3 Ceramics Part II - ACS Publications - American Chemical Society

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NMR Investigations in Li1.3Al0.3Ti1.7(PO4)3 Ceramics Part II: Lithium Dynamics, Experiments, and Model Joel̈ Emery,*,† Tomas Šalkus,‡ and Maud Barré*,† †

Institut des Molécules et Matériaux du Mans IMMM, UMR CNRS 6283, LUNAM, Université du Maine, 72085 Le Mans Cedex 9, France ‡ Faculty of Physics, Vilnius University, Saulėtekio al. 9/3, LT-10222 Vilnius, Lithuania S Supporting Information *

ABSTRACT: Often the paramagnetic defects have important impact on the nuclear relaxation even in the “nominally” pure samples. These effects show themselves either by the nearestneighbors only or by the dipolar long-range interaction. The former are addressed in this paper in which we built a model for calculation of T1, T1Q, T2, T1ρ when there isor nota residual quadrupolar interaction. This model shows that the maxima of R2 = 1/T2 and R1ρ = 1/T1ρ are shifted toward the low frequencies (low temperatures with regards to the one of R1 = 1/T1), the maximum of R1ρ is shifted toward the low frequencies with regards to the one of R2, and this model is general and can be applied to any system which relaxes by transferred hyperfine interaction. This model was applied to Li1.3Al0.3Ti1.7(PO4)3. It accounts for all the experimental results of T1 (6Li), T1 (7Li), T1Q (7Li), T2 (7Li), and T1ρ (7Li) which were used to highlight local dynamic properties of the lithium versus temperature. The analysis of these results allows us to conclude that inside the natrium superionic conductor (NASICON) framework the lithium undergoes an anisotropic motion, evidenced both with the residual quadrupolar interaction and the measures of the relaxation times, and the relaxation of the lithium is mainly due to the hyperfine transferred fluctuations. Thus, the motion drives the lithium ions in the neighborhood of the oxygens, and these dynamical results allow confirming structural aspects concerning the conducting pathways. along the c ⃗ axis. This motion gives a so-called axis 3 to the 7Li quadrupolar tensor. This part II is devoted to the analysis of lithium dynamics in the Li1.3Al0.3Ti1.7(PO4) sintered ceramic sample studied in part I. These experimental studies mainly consist in the measurements, versus temperature, on one hand of the different relaxation times of the Zeeman energy (Zeeman order) T1Z = T1, T2, T1ρ, for 7Li and T1 for 6Li, and on another hand, of the quadrupolar energy (quadrupolar order) characterized by the relaxation time T1Q. In a first step we show that the relaxation is due to hyperfine transferred fluctuations. In the second step we build a model which allows calculating T1, T2, T1ρ, T1Q. This set of dynamical parameters was used to bring new insight on the local lithium properties. These dynamical results allow confirming structural aspects established in part I: the motion of lithium ions in the neighborhood of oxygen.

1. INTRODUCTION In part I of this work (10.1021/acs.jpcc.6b06764), structural aspects of the modified system Li1+xAlxTi2−x(PO4)3 (with the acronym LATP) was reported. Li1.3Al0.3Ti1.7(PO4)3 belongs to the rhombohedral symmetry (space group R3̅c, Z = 6) which is typical for NASICON (natrium superionic conductor) structure. The lattice parameters of Li1.3Al0.3Ti1.7(PO4)3 are a = 8.5098 Å and c = 20.8305 Å.1−3 The main results of the first part were (i) the synthesis of samples with Al3+/Ti4+ substitution exclusively on the octahedral site of the NASICON framework (Figure 2b of part I) and (ii) the characterization of the environment of the ions in the lattice. We have shown that the aluminum and phosphorus environments are disordered and these two ions are sensitive to the motion of the M(IV)(PO4)3 NASICON skeleton. In both static and magic angle spinning (MAS) modes, a well-resolved quadrupolar structure is observed on 7Li spectra evidencing only one chemical site, while the chemical formulation enforces at least two crystallographic sites. This phenomenon together with the increase at increasing temperature of the quadrupolar parameter CQ (Figure 4d of part I) is explained by the motion of lithium ions in the neighborhood of oxygen planes formed between the M(IV)(PO4)3 columns © XXXX American Chemical Society

2. EXPERIMENTAL SECTION The details of experimental procedure are reported in the Supporting Information file of part I, and in refs 4 and 5. Received: October 14, 2016 Revised: October 21, 2016 Published: October 21, 2016 A

