Interchannel Hopping in Single Crystalline Lithium Triborate Probed

Article pubs.acs.org/JPCC

Interchannel Hopping in Single Crystalline Lithium Triborate Probed by 7Li NMR: Spin Relaxation, Line Shape Analysis, Selective-Inversion Spin Alignment, and Two-Dimensional Exchange Spectra M. Storek and R. Böhmer* Fakultät Physik, Technische Universität Dortmund, 44221 Dortmund, Germany ABSTRACT: The ion dynamics of single crystalline lithium triborate, LiB3O5, an important material in nonlinear optics, is studied using various 7Li and 11B nuclear magnetic resonance (NMR) techniques at temperatures from about 480 to 780 K in order to elucidate the apparent discrepancies underlying previous interpretations of NMR line shape analyses and results from dielectric spectroscopy. Rotating frame spin−lattice relaxation as well as line shape measurements are carried out and are combined with selective-inversion spin alignment as well as two-dimensional chemical exchange spectroscopy to track the temperature-dependent Li ion motion. From symmetry considerations the latter is clearly identified as interchannel hopping. By combining the present results with those from the published, yet so far not fully analyzed, dielectric loss spectra, it is shown how seeming differences in energy barriers hindering the ion motion and in the evolution of the distribution of correlation times can be reconciled.

1. INTRODUCTION There is a large interest in understanding the properties of crystals such as LiB3O5 (also called Li2O·3B2O3 or LBO) that are of importance in nonlinear optical equipment.1,2 Due to LBO’s favorable chemical and mechanical stability, high radiation damage resistance,3 and moderate piezoelectricity4 it is often used for higher harmonic generation particularly in the visible to ultraviolet spectral ranges, for parametric optical units, and for a large number of further laser applications in the medical as well as in the material sciences.5 It has been argued that crucial optical properties, such as the radiation damage resistance of nonlinear optical materials, are related to their electrical conductivity.6 As a lithium-containing crystal, LBO displays a strongly anisotropic cation mobility which has been studied using dielectric relaxation7−9 and via the 7Li isotope in the framework of nuclear magnetic resonance (NMR) spectroscopy.10−14 It is natural to link the anisotropic charge transport in LiB3O5 with its orthorhombic unit cell crystal (space group Pna21)15 that has been examined at room temperature16−20 and up to 650 K.21 LBO features (B3O5)− groups where two trigonally and one tetrahedrally oxygen coordinated boron atoms form a network of chains spiraling along the c-direction leaving two kinds of channels (see Figure 1), which promotes fast Li+ ion transport along them. Along the channels the shortest Li+ hopping distances are 3.13 Å, while between two adjacent channels the closest distance is 5.01 Å. This may rationalize why at 373 K the intrachannel conductivity (along the c-axis) is more than an order of magnitude higher than the interchannel motion (along the perpendicular a- and b-axes).7,9 Dielectric © 2016 American Chemical Society

Figure 1. View along the c-axis of the orthorhombic crystal structure of lithium triborate according to X-ray data taken at 293 K (lattice constants a = 8.444 Å, b = 7.378 Å, c = 5.146 Å).21 Boron (green) and oxygen atoms (red) form a rigid lattice of trigonal BO3 and tetrahedral BO4 groups providing channels along the c-axis in which two distinguishable Li sites (light and dark blue) are located. This image was created using VESTA.22

measurements carried out at T > 400 K along the a- and b-axes revealed non-Debye-type dielectric loss peaks, indicative for a broad distribution of motional correlation times.7,9 Analogous effects are not discernible from dielectric measurements along the c-direction because the strong electrical conductivity masks any underlying dielectric loss peaks. Received: February 8, 2016 Revised: March 21, 2016 Published: March 21, 2016 7767

DOI: 10.1021/acs.jpcc.6b01347 J. Phys. Chem. C 2016, 120, 7767−7777

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The Journal of Physical Chemistry C The dielectric finding of non-Debye ion dynamics contrasts with an analysis of 7Li NMR line shapes of LiB3O5 powder crystals by Matsuo et al. which display significant temperature dependence in the 500−700 K range.10 That work tacitly assumed that a correlation time distribution is absent. Although not pointed out in previous NMR work on LBO, it is of central importance to realize that in this crystal Li NMR is sensitive solely to interchannel hopping processes. This is because the local symmetries of the two crystallographically inequivalent Li sites in each channel and thus their electrical charge environments are connected by rotations about 21-screw axes. Such rotations leave the electric field gradient (EFG) at the Li nucleus and thus the corresponding NMR frequency invariant which is by far dominated by the orientation of the EFG. The sites in dif ferent channels are related to each other via mirroring operations at different glide planes. Glide plane operations generally lead, however, to distinguishable NMR frequencies. Hence, in single-crystal Li NMR only two (pairs of) resonance lines are observed that one may call Li(A) and Li(B), depending on the channel in which the nuclei reside. Apart from the work of Matsuo et al.10 all available NMR articles on LBO focus either on structural aspects using 7Li and/or 11B NMR or on the temperature range below 400 K.10−14,23 In this temperature range simple spin−lattice relaxation times are hardly sensitive to the ion dynamics in LBO but governed by single-phonon processes.12 This calls for a study using more advanced methods including spin alignment and two-dimensional NMR spectroscopy in order to examine the interchannel ion hopping in LBO. Such NMR techniques have successfully been applied for a range of other Liconducting materials using Li NMR;24−29 for reviews see, e.g., refs 30 and 31. In the present work, we focus on the spin I = 3/2 nucleus 7Li which, owing to its strong quadrupolar interaction, represents a highly sensitive probe to unravel ion hopping processes. It is important to note that the present work also exploits a novel methodological approach because in all of the previous two-dimensional Li NMR work24−26,28 exchange spectroscopy was performed using 6Li and/or sample spinning at the magic angle, thereby suppressing any orientationdependent labeling of different sites. Here, we rather exploit the orientation dependence of the quadrupolar satellite frequencies.32

integer spins like 11B the use of a φ = 180° refocusing pulse was established long ago,36,37 the choice of the flip angle for the nonselective excitation of I = 3/2 nuclei like 7Li depends on the property which is to be optimized. To maximize the echo amplitude one should set φ = 64°.38 For the present work we applied φ = 54.7° which, as shown in previous work, faithfully reproduces the 6:4 intensity ratio of the FID spectrum for the satellites and the central line.39,40 The interpulse delay was tp = 20 μs, and Fourier transformation started at the echo maximum at ta = tp. Spin−lattice relaxation times T1 and spin−spin relaxation times T2 were monitored using the sequence X90°− tp−Yφ−ta−acq as well. Rotating frame relaxation times T1ρ were recorded using a X90°−Ylock−ta−acq sequence.41 The locking frequency ω1 was determined in an independent T2ρ experiment corresponding to X90°−Xlock−ta−acq.52 We obtained ω1 = 2π × (15 ± 2) kHz. Magnetization curves of T1 and T1ρ were analyzed by means of a Kohlrausch function ⎡ ⎛ ⎞1 − μ⎤ t M(t ) = M 0 + (Mi − M 0)exp⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ T1 ⎠ ⎥⎦

