form above 200 K, the magnetization recovery is described by a single exponent, and the relaxation time *T*1<sup>ρ</sup> rapidly decreases with temperature decreasing; see Figure 7a. However, as was mentioned above, below 200 K the character of the magnetization recovery changes, and a second exponent with a longer relaxation time appears. With further temperature decreases, the values of the both short (*T*1ρ ) and slow (*T*1ρ") components do not change much; nevertheless, the contribution of the *T*1ρ" component becomes more important and achieves about 44% at 145 K; see Figure 7d. It should be noted that, according to 1H MAS NMR spectra (Figure 5a), below 200 K there is only one spectral line at about 7 ppm. Such temperature dependencies of the both

relaxation times and spectral parameters implicitly show the changes in dynamics of protoncontaining species at 200 K. To estimate parameters of the proton motion in α-form, we applied the abovementioned KWW model to the high temperature branch of the *T*1ρ (1/*T*). This results in the parameters *<sup>E</sup>*<sup>a</sup> = 0.210(2) eV, *<sup>τ</sup>*<sup>0</sup> = 9.0(1) × <sup>10</sup>−<sup>12</sup> s, which can be associated with the translational motion of H3O+. Although above 200 K the magnetization recovery curve is described by a single exponential decay, 1H MAS NMR spectra exhibit the existence of different hydrogen-containing species in <sup>α</sup>-HCa2Nb3O10·*y*H2O (H3O+, H2O and other); see Figure 6a. This suggests a fast exchange between the species involved in the translational motion. Below 200 K, with the slowing down of the translation, other types of motion (e.g., reorientation) became more important.

The relaxation times for the β-form of HCa2Nb3O10·*y*H2O, Figure 7b, show a complex temperature dependence: within the studied temperature range there are at least two characteristic points (224 and 176 K) at which the proton dynamics change. These changes can also be observed in the temperature dependence of the proton linewidth, but they are less significant.

Therefore, the state of protons and water molecules located in the interlayer space, as well as their dynamics, are determined by the level of hydration. It is noteworthy that the formation of H3O<sup>+</sup> is confirmed for the most hydrated α-form only, in which it is quite mobile even at low temperatures. For the γ-form, water molecules are not involved in the formation of hydronium ions; protons, behaving as lattice cations, can occupy at least two nonequivalent positions and participate in translational motion. The β-form is the most mysterious. The isotropic chemical shift of only the 1H spectral line indicates an exchange between different species, but it is not possible to unambiguously identify these species from the data obtained. The presence of two characteristic points on the temperature dependence of the proton relaxation time indicates that the mechanism of this exchange is temperature-dependent.