DOI: 10.1021/acs.jpcc.6b10392 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

= 2πνQ) are well-defined: ωQ(0) at the end of the preparation period of duration tp, ωQ(tm) at the end of the dephasing (or mixing) period of duration tm. The signal (recorded during t) depends upon this last frequency. This means that during these periods ωQτQ ≫ 1, τQ being the correlation time of the quadrupolar interaction. The decay of the SAE amplitude (eq 5) generally proceeds in two steps. The first one reflects the hopping and can be described by a stretched exponential function. This decay is a direct measurement (i.e., model-independent) of the correlation time of the slow motion acting on quadrupolar or dipolar order.7−13 The second decay, characterized by T1Q (respectively, T1D), accounts for the relaxation of the quadrupolar energy or quadrupolar order (respectively, dipolar energy or dipolar order) by spin−lattice relaxation. Thus, the experimental data are analyzed by using the stretched exponential function:

Nuclear magnetic resonance (NMR) experiments were performed on an Avance DSX300 spectrometer (Bruker) working at Larmor frequencies ν0 = 116.642 MHz and ν0 = 44.17 MHz, for 7Li and 6Li nuclei, respectively. 6Li, 7Li are quadrupolar nuclei with I = 1, I = 3/2, respectively. Dynamical studies were performed on static samples (static mode, i.e., without MAS), in the temperature range of 120−420 K. The number of transients was experiment-dependent (N = 16 for single pulse and T1 sequences, N = 64 for solid echo sequence, N = 128 for Jeener−Brokaert and T1Q sequences, N = 32 for T2, T1ρ sequences); prior to each transient a saturation comb (16 π/2 pulses) was applied. The delays between the pulses of the saturation comb were adjusted by canceling the intensity of spectra obtained with a wait time of 10 μs after the end of the saturation comb. The T1 values versus temperature were obtained using the saturation−recovery sequence. All the transitions are saturated, then observed. Pulse durations were t90(7Li) = 3.5 μs, t90(6Li) = 16 μs. The delay between transients was taken as tdel = 10T1 (in the case of T2, T1ρ, T1Q measurements) and the one-dimension acquisition (single pulse, solid echo, and spin alignment echo spectra). Spin alignment echo (SAE) experiments allow obtaining information about slow motion and relaxation of quadrupolar (T1Q relaxation time) and/or homonuclear dipolar (T1D relaxation time) energy.6−13 SAE is sampled employing the Jeener−Broekaert sequence (π/2)x−tp−(π/4)y−tm−(π/2)θ− tp−Acq with a t90 liquid pulse duration of 3 μs.6 A short preparation time tp = 10 μs allows one to select the largest interaction between dipolar and quadrupolar ones. In the absence of quadrupolar interaction it is impossible to create quadrupolar order, but it is possible to obtain dipolar order. Let us consider a system evolving under the quadrupolar interaction with Hamiltonian:

S20(t p , tm , t ) = (S0 + S1 exp(− [tm/τSAE]βSAE(t p) ))exp(− (tm/T1eff )) (6)

The offset S0, the amplitude S1, the delay constant τSAE, and the stretching exponent βSAE in general depend on tp. The second term, in parentheses, accounts for the loss of intensity in the correlation due to ionic motion and is well-described by a Kohlrausch−Williams−Watts function with the Kohlrausch exponent βSAE(tp).14 The damping due to relaxation is also expressed by a stretched exponential function with stretching parameter β0. If the ratio S∞ = S0/(S0 + S1) is S∞ ≠ 0, usually the amplitude S02 decay proceeds in two steps at fixed tp and variable tm. In the first step (short tm) the decay is due to individual jumps of the ions, and the second one is due to the spin−lattice relaxation of the quadrupolar (or dipolar) energy characterized by T1eff = T1Q (or T1eff = T1D). Thereby, it will be shown in paragraph 4 that any kind of fluctuation (and in particular the hyperfine fluctuations) can affect the signal of alignment by the relaxation mechanism. The delays were set up at high temperature and maintained constant at any temperature so that any change in the interactions is observable on the SAE spectrum. A complete description of this use of the SAE experiment can be found in ref 13. Slower ionic motion can also be probed by studying the line width, the transverse spin−lattice relaxation T2, and the spin− lattice relaxation in the rotating frame T1ρ. Line width versus temperature is given in Supporting Information. T2 values are obtained from the CPMG (Carr−Purcel−Meiboom−Gill) modified sequence.15 T1ρ values are obtained by using the classic pulse sequence {t90−(spin lock)τ−acquisition}. In such an experiment the characteristic frequency v1 = γB1/2π is given by the amplitude of the radio frequency field B1, and γ is the nucleus gyromagnetic constant. This characteristic frequency can vary from few hertz to several kilohertz. These experiments were performed with nonselective pulse and v1 = 62.5 kHz. 2.1. Remark I. It is important to note that each parameter Aq has its own symmetry given by the corresponding spherical harmonics. So it is possible to report anisotropic phenomena by differentiating the amplitude and the correlation times of these parameters. 2.2. Remark II. In absence of rapid spin exchange transition the relaxation of a quadrupolar spin with I spin value is characterized by several relaxation times.16−26 Due to symmetry properties this number is reduced to 2 for I = 3/2. Furthermore, in the case of half-integer spin, the homodipolar