(1)

Here Mi and M0 denote the initial and equilibrium magnetizations, respectively, and μ quantifies the deviation from a monoexponential relaxation. 7 Li sin−sin stimulated-echo measurements were carried out by employing the Jeener−Broekaert42 sequence (X90°−tp− Y45°−tm−X45°−ta−acq) with appropriate phase cycling.43 Spinalignment spectra were recorded using this pulse sequence for pulse-length-corrected evolution times from tp = 1.9 to 513.9 μs in increments of 8 μs. Owing to receiver overload following the last pulse, approximately 14 μs of the time signals are lost, and they were back extrapolated to ta = 0 as using the f 2 function defined in Section 3.3, below. For the spin-alignment spectra a Gaussian apodization with 800 Hz spectral width was applied, and a Fourier transformation was carried out starting from the time origin. For the two-dimensional spectra a Gaussian apodization window of 2 kHz was chosen. For the powdered sample also some cos−cos (X90°−tp−X45°−tm−X41.8°−ta−acq) echo experiments were performed. The recently invented 7Li cos−cos experiment is described in ref 39 in detail.

3. RESULTS AND ANALYSES 3.1. Quadrupole Echo Spectra. The acquisition of solidecho spectra is a well-established technique to study motions of mobile nuclei as well as fluctuations in their local environment. In LiB3O5 the available Li sites are characterized by two different EFG orientations, and hence, two distinct precession frequencies ωQ exist in a single crystal. The translational motion of mobile 7Li ions among these sites then renders the precession frequency ωQ time dependent. Here the index Q indicates that for 7Li spins the largest internal interaction is quadrupolar in nature which leads to

2. EXPERIMENTAL DETAILS Single crystals of LiB3O5, 3 × 3 × 3 mm3 in size, were obtained from Altechna as well as from Photon LaserOptik. A polycrystalline sample was prepared by grinding the crystal from Photon LaserOptik to fine powder in dry atmosphere. The powder was filled into an NMR quartz tube and then flame-sealed. All measurements were conducted on home-built NMR spectrometers. The Larmor frequencies were ωL = 2π × 149.7 MHz and 2π × 123.6 MHz for 7Li and 11B, respectively. For the 7 Li experiments the π-flip lengths were 4.5−15 μs, at temperatures below and above ∼700 K, respectively. The central-line magnetization of 11B was selectively inverted by a 12 μs long pulse. Using a high-temperature probe head33 the range from ambient to 847 K was covered. The temperature calibration was checked against the melting temperature of NaNO3.34 Solid-echo spectra of 7Li and 11B were recorded using the two-pulse sequence X90°−tp−Yφ−ta−acq with appropriate phase cycling.35 While for the selective excitation of half-

ωQ =

1 δQ (3 cos 2 θ − 1 − η sin 2 θ cos 2ϕ) 2

(2)

1

Furthermore, δQ = 2 e 2qQ /ℏ and η denote the anisotropy and asymmetry parameters, respectively, that characterize the EFG tensor at the nuclear site. Lim et al.10 reported that at room temperature δQ = 2π × (71.8 ± 0.5) kHz and η = 0.6 ± 0.1. In view of the anisotropic thermal expansion9,44 of LiB3O5 these parameters can be expected to change somewhat with temperature. The polar angle θ and the azimuthal angle ϕ 7768


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The Journal of Physical Chemistry C

In analogy to the quadrupolar broadening also dipolar interactions vanish as the fast motion of the nucleus tends to average the dipolar anisotropy. The central transition’s full width at half-maximum (fwhm) Δω was read out from the single-crystal spectra and is plotted vs temperature in Figure 3.

encode the orientation of the principal axis system of the EFG with respect to the laboratory frame. The 7Li NMR singlecrystal spectrum consists of a central line and a pair of satellite lines for each site that correspond to a differently oriented EFG tensor. The central, −1/2 ↔ +1/2, transition is not affected by the (first-order) quadrupolar interaction, and its width is typically determined by nuclear dipole interactions. Single-crystal solid-echo spectra of lithium triborate are presented in Figure 2 for temperatures in the range from 491 to

Figure 3. Widths, Δω/2π, of the 7Li central line for the single crystal (diamonds) of LiB3O5. Widths from 11B central-line NMR spectra for powdered samples are indicated by the green squares. The solid lines represent a joint fit using eqs 3 and 4. For the 11B data curves were calculated also using significantly different activation energies, EA ± 0.15 eV, and are indicated by the dashed and the dotted green lines. For details see the text. Data for powdered LiB3O5 by Matsuo et al.10 are represented by black circles.

One recognizes that upon heating the line width changes from 2π × 6.3 kHz (slow motion regime) to about 2π × 1.0 kHz (high-temperature limit). In this respect our single-crystal data essentially confirm the data obtained by Matsuo et al.10 obtained for powdered LBO. We are, however, unable to reproduce the apparent two-step behavior seen in the powder. To further check for a possible occurrence of a two-step decay we recorded 11B central-transition spectra of LiB3O5 powders (cf. Figure 3), thus monitoring the mobile ion motion “from a distance”. In general, the observed 11B line shapes are broad and featureless (not shown). They display considerable residual high-temperature broadening which hints at significant second-order quadrupolar contributions arising from the trigonal BO3 groups. Overall, the Δω(T) dependence from 11 B NMR is similar to that from 7Li NMR45 and thus also the temperatures at which the line widths Δω show inflection points. At the inflection point an estimate of the correlation time is possible according to an empirical relation46 which states that within a factor of two τ1/2 ≈ 0.3⟨δω2⟩−1/2 which yields τ1/2,Li ≈ 20 μs at 624 K (7Li NMR) and τ1/2 ≈ 10 μs at 657 K (11B NMR). For another analysis of the line width (fwhm = Δω/2π), obviously called ΔH1/2 by Matsuo et al.,10 we used their approach which assumes that a distribution of correlation times is absent. For Gaussian line shapes Δω is related to the second moment via δω2 = Δω2/(8 ln 2) which enables one to calculate the second moments δω20 and δω2∞ in the low- and in the hightemperature limit, respectively. As used by Matsuo et al.10 it is common to relate the temperature-dependent dipolar second moment δω2 of the NMR line to the correlation time τ by47 2 δω 2 = δω∞2 + (δω02 − δω∞2 )tan−1(τ δω 2 ) (3) π

Figure 2. 7Li NMR quadrupole-echo spectra of a lithium triborate single crystal at various temperatures. The external magnetic field was parallel to the (c+5°)-axis in the bc plane (with an estimated uncertainty of 3°). The evolution time was tp = 20 μs. To avoid overlap the spectra are shifted vertically. For increasing temperatures the two quadrupole frequencies on each side of the spectra merge into a single line. The dashed spectrum was calculated as described in the text39 and is seen to reproduce the major features of the experimental low-temperature spectra.