+2

HQk = ωQk

( −1)m V −k mTmk

∑

for the k th site

m =−2

(1)

We define CQk e 2q k Q = ωQk = = 2πνQk 2I(2I − 1)ℏ 2

(2)

The secular part of the first-order quadrupolar Hamiltonian is written HQk = =

δQk 2

2

2 (3 cos2 θQ − 1 − ηQ sin 2 θQ cos 2ϕQ )[3Izk − Ik⃗ ]

0 (2) 1/6 ωQ0kT0(2) = ΩQ k T0

(3)

with T0(2)(k) =

2 1 2 [3Izk − Ik⃗ ] 6

(4)

In the absence of spin relaxation effects, the signal amplitude generated by the Jeener−Broekaert sequence is given by the correlation function S20(t p , tm , t ) =

9 ⟨sin(ωQ (0)t p)sin(ωQ (tm)t )⟩ 20

(5)

where the bracket ⟨···⟩ indicates the powder averaging.13 Relation 5 assumes that the quadrupolar frequencies νQ (or ωQ B


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Figure 1. 7Li low-frequencies results. Experimental results were obtained in the static mode. (a) 7Li spectra vs temperature. We can observe the increase of the quadrupolar splitting and the improvement of the resolution when the temperature increases. (b) Spin alignment echo amplitude vs the evolution time tm. We observe an important change in the SAE behavior change between 200 and 190 K. The inset gives an example of data fitted by using eq 6 with S1 = 0 (see also Table 7 of part I).

Figure 2. 7Li transverse relaxation rate R2 (full square) measured on the central transition and relaxation rate in the rotating frame R1ρ (open square) vs reciprocal temperature: experimental results (symbols) and calculated ones (lines). RLL is the rigid lattice limit. The connection to the rigid lattice limit is realized with an ad hoc function. For the meaning of the temperatures Tc1M, Tc2M, Tc1ρ, Tc1, and Tc0, see text. The calculated values are obtained with the parameters given in Table 2. (a) Isotropic transferred hyperfine model (EaPAR = EaPERP, τc0PERP = τc0PAR). Both calculated curves are shifted with regard to those of R1 and R1Q in the right direction, but their shifts are slightly too hard. EaPERP and τc0PERP and the concentration of defects are found by adjusting the curve of R1. (b) The same as panel a but with an anisotropic transferred hyperfine model (EaPAR ≠ EaPERP, τc0PERP ≠ τc0PAR; see Table 2).

and quadrupolar fluctuations do not bring any adiabatic contribution to the transverse relaxation rates R2 of the central line. Thus, for this transition, R2 has a maximum at the same temperature position as R1. The DMFIT software is used to fit the spectra and to obtain the peak’s line width, peak positions (in Hz or ppm), percentage, and quadrupolar splitting.27 The spectra of 7Li and 6Li nuclei are referenced from LiCl. Results are expressed either in hertz or in parts-per-million [X(Hz) = X(ppm)ν0(MHz)]. In the different tables, the quadrupolar fluctuation = Aq are expressed in kilohertz, while the amplitudes A(2) q dipolar amplitude fluctuations are without dimension thanks to the introduction of the parameter CIJ =

μ0 γ γ ℏJ(J 4πa03 I J

3. RESULTS Dynamical NMR studies can be separated into a low-frequency regime evidenced by R2 = 1/T2 and line width, R1ρ = 1/T1ρ, spin alignment, and high frequency given by R1 = 1/T1 and R1Q = 1/T1Q. T2 and the line width are intimately connected, but this last one can be perturbed by contributions which have nothing to do with the dynamics. Thus, only T2 will be analyzed. 3.1. Low-Frequency Range. 3.1.1. 7Li One-Pulse and SAE Spectra. Two particular features are clearly evidenced on Figure 1 when temperature increases: (i) Figure 1a shows that the satellite transitions are more and more resolved and the central transition narrows (see also part I). This behavior was already observed.3,4 This is due to the motional averaging of the dipolar broadening. Below 180 K, the satellite transitions are smoothed by the dipolar interaction and disorder. Then in the temperature range of 180−420 K the fluctuation correlation

+ 1) with

a0 = 1 Å. C


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Figure 3. 7Li high-frequency relaxation rates R1 and R1Q vs reciprocal temperature: all the results were obtained in the static mode, experimental results (symbols) and calculated ones (lines). The calculated results are obtained with parameters given in Table 1 and transferred hyperfine model. The vertical line at Tc1M indicates the position of R1 and R1Q maxima. (a) 7Li spin−lattice relaxation rate (R1). The inset gives the ratio R2/R1. (b) Quadrupolar order relaxation rate (R1Q) of 7Li. The inset gives the ratio R1Q/R1.

time τc of the dipolar fluctuation verifies ωDτc < 1 with ωD the rigid lattice broadening. The abnormal behavior of the CQ parameter (CQ increases at increasing temperature) was analyzed in part I where we found it is due to the distortion of the M(IV)(PO4)3 NASICON skeleton which induces anisotropic deformations. (ii) Figure 1b gives several examples of the SAE amplitudes versus the mixing time tm obtained in the temperature range where SAE amplitudes undergo drastic changes. Above 200 K (see Table 8 of part I) only the second decay due to spin−lattice relaxation (T1Q) was observed (quadrupolar relaxation or homodipolar relaxation). Nevertheless, below 200 K a change in the shape and the behavior of the SAE amplitude is clearly evidenced. In part I it was shown that this change is partially due to temperature effects, but it accounts also for some modifications in the interactions. Above 220 K, where only one breakdown is observed, the T1Q values are obtained by using the function

at TC2M, then it decreases down to a relative minimum at Tc1, and it finishes by an abrupt growth up to its rigid lattice limit. The decrease of R2 above T ≈ 142 K (1000/T ≈ 7) is due to the well-known motional averaging which appears when the lithium ions undergo motion at increasing temperature. The correlation time τc of the motion verifies Δωτc ≤ 1. (ii) R1ρ exhibits an apparent activated law due to activated motion which correlation time should be given by τcρ = τc0ρeEa/kT. At this maximum ω1τcρ ≈ 1 gives a correlation time τcρ(R1ρmax) ≈ 2.3 × 10−6 s. (iii) Around its maximum R1ρ is nearly symmetrical, and we could deduce an activation energy Ea1 = 0.15 eV and τc0ρ ≃ 1.410−9 s. (iv) R1ρ and R2 have their maximum strongly shifted from the R1 maximum. Furthermore, it is worth noting the unexpected position of the maximum of R1ρ, which is shifted toward the low-temperature range (low frequency) with regards to that of R2. 3.2. High-Frequency Range. 3.2.1. 7Li Zeeman Order (T1) and Quadrupolar Order (T1Q). In Figure 3 are sketched, versus temperature, the relaxation rates of the Zeeman energy R1 = 1/ T1 (Figure 3a) and quadrupolar one R1Q = 1/T1Q (Figure 3b). It is worth noting the shift of the maximum of R1(7Li) with respect to that of R2(7Li) (see Figure 2a). It is also worth noting that the R1 values are not very different from the ones obtained in refs 3 and 4. R1 and R1Q probe the motion with frequency in the range of the Larmor frequency; the relaxation time of the quadrupolar/dipolar energy brings further information on the relaxation process. The relaxation rates for R1, R1Q, are monoexponential and present a maximum at nearly the same temperature Tc1M: 1000/Tc1M ∼ 3.1. As for R1ρ(7Li), the experimental results show that R1(7Li) and R1Q (7Li) follow an apparent activated law due to activated motion, which correlation time is given by τcq = τc0qeEa/kT. Around their maxima R1 and R1Q are symmetrical down to Tc1, and we can deduce the activation energy Ea = 0.14 eV that leads to τc01 = 1.2 × 10−11 s. Nevertheless, a weak asymmetry can be observed on the low-temperature side. The symmetry around the maxima is a mark of the exponentiality of the correlation function. These two correlation times obtained from R1 and R1Q on one hand and R1ρ on another hand could indicate the presence of two motions with a difference of nearly 2 decades in frequency: a rapid one detected by T1 and the other slow one detected by T1ρ. Although this situation was previously observed in other systems as Li3xLa2/3−x□1/3−2xTiO3,5 in the