684 K. At the lowest temperature a central transition at ω = ωL and two pairs of quadrupole satellites at ωQ,A ≈ ±2π × 45 kHz and ωQ,B ≈ ±2π × 20 kHz are readily recognized. Slight undershoots close to the central resonance mark quadrupolar transients.39 Using the frequencies ωQ,A and ωQ,B and taking into account a Gaussian-like dipolar broadening of σD ≈ 2π × 2.3 kHz we modeled the NMR spectrum using eq 13 of ref 39. The result is shown as a dashed line at the bottom of Figure 2 and demonstrates that the resonance peaks as well as the undershoots are properly reproduced. At 491 K the hopping rate of ionic motion is much smaller than the width of the 7Li NMR satellite spectrum, but as the temperature increases ionic hopping motion becomes faster and leads to motional narrowing of the spectra. When the hopping correlation time, τ, among two sites A and B becomes shorter than the inverse of the difference between their NMR frequencies, τQ ≈ |ωQ,A − ωQ,B|−1, here (2πτQ)−1 ∼ 25 kHz (or τQ a few microseconds), both peaks coalesce to a single line at a residual frequency ωQ̅ ,AB ≈ 2π × 27.8 kHz (see Figure 2 for T ≥ 635 K). 7769


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The Journal of Physical Chemistry C The only free parameter in this equation is the time scale τ for the ionic jump motion which in crystalline electrolytes typically follows an Arrhenius law ⎛ E ⎞ τ = τ0 exp⎜ A ⎟ ⎝ kBT ⎠

51 and 54. In Figure 4 we present magnetization curves s(tlock) for two temperatures. Except for some scatter at short locking

(4)

To describe our line width data we used an activation energy of EA = (0.71 ± 0.03) eV and a pre-exponential factor τ0 = (1.1 ± 0.5) × 10−10 s. Figure 3 shows that using these parameters (which incidentally describe also all other measurements recorded in the course of this work) eqs 3 and 4 provide a very good description of our 7Li and 11B data if the limiting line widths (δω20 and δω2∞) are properly chosen for each set of data. The fact that the Δω(T) data are well captured hints at an absence of a distribution of relaxation timeswhich otherwise would blur the sigmoidal curve46,48,49at least for 550 K < T < 750 K, i.e., the range in which Δω is significantly temperature dependent.50 Equation 3 would not account for a kink, casting doubts on the reliability of correlation times determined previously10 for temperatures larger than 690 K. Furthermore, to test the reliability of activation energies EA obtained by eq 3, for the two fixed values EA = (0.71 ± 0.15) eV we adjusted solely τ0 to fit our Δω data. These curves are shown in Figure 3 for the 11B NMR as a green dashed line (EA = 0.86 eV, τ0 = 6 × 10−12 s) and as a dotted line (EA = 0.56 eV, τ0 = 1.4 × 10−9 s). Apparently, albeit calculated from substantially different activation energies, these lines still yield acceptable descriptions of the experimental data and thus demonstrate that without further restrictions to τ0 the parameters obtained from eq 3 are rather insensitive to the time scale of ion motion away from the inflection point in the Δω(T) curve. Therefore, in a next step (Section 3.2) we measured rotating frame spin−lattice relaxation times to probe the dynamics at temperatures of 750 K and higher. Then, in subsequent sections, we exploit spin-alignment measurements and twodimensional exchange spectra to examine the temperature range also below 550 K, at which the line shape analyses of LBO are of limited accuracy. The alignment and exchange methods will also allow us to address the as yet unresolved issue concerning the width of the dielectric spectra. 3.2. Spin−Lattice Relaxation in the Rotating Frame. Measurements of spin−lattice relaxation in the rotating frame provide a reliable means to access the time scale of ionic motions in the regime of microseconds. Detailed theoretical treatments of quadrupolar relaxation of spin-locked I = 3/2 nuclei have been given,51,52 and their applicability to 7Li has been discussed.53 In such experiments, one first generates transverse magnetization using an intense and short (few μs) nonselective 90° pulse, and afterward one monitors its relaxation under the influence of a (here: up to 50 ms) long and moderately strong radio frequency field. The remaining transverse relaxation is subsequently acquired. However, due to the finite dead time of the receiver only the magnetization originating from the central transition can be observed experimentally. It can be shown that this magnetization involves three contributions51

Figure 4. Intensities observed in T1ρ experiments as a function of the lock-pulse duration tlock at two temperatures. The solid lines represent fits to the data by stretched exponentials (μ = 0.32 ± 0.06). Lower inset: Intensities observed in a T2ρ experiment carried out at 680 K as a function of the lock-pulse duration. The data are well described by a damped oscillation with frequency 2ωlock. Upper inset: Temperature dependence of the rotating frame spin−lattice relaxation times T1ρ. The line is based on eq 6 in conjunction with eq 4. A good description of T1ρ(T) was found when setting AS in eq 6 to zero. At the temperature of the T1ρ minimum the correlation time is τρ ≈ (4 ± 2) μs.

times for tlock ≥ 100 μs the data are well described analogous to eq 1 (cf. the solid lines in Figure 4). Therefore, like in a previous study,54 it is convenient to define the rate 1/T1ρ as δQ2 ⎛ η2 ⎞ 1 = ⎜1 + ⎟[(R S)AS (RF +)AF+ (RF −)AF− ]1/ AS + AF+ + AF− T1, ρ 10 ⎝ 3⎠ (6)

Here, the rate RS = J1 + J2 depends on the spectral densities Jn(nω) = 2τ/[1 + (nωτ)2] at multiples of the Larmor frequency ω = ωL. Hence, similar to relaxation in the laboratory frame, RS displays a maximum at ωLτ ≈ 0.81, i.e., when τ is in the nanosecond regime. Slower motion is probed by the rates RF± which are sensitive to the spectral densities Jλ± = 2τ/(1 + λ2±τ2) at the effective evolution frequencies λ± = (ω2Q ± 2ωlockωQ + 4ω21)1/2 in the rotating frame. Thus, depending on the lock field ω1 and on the mean quadrupole frequency ⟨ω2Q⟩1/2 for powders the relaxation rates RF± are maximum for λ±τ ≈ 1, i.e., for 7Li typically when τ is in the microsecond regime. The strength of the lock field, ωlock, can be determined in a so-called T2ρ experiment which is similar to the T1ρ experiment except that the 90° pulse and the lock pulse now have the same relative phase (rather than a 90° phase shift between them for the T1ρ experiment). Monitoring the magnetization as a function of tlock yields a damped oscillation with frequency 2ωlock. A T2ρ measurement performed at 680 K is presented in the lower inset of Figure 4. The solid line represents a fit to the data by means of eq 1 multiplied by a cosine function. We obtain ω1 = 2π × (15 ± 2) kHz and a time constant T2ρ = 40 ± 3 μs (stretching exponent ≈ 1.2). Then, in the upper inset of Figure 4 we summarize the rotating frame spin−lattice relaxation times T1ρ. A minimum is observed at a temperature of T ≈ 750 K which implies that the rate of the Li(A) ↔ Li(B) exchange is approximately given by55 τ−1 ρ ≈ λ± ≈ 2π × (40 ± 10) kHz.

s(tlock ) = AS exp( −R Stlock ) + AF + exp( −RF +tlock ) + AF − exp( −RF −tlock )

(5)

Explicit expressions for the amplitudes AS and AF± and for the relaxation rates RS and RF± appearing in eq 5 are given in refs 7770


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The Journal of Physical Chemistry C We also determined spin−lattice relaxation times, T1 ∝ J1 + J2, in the laboratory frame and found them to be relatively long: T1 is 227 s at T = 387 K and it decreases monotonously for increasing temperatures reaching T1 ≈ 5 s at T = 829 K, the highest temperature accessible in the present study. These findings agree with those by Matsuo et al.10 However, in the absence of an observed minimum in T1(T) a reliable assessment of correlation times is difficult unless additional assumptions are made. Unambiguous determinations of τ are, however, possible using the three-pulse techniques that we exploit in the following. 3.3. Selective Inversion Spin-Alignment Spectra. In the regime of ultraslow motions, with τ in the milliseconds to seconds range, direct access to ion hopping correlation functions can be gained by means of stimulated-echo spectroscopy. Application of a suitable pulse sequence39,56 generates a spin-alignment echo which provides a measure for the correlation of the spin phases acquired during the evolution interval tp and the detection interval ta. These two intervals are separated by the so-called mixing time tm which can be adjusted by the experimenter. During tm (often chosen ≫ tp) the phase information is stored in a long-lived spin population state. Assuming optimal flip angles the explicit time-domain signal reads56 F2sin(t p , tm , ta) =

b(tp), and c(tp) allowed us to obtain f 2(tp, ta) over the full ta range. Fourier transformation then yielded the corresponding spin-alignment spectra. Figure 5 presents the spectra that we thus recorded for a very short mixing time of 10 μs (during which no exchange

9 ⟨sin[ωQ (0)t p]sin[ωQ (tm)ta]D(t p , ta)⟩ 20 (7)

Here D(tp, ta) is a function that accounts for dipolar dephasing processes, and ωQ(0) and ωQ(tm) denote the quadrupolar precession frequencies before and after the mixing time, respectively. Because molecular motion can render ωQ time dependent, monitoring the amplitude of the echo at ta = tp for variable tm yields the time scale of the motion. In eq 7 the angle brackets indicate an ensemble average covering all possible precession frequencies. To carry out a one-dimensional spin-alignment experiment one first has to identify an optimum tp at constant tm, and then one records Fsin 2 (ta) for a number of suitably chosen tm times. For the present case of single crystalline LiB3O5 exchange takes place between only two 7Li sites Li(A) and Li(B) with the precession frequencies ωQ,A and ωQ,B, respectively. Under these conditions for fixed tm, eq 7 becomes

Figure 5. 7Li spin-alignment spectra of a single crystal of LiB3O5 acquired at 539 K for increasing evolution times. The mixing time was tm = 10 μs. All peaks are subjected to a Gaussian broadening with σD ≈ 2π × 3.3 kHz. The amplitude modulation of the satellite peaks follows a dependence according to sin(ωQ,Atp) or sin(ωQ,Btp). Furthermore, the intensity of the central transition increases markedly due to a buildup of dipolar correlations on a time scale of tp ≤ 1/σD ≈ 48 μs.

processes can take place). The spectra feature two pairs of absorptive antiphase peaks at ω = ±ωQ,AB and a dispersive central peak. The goal of these experiments was to identify an evolution time at which the intensity difference of the peaks, x(tm = 10 μs) = |a(tp) − b(tp)|, is maximized in order to ascertain optimum initial conditions for the subsequent exchange experiment. One recognizes that when going from tp = 1.9 to 17.9 μs the outer satellites have become selectively inverted.58 Thus, for the following experiments an evolution time tp = 17.9 μs was chosen, and the equilibration of the peak amplitudes at ωA and ωB was tracked for mixing times ranging from 10 μs to 15 ms. Corresponding spectra are shown in Figure 6 where one recognizes that for increasing tm intensity is transferred from the outer satellite peaks (at ωQ,A) to the inner ones (at ωQ,B). Since these two frequencies correspond to lithium sites in the two different (A and B) channels, this experiment visualizes the interchannel hopping of the mobile ions most directly. At tm = 15 ms both peaks display about equal amplitudes indicating complete equilibration of the Li(A) and Li(B) sites in LiB3O5. To quantify the tm dependence of the observed populations in the next section we will analyze the ratio r(tm) = x(tm)/x(tm → 0) where x(tm → 0) was obtained from a Kohlrausch fit to x(tm). 3.4. Slow Dynamics: Two-Dimensional Exchange Spectra and Stimulated Echoes. An even more impressive visualization of the interchannel hopping is possible via twodimensional exchange spectra S(ω1,ω2). Here ω1 and ω2

f2 (t p , ta) = [a(t p)sin(ωQ ,A ta + ϕ) + b(t p)sin(ωQ ,Bta + ϕ) ⎧ ⎛ ⎛ t − t ⎞κ ⎞ ⎪ a p + c(t p)] × ⎨exp⎜⎜ − ⎜ ⎟ ⎟⎟ ⎪ * ⎩ ⎝ ⎝ T2 ⎠ ⎠ ⎛ ⎛ t + t ⎞ κ ⎞⎫ ⎪ a p + exp⎜⎜ − ⎜ ⎟ ⎟⎟⎬ ⎪ * ⎝ ⎝ T2 ⎠ ⎠⎭