S20(t p , tm , t p) = S2Q (t p , tm , t p) = S0 − S1 exp[( −tm/T1Q )] (7)

Below this temperature the SAE amplitude was analyzed (see part I) with eq 6. The Kohlrausch exponent of the relaxation (T1Q) decreases a little when this exponent of the first part of the SAE decay is small (see Table 8 of part I). This behavior indicates that dynamics of the process is characterized by a distribution of correlation time. 3.1.2. 7Li R2 Transverse Relaxation Rate and R1ρ Transverse Relaxation in the Rotating Frame. Figure 2 gives, versus reciprocal temperature, experimental results concerning the 7Li transverse relaxation rate R2 (R2 = 1/T2 Figure 2a) and transverse relaxation in the rotating frame R1ρ (R1ρ = 1/T1ρ, Figure 2b) with locking field amplitude v1 = 62.5 kHz. These results are obtained on the central line, and a single relaxation time was observed. In these figures several temperatures, Tc0, Tc1, Tc1M, TcρM, TcρM, are identified. Above Tc0 = 170 K central transitions (CT) and satellite transitions (ST) are welldistinguished; Tc1 ≈ 195 K identifies the temperature of the relative minimum of R2. TcρM = 225 K identifies the temperature of the R1ρ maximum, while Tc2M = 250 K indicates the temperature of the relative maximum of both R2 and of the line width (see Supporting Information). Tc1M = 320 K indicates the temperature of the maximum in R1 and R1Q. This figure gives evidence to several features: (i) At decreasing temperature, R2 increases up to a relative maximum D


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be respected if the anisotropy of the motion acted differently on the quadrupolar fluctuations A(k) q , and (c) when there is a residual quadrupolar interaction and if the central transition and the satellite ones are well-separated, the central resonance relaxes with three relaxation times in the presence of a locking magnetic field13. None of these conditions is verified and we can rule out this interaction. (ii) Now, let us consider the nuclear dipolar relaxation. At the maximum of R1(7Li) and if the relaxation is due to

present case, we cannot reproduce the same analysis for reasons which will be developed below. 3.2.2. 6Li Zeeman Order (T1). The results of 6Li relaxation times are sketched in Figure 4. R1(6Li) is activated, and its

homonuclear we have R1 ≈

(Δωd)2 with ω0

Δωd the dipolar

fluctuation amplitude. By taking this amplitude equal to the experimental rigid lattice limit Δωd = 2πδ with δ = 2880 Hz we obtain R1 = 2.8 s−1 that is too weak with regard to the experimental value. In the case of heterodipolar fluctuations the main contribution is obtained by using the maximum of

R1 ≈

(Δωd)2 , |ω0Li − ω0Al|

and with the same approximations as above we

obtain R1 ≈ 10 s−1. Thus, the dipolar coupling has to be ruled out. (iii) Exchange between two sites. As reported in part I, a single value of the quadrupolar CQ(7Li) parameter is evidenced by using SAE and solid echo experiments. So exchange experiments cannot bring any information. The Goldman− Shen sequence,28 performed at different temperatures, did not allow us to evidence any exchange phenomena through the typical behavior of free induction signal. From these results we have also to rule out the mechanism proposed in ref 3. (iv) A correlation function, containing a frequency of coherent oscillation, which, in the secular case (frequency of measure = 0) would lead to a characteristic frequency different of zero. Although we studied this case, we shall not present it here considering the electron spin resonance (ESR) results. Indeed, the model of hyperfine fluctuations turns out to be the one which uses the least hypothesis and takes into account the presence of paramagnetic defects. 4.1. Hyperfine Interaction. The ratio R1/R2 ≈ 8 for 7Li (see the inset of Figure 3b) which is observed in relatively high temperature indicates that the relaxation is due to paramagnetic defects. The results of the 6Li relaxation rate are also a strong indication for dipolar relaxation for both 7Li and 6Li. However, it is not a cross-relaxation mechanism between these two nuclei which monitors the 6Li relaxation because R1 maxima of both nuclei are of the same order. This result is at a variance of the one invoked in refs 3 and 4 (in ref 4 the sample was not the same, and no paramagnetic defect was detected by ESR at RT, but it could exist and only be observable at low temperature for reasons of relaxation). Paramagnetic defects were evidenced in the sample under consideration. The ESR powder spectrum recorded at RT is