(8)

Here, the terms in the square brackets contain the precession frequencies, a phase offset ϕ, and the amplitude modulation factors a(tp) and b(tp). Because dipolar contributions give rise to a central line at ω = ωL which is not accounted for in eq 7, we added the ωQ-independent term c(tp). The expression in the curly brackets represents an empirical variant of D(tp, ta).57 Next, at a temperature of 539 K, here ωQ,A = 2π × 42 kHz and ωQ,B = 2π × 23 kHz, we recorded time-domain signals f 2(tp, ta)for fixed tp = 1.9 up to 41.9 μs (in steps of 8 μs) and tmwhich were fitted with a common ϕ (close to zero) and transverse relaxation time T2*. The fitting parameters a(tp), 7771


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of the spectrum recorded for tm = 15 ms (see Figure 7(c)) reveals that the intensity at (ωA, ωA) is somewhat higher than the intensity of the peak at (ωB, ωB). To elucidate whether this finding could be caused by an unequal population of the Li sites, although not obvious from the crystal structure determinations of LiB3O5, we performed dedicated stimulated-echo experiments which are very sensitive to the populations of the Li sites. As we show in Appendix A these measurements reveal that the two channels are equally populated. From the diagonal intensity IA and the cross intensity IE of the corresponding peaks in the two-dimensional spectra acquired at 539 K (cf. Figure 7), we extract the normalized exchange intensity60 Iex(tm) = IE(tm)/[IE(tm) + IA(tm)]

(10)

and show the result as open symbols in Figure 8. One recognizes that for increasing mixing times Iex(tm) evolves from Figure 6. 7Li spin-alignment spectra for a single crystal of LiB3O5 as recorded for various mixing times at 539 K. The evolution time was set to tp = 17.9 μs. Starting at tm = 10 μs with opposing amplitudes the spin populations corresponding to the satellite resonances at ωA and ωB equilibrate to a common level as tm increases. This observation directly reflects the ion exchange among the A and B channels in this ion conductor.

correspond to the NMR frequencies of a mobile Li ion before and after the mixing time tm, respectively. Treating the sin−sin data as described in the previous section and subjecting them to a two-dimensional Fourier transformation with respect to t1 ≡ tp and to t2 ≡ ta yields

∫0

S(ω1 , ω2) = Re{

∞

∫0

i Im[

∞

sin[ωQ (0)t1]sin[ωQ (tm)t 2]

exp( −iω2t 2)dt 2]exp(−iω1t1)dt1}

(9) Figure 8. Comparison of normalized exchange intensity, Iex(tm) (cf. eq 10), from two-dimensional spectra (cf. Figure 7) and the ratio r(tm), both obtained at 539 K, as well as sin−sin stimulated-echo amplitudes measured for tp = 38.7 μs for several temperatures. The stimulatedecho curves, shown as Fsin 2 (tm)/M2(tm), were normalized to evolve from 0 to 0.5. The comparison of all data sets at 539 K demonstrates good agreement. The inset shows a stimulated-echo decay Fsin 2 (tm) prior to normalization. The solid lines represent Kohlrausch fits (cf. eq 11).

In Figure 7 we present spectra for the LiB3O5 single crystal that we thus obtained at 539 K and for mixing times of tm = 10 μs, 900 μs, and 15 ms.59 During the short mixing time of tm = 10 μs the spectrum features only quadrupolar diagonal peaks at (ω1, ω2) = (ωA, ωA) and (ω1, ω2) = (ωB, ωB) confirming that no exchange takes place. For increasing mixing times (see Figure 7(b) and (c)), exchange peaks arise. These cross peaks indicate the occurrence of interchannel diffusion. A closer view

Figure 7. Two-dimensional 7Li exchange spectra of single-crystalline LiB3O5 recorded for various mixing times tm at 539 K. All spectra were normalized to a common integrated intensity. Only that section of the spectrum is shown from which the exchange between the channels Li(A) and Li(B) is readily recognized. For increasing tm the intensity of the exchange peaks increases, while simultaneously the intensity of the diagonal quadrupole peaks decreases. 7772


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The Journal of Physical Chemistry C about 0 to 0.5. From Figure 8 one also infers that at 539 K the interchannel diffusion is characterized by a time scale of about 0.4 ms. Since at this temperature the spin−lattice relaxation of the Li probe is quite long (T1 ≈ 140 s) measurements of the type presented in Figure 7 are rather time-consuming. Therefore, in order to be able to efficiently study the ion dynamics in a wide range of temperatures, we carried out stimulated-echo experiments from 487 to 632 K to measure Fsin 2 (tp,tm,ta) (cf. eq 7) for ta = tp. As the ion hopping information is most clearly encoded in the satellite part of the sin−sin signal (see, e.g., Figure 6) we evaluated the stimulatedecho spectra in the frequency ranges of ± (35 ± 17.5) kHz, thereby excluding intensity originating from the central line. At all currently examined temperatures a two-step decay is observed in Fsin 2 (tm), and an example for such a measurement (for tp = 38.7 μs) is presented in the inset of Figure 8. As one recognizes from the solid line shown there, the data are well described by a Kohlrausch function of the form F2 = {(1 − Z)exp[−(tm/τ )β ] + Z}M 2(tm)

Figure 9. Arrhenius representation of correlation times from various NMR experiments carried out are detailed in the upper left corner of the figure. The bar at the symbol referring to the time constant determined from the T1ρ minimum indicates its uncertainty implied by the temperature deviation of the shortest T1ρ value from the fitted curve (cf. the inset of Figure 4). Symbols in the lower right corner refer to line width data by Matsuo et al.10 and dielectric data by Kim et al.7 as determined using eqs 12 and 13. The solid lines refer to Arrhenius laws, eq 4, with the given activation energies and the preexponential factors discussed in the text.

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Here, an initial and a second stretched exponential decay, M2(tm) ∝ exp[−(tm/τ2)β2], occur that are characterized by the time constants, τ and τ2, respectively. The initial decay corresponds to the interchannel diffusion, while the origin of M2(tm) which is of minor importance in the present context is not entirely clear.61 Finally, Z denotes the final-state correlation of the initial decay and can be read off from the plateau in the F2 curve. To compare the initial decay of the F2 curve with the results from the two-dimensional exchange experiments, we rescaled Fsin 2 (tm)/M2(tm) so that, analogous to Iex(tm), this ratio evolves from 0 to 0.5. The ratios r(tm) determined on the basis of the spin-alignment spectra (see Section 3.3) were normalized analogously. The outcome of this procedure is shown in Figure 8 and reveals a good agreement between the three different experimental approaches (see the closed, open, and half-filled red circles). At 539 K we checked for tp = 21.6−41.6 μs whether τ or β depend on the evolution time. Since this was not the case, we conclude that the loss of phase correlation is governed by large changes in ωQ upon mobile ion hopping.62 Hence, we identify τ with the interchannel exchange times which we are able to track as they evolve from ∼4 ms to ∼30 μs in the temperature range from 487 to 632 K. In this range the stretching parameters β increase from 0.4 to 0.8 (±0.05). Furthermore, for tp = 38.7 μs the final-state amplitudes Z were found to decrease from 0.7 to approximately 0.5.

joint fit yielded an activation energy EA = (0.71 ± 0.03) eV and a pre-exponential factor τ0 = (1.1 ± 0.5) × 10−10 s; i.e., the parameters already listed below eq 4. Correlation times from 7Li NMR line width measurements by Matsuo et al.10 are seen to match our results for T ≤ 675 K, but they deviate markedly at higher temperatures where we collected data using 7Li and 11B line shape analyses as well as via 7Li T1ρ measurements. Nevertheless, the Arrhenius parameters reported by Matsuo et al.10 (EA = 0.74 eV, τ0 = 6.2 × 10−11 s) are in good agreement with ours. Now let us compare these results with those from dielectric spectroscopy. All of the pending work reports on conductivitybased intrachannel activation energies Ec albeit the given values deviate considerably from each other (Kim et al.,7 503 to 643 K: 0.31 eV and 643 to 773 K: 0.75 eV; Guo et al.,8 370 to 490 K: 0.43 eV; Kannan et al.,9 300 to 570 K: ∼ 0.20 eV). However, while two of the articles7,9 present dielectric loss spectra they are not analyzed. For instance, Kim et al.7 show dielectric loss spectra from 573 to 773 K that exhibit maxima in the kHz to MHz regime. Using the standard relation64

τε ≈ (2πνmax )−1

4. DISCUSSION In order to provide an overview on the dynamic properties of the interchannel diffusion in lithium triborate LiB3O5 we collect the temperature-dependent time constants determined in the current work in Figure 9, and then we compare them with data from the literature. In the range from ∼30 μs to ∼4 ms time constants τ from sin−sin and cos−cos39,63 stimulated echoes of single-crystalline LiB3O5 are corroborated by experiments on powdered samples of LiB3O5 and are seen to follow an Arrhenius law (cf. eq 4). Time constants from 7Li and 11B NMR line width and 7Li T1ρ measurements enhance the accessible times and temperature window considerably. In a joint analysis the temperature dependences of the various quantities were subjected to a simultaneous fit using eqs 3, 4, and 6 where applicable. This

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we determined dielectric relaxation times τε from the loss peak frequencies νmax. Figure 9 shows that the time constants τε,b thus assessed from measurements along the b-axis agree well with the NMR results. In particular, the high-temperature result is fully compatible with the outcome of our T1ρ experiment. The time constants τε,a measured along the a-direction are roughly 3 times shorter than τε,b but compatible with the Arrhenius law determined from NMR. One should note, however, that τε,b as determined at 573 K is much shorter than one may have anticipated from the other data. This could indicate a decrease of the effective energy barrier as the temperature is lowered. In fact, analyzing the ε″ peaks that Kannan et al.9 obtained from measurements along the a-axis at 525, 475, and 425 K, we find an activation energy of (0.52 ± 7773


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The Journal of Physical Chemistry C 0.05) eV. Stunningly, the resulting τε,b, when extrapolated to 575 K, is about 220 times shorter than τε,b from Kim et al.7 There is another important piece of information which is not analyzed by Kim et al.7 who report dc conductivities σa from measurements performed along the a-axis. We digitized their data in the range from 680 to 760 K, transformed them to conductivity relaxation times according to65 τσ = 1/(ε0ε∞σa)

5. CONCLUSIONS In the present article we employed an array of NMR techniques ranging from line shape analysis and spin-lock experiments to one- and two-dimensional exchange NMR in order to study the interchannel exchange of Li ions in LiB3O5. We were able to track its time scale over 3 orders of magnitude from μs to ms yielding an activation energy of EA = (0.71 ± 0.03) eV and, in addition, to scrutinize the seeming difference resulting from comparisons of previous NMR and dielectric results on LBO. As seen most directly from 7Li spin-alignment experiments, upon lowering the temperature we observe a broadening of the distribution of interchannel Li hopping correlation times. At temperatures T > 600 K a near-exponential behavior is observed which reflects the slight motional anisotropy in the ab-plane of LBO’s orthorhombic crystal structure. At T < 500 K the ion transport is characterized by a 2 orders of magnitude wide correlation time distribution. We have shown that our findings rationalize the seeming discrepancy among different previous results. Finally, we demonstrated that in nonrotating single-crystalline samples two-dimensional exchange as well as selective-inversion spin-alignment spectroscopy is possible using the highly abundant 7Li probe which allows one to track the ion hopping between the Li(A) and Li(B) sites in LBO most directly. These approaches should be useful in studies of other Li-containing single crystals where the application of fast magic-angle spinning is not practical.

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using ε∞ ∼ 11, (and the permittivity of free space, ε0), and show the data in Figure 9. A fit to these data yields EA,σ = (1.50 ± 0.02) eV. At first glance, this large energy barrier is surprising. Yet, it is well-known that for strong electrolytes, i.e., when the number of mobile charge carriers is temperature independent, the barrier EA can be smaller than EA,σ by a factor of (1 − s) (see, e.g., refs 66 and 67). Here, s is an exponent that characterizes the frequency-dependent conductivity that is usually parametrized as68 σ(ω) = σa + Gωs where G is a temperature-dependent coefficient. In turn, σ(ω) can be obtained from the published dielectric loss curves via σ(ω) = ωε0ε″(ω). Analyzing the data by Kim et al.7 using these considerations we find that 1 − s ∼ 0.4 which very roughly agrees with the ratio EA/EA,σ ≈ 0.47. We suspect that the discrepancy noted above for the intrachannel activation energies Ec can at least partially be rationalized along these lines. However, we are not in a position to assess this conjecture critically. Let us finally turn our attention again to the line shape analyses of Matsuo et al.10 that, on the one hand, tacitly assume that a distribution of correlation times, g(ln τ), is absent. Indeed the analyses of the present 7Li and 11B line shapes as well as of the T1ρ measurements also provide no hints for such a distribution. As may have become clear in Section 3.1, the statement regarding the line shape analysis should be met with some caution when temperature ranges are concerned in which the line widths level off. On the other hand, the dielectric results7,9 indicate strong deviations from non-Debye behavior. The breadths of the dielectric loss peaks, which is most clearly resolved near 600 K, are about 2.2 decades which means69,70 that here g(ln τ) is roughly two decades wide. Let us recall that in order to describe our sin−sin data at 487 K we used a stretching exponent β ≈ 0.4. This parameter corresponds to a distribution width of about 2.2 decades,71 in harmony with the dielectric results. Conversely, at 632 K we find that β is about 0.8 indicating that the time scale distribution is very narrow (covering less than half a decade, corresponding to a factor of 2.7)72 for the fwhm of g(τ). In view of the orthorhombic crystal structure of LBO some hopping anisotropy in the ab-plane is expected and therefore should be picked up by the current exchange experiments. Indeed, a factor of 3 is compatible with the dielectrically observed7 τε,b/τε,a ratio (cf. Figure 9). Such a minor deviation from exponential loss of correlation is hardly resolvable using line shape experiments. Overall, a narrowing of the effective correlation time distribution, g(ln τ), indicates that it is rather the energy barriers that are distributed. Assuming that the width of the barrier distribution g(E) is temperature independent and that eq 4 holds, a narrowing of the correlation time distribution follows from g(ln τ) = kBTg(E), and hence, an increase of β arises for increasing temperatures.73−75 In future studies it would be worthwhile to examine whether and to which extent the marked changes in the anisotropic expansion9,44 of LBO contribute to the variation of g(ln τ).

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APPENDIX A. INITIAL AND FINAL STATE CORRELATIONS As we will show here the evolution time dependence of the stimulated-echo amplitude is very sensitive to the occupancy of the quadrupole frequencies associated with the Li sites that are visited during the ion hopping. For single crystalline LiB3O5 the quadrupole frequencies ωQ,A and ωQ,B of two 7Li sites Li(A) and Li(B) have to be considered. The signal F2(tp,tm,ta) detected after the stimulated-echo sequence (cf. eq 7) consists of nonexchanged intensity F2,AA,BB, i.e., from ions that show the same frequency ωQ,A,B before and after the mixing time and of exchanged intensity F2,AB, i.e., from ions that changed from Li(A) to Li(B) sites or vice versa during tm. Taking into account a theoretical treatment of dipolar interactions (cf. eq 19b in ref 40), one may write explicitly F2(t p , tm , ta) = F2,AA,BB + F2,AB

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where F2,AA,BB ∝ [

∑

pA sin(ωQ , it p)sin(ωQ , ita)]d+ + C1d− i 1, i

i ∈ {A,B}

+[

∑

pD cos(ωQ , it p)cos(ωQ , ita)]d− i 1, i

i ∈ {A,B}

(15a)

and F2,AB ∝ pA pB [A 2,A sin(ωQ ,Bt p)sin(ωQ ,A ta) + A 2,B sin(ωQ ,A t p) sin(ωQ ,Bta)] × d

Here d± =

1 2

exp⎡⎣ − 2 σD2(ta − t p)2 ⎤⎦ ± 1

1 2

(15b)

exp⎡⎣ − 2 σD2(ta + t p)2 ⎤⎦ 1

1 and d = exp⎡⎣ − 2 σD2(t p2 + ta2)⎤⎦ are dipolar correlation functions

7774


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The Journal of Physical Chemistry C with Gaussian standard deviations σD. The d functions account for a decay of the sine modulated correlations, A1,AB, and additionally quantify the emergence of cosine modulated correlations, D1,AB, and of a central-transition contribution, C1,AB, in eq 15a.40,76 The sums in eq 15a extend over the two sites with quadrupole frequencies ωA and ωB which refer to the weighting factors pA and pB = 1 − pA, respectively. If chemical exchange does not take place, i.e., for tm → 0, for the so-called initial correlation I(tp) of the stimulated echo at ta = tp, eq 15a reduces to I(t p) = I0

∑

pi {Ad+ sin 2(ωit p) + Cd−+Dd−cos2(ωit p)}

i ∈ {A,B}

(16) 1

where, for ta = tp, we now have d± = 2 [1 ± exp( −2σD2t p2)]. For ideal flip angles the coefficients are A = 72/160, C = 29/160, and D = 45/160.40 I0 denotes the overall signal strength. In order to describe the experimental data the initial-correlation amplitude, eq 16, has to be multiplied by exp[−(2tp/T2)μ2] to account for an overall spin−spin relaxation. At long mixing times, tm → ∞, when the ions will have hopped many times from site to site, an end correlation remains which is given by E(t p) = I0A exp( −σD2t p2) × {pA2 sin 2(ωA t p) +

pB2

Figure 10. (a) Initial state correlations I(tp)/I0 and (b) final state correlations Z(tp) for single-crystalline LiB3O5 recorded in stimulated sin−sin experiments at 539 K as a function of the evolution time. Open and closed circles refer to full-decay experiments and experiments using a time-saving method, respectively. The open stars mark correlations obtained from the two-dimensional exchange spectra, cf. Figure 7. The solid lines represent a joint fit to I(tp) and Z(tp) using eqs 16 and 17. The excellent agreement of experiment and theory demonstrates that the Li(A) and the Li(B) sites show the same occupancy.

2

sin (ωBt p)+ 2pA pB sin(ωA t p)sin(ωBt p)} (17)

When comparing eqs 17 and 16 one finds that while the C and D terms of the initial correlation will decay due to dipolar broadening, the first term evolves into three separate contributions. The first two contributions in E(tp) originate from eq 15a and account for the probability that an ion has the same precession frequency (ωA or ωB) before and after the mixing time which cannot be distinguished from the static (tm → 0) case. The third contribution stems from eq 15b and represents the exchange between two different frequencies ωA and ωB. From eqs 16 and 17 the final state correlation is defined as the ratio Z(tp) = E(tp)/I(tp). In the limit of short evolution times, tp → 0, the final correlation becomes Z(t p → 0) =

To obtain a coherent description we performed a joint fit of eqs 16 and 17 to the initial and to the final correlations yielding a set of common parameters: ωA = 40.9 ± 0.2 kHz, ωB = 21.8 ± 0.2 kHz, pA = (0.52 ± 0.02), σD = 2π × (2.44 ± 0.09) kHz, and T2 = 350 ± 70 μs. The resulting I(tp) and Z(tp) traces are plotted as solid lines in Figure 10 and reveal a very good agreement with the data. This agreement demonstrates the applicability of eqs 16 and 17 to the stimulated-echo intensities before and after equilibration due to atomic exchange in LiB3O5. For the initial correlations substantial deviations from the fit become visible not before evolution times of tp = 60 μs. The final correlations approach Z = 0.92 for tp → 0 while due to dipolar dephasing Z tends to vanish at very long evolution times. Most importantly, we find that pA is close to 0.5 so that from the sensitive test presented in this Appendix we conclude that in LiB3O5 the Li(A) and the Li(B) sites are populated with equal weight.

pA2 ωA2 + pA2 ωA2 + 2pA pB ωA ωB pA ωA2 + pB ωB2

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After having outlined the theoretical framework, we now describe how the final-state ratios Z(tp) were determined experimentally. First, at 539 K we recorded full stimulated-echo decays from tm = 15 μs to 50 s at selected evolution times. Fits to these full-decay experiments by means of eq 11 yielded the Kohlrausch parameters τ = 0.45 ± 0.02 ms and β = 0.78 ± 0.07, independent of the evolution time. Then, to reduce the acquisition time which is governed by slow spin−lattice relaxation, we performed experiments only at tm = 15 μs and 15 ms for evolution times ranging from tp = 13.7 to 81.7 μs. By using τ and β from the full decays the initial and final correlations for all these evolution times could be determined and are shown in Figure 10(a) and (b), respectively. All data were shifted to the right by ∼1.7 μs to account for finite pulse lengths. The data from the full decays are seen to nicely match those recorded with the time-saving method as well as those from the two-dimensional exchange spectra (cf. Figure 7).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +49-231-7553514. Fax: +49-231-755-3516. Notes

The authors declare no competing financial interest. 7775


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ACKNOWLEDGMENTS We thank the Deutsche Forschungsgemeinschaft, Grant No. BO1301/10-1, for the financial support provided for this project.

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Superionic Conductor Lithium Indium Phosphate. J. Magn. Reson. 2015, 260, 116−126. (40) Storek, M.; Jeffrey, K. R.; Böhmer, R. Local-Field Approximation of Homonuclear Dipolar Interactions in 7Li NMR: DensityMatrix Calculations and Random-Walk Simulations Tested by Echo Experiments on Borate Glasses. Solid State Nucl. Magn. Reson. 2014, 59−60, 8−19. (41) Slichter, C. P.; Ailion, D. Low-Field Relaxation and the Study of Ultraslow Atomic Motions by Magnetic Resonance. Phys. Rev. 1964, 135, A1099−A1110. (42) Jeener, J.; Broekaert, P. Nuclear Magnetic Resonance in Solids: Thermodynamic Effects of a Pair of rf Pulses. Phys. Rev. 1967, 157, 232−240. (43) Qi, F.; Diezemann, G.; Böhm, H.; Lambert, J.; Böhmer, R. Simple Modeling of Dipolar Coupled 7Li Spins and Stimulated-Echo Spectroscopy of Single-Crystalline β-Eucryptite. J. Magn. Reson. 2004, 169, 225−239. (44) Wei, L.; Guiqing, D.; Qingzhen, H.; An, Z.; Jingkui, L. Anisotropic Thermal Expansion of LiB3O5. J. Phys. D: Appl. Phys. 1990, 23, 1073−1075. (45) Our 7Li powder spectra showed the same temperature trend. However, their widths were significantly larger than those for the single crystal. With a focus on dynamics we refrained from exploring the detailed cause of this increased width. (46) Bjorkstam, J. L.; Listerud, J.; Villa, M.; Massara, C. I. Motional Narrowing of a Gaussian NMR Line. J. Magn. Reson. 1985, 65, 383− 394. (47) Abragam, A. The Principles of Nuclear Magnetism; University Press: Oxford, 1961; p 456. (48) Hendrickson, J. R.; Bray, P. J. A Phenomenological Equation for NMR Motional Narrowing in Solids. J. Magn. Reson. 1973, 9, 341− 357. (49) Sergeev, N. A.; Olszewski, M. Dynamic Disorder and Solid State NMR. J. Phys.: Condens. Matter 2008, 20, 175208. (50) Calculations using Gaussian distributions of correlation times g(ln τ) with various widths reveal that the mentioned blurring leaves the inflection point of the δω2 vs T curve at δω2 ≈ δω20/2 almost unaffected. From eq 3 one thus finds that the correlation time is τ = √2/δω0 at the temperature at which the second moment has reduced to half its rigid-lattice value, i.e., at which δω2 = δω20/2. The advantage of this estimate is that an elaborate analysis of the full Δω(T) curve is not required if information about a distribution of correlation times is not sought. (51) van der Maarel, J. R. C.; Jesse, W.; Hancu, I.; Woessner, D. E. Dynamics of Spin I = 3/2 under Spin-Locking Conditions in an Ordered Environment. J. Magn. Reson. 2001, 151, 298−313. (52) Porion, P.; Faugère, A. M.; Delville, A. Long-Time Scale Ionic Dynamics in Dense Clay Sediments Measured by the Frequency Variation of the 7Li Multiple-Quantum NMR Relaxation Rates in Relation with a Multiscale Modeling. J. Phys. Chem. C 2009, 113, 10580−10597. (53) Kuhn, A.; Kunze, M.; Sreeraj, P.; Wiemhö fer, H.-D.; Thangadurai, V.; Wilkening, M.; Heitjans, P. NMR Relaxometry as a Versatile Tool to Study Li Ion Dynamics in Potential Battery Materials. Solid State Nucl. Magn. Reson. 2012, 42, 2−8. (54) Storek, M.; Adjei-Acheamfour, M.; Christensen, R.; Martin, S. W.; Böhmer, R. Positive and Negative Mixed Glass Former Effects in Sodium Borosilicate and Borophosphate Glasses Studied by 23Na NMR. J. Phys. Chem. B, submitted. (55) Here, we calculated an average squared quadrupole frequency ⟨ω2Q⟩ = (1/5)δ2Q(1 + η2/3). See p 30 of Haeberlen, U. High Resolution NMR in Solids: Selective Averaging. Adv. Magn. Resonance, Supplement 1; Academic Press: New York, 1976. (56) Böhmer, R. Multiple-Time Correlation Functions from Spin-3/ 2 Solid-State NMR Spectroscopy. J. Magn. Reson. 2000, 147, 78−88. (57) The expression in the curly brackets of eq 8 accounts for an echo at ta = tp that reflects the refocusing of dipolar dephasing and of B0 inhomogeneities. The transverse relaxation is characterized by T*2 ≈ 120 μs and κ ≈ 1.9. With κ close to 2 the broadening is approximately 7777


Interchannel Hopping in Single Crystalline Lithium Triborate Probed

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