6

Figure 4. Li spin−lattice relaxation rate (R1) obtained in the static mode: experimental (full square) and calculated (solid line) results. The calculated values are obtained with the parameters given in Table 2: the vertical lines at Tc1M (respectively, Tc2M), the position of the maximum of R1(7Li) (respectively, R2(7Li).

maximum is around the one of 7Li. Their ratio is R1(7Li)/ R1(6Li) ≈ 3 in the range of [γ(7Li)/γ(6Li)]2. This result is a strong indication for relaxation by dipolar interaction for both 6 Li and 7Li.

4. MODEL FOR LITHIUM RELAXATION Two results deserve to be analyzed more in detail: on one hand the positions of the maxima of R2 and R1ρ with regard to the position of the R1 maximum, and on another hand the position of R1ρ maximum with regard to the R2 one. Before analyzing more in detail our results, we have to look at different processes leading to this relative maximum of R2: (i) The specific R2 behavior for half-integer quadrupolar spins when the relaxation is driven by the quadrupolar fluctuations (see Remark II). This maximum is due to the nonexistence of adiabatic contribution to the transverse relaxation of the central transition.6−15 Furthermore, this effect would lead to the observation of the maximum of R2 at the same temperature as for R1. This case has to be ruled out because the R2 maximum is clearly shifted with regards to the R1 maximum. Moreover, relaxation by quadrupolar fluctuations requires (a) two contributions to the longitudinal and transversal relaxations at low temperature, (b) at high temperature the ratio R2/R1 = 1, which is not observed as can be seen on inset of Figure 3a; however, this ratio could not Table 1. 7Li Fluctuating Interactionsa dipolar 7

Li−27Al hyperfine

D20

D21

D22

τc020

τc021

τc022

Ea20

Ea21

Ea22

0.350 Ahyp00

0.230

0.230

1.8 × 103 Ahyp11 Ahyp1−1

2

2

0.13

0.1

6.7 × 103

Ahyp10 3.8 × 103

ND

ND

τc0PAR

τc0PER

EaPAR

0.1 EaPER

45.5

2.5

0.17

0.18

Hyperfine transferred fluctuation parameters Ahypαβ and heterodipolar D2q fluctuations parameters. Ahypαβ is the amplitude of the αβ component of the spatial averaged transferred hyperfine fluctuation amplitude tensor (α,β = 0, ±1, 0 for z), with activation energy EaPER = Ea00, EaPAR = Ea10 = Ea01 and the correlation times τc0PAR = τc00, τc0PERP = τc021 = τc022 in 10−12 s. A11 and A1−1 have no effect on the NMR relaxation. D2q correspond to the qth component of the irreducible dipolar tensor of rank 2. The amplitudes of the interaction are given in 103 rd s−1. ND means “not determined” because the NMR results are not sensitive to these parameters. τc0PAR and EaPAR are evaluated from R1, when τc0PAR and EaPAR are indifferently estimated by means of R1ρ or R2. T1e = 8.5 × 108 s. a

E


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Z is the number of host sites which are nearest neighbors to nuclear spin site, c is the impurity concentration, and W is the hopping rate from site with hyperfine interaction

characterized by three bands (see Supporting Information) and is reconstructed with the following parameters: the eigenvalues of the Landé tensor, g11 = 2.014, g22 = 2.0159, g33 = 2.0098; the hyperfine tensor (in MHz), A11= 61.886, A22 = 50.634, A33 = 16.878; the line widths S 11(MHz) = 8.439, S 22 = 11.252, S 33 = 19.69. This paramagnetic center was identified as being a hole (missing electron) trapped on oxygen in a PO4 group. The concentration of these paramagnetic defects is c ∼ 5.5 × 10−4 that is too weak to be observed on the NMR spectrum but which can monitor the nuclear relaxation. The anisotropic line width can be due to an anisotropic relaxation mechanism. Let us now consider the hyperfine coupling. In a first step we eliminate the dipolar hyperfine interaction by proceeding as follows. We calculate the relaxation parameters for R1(7Li), then we use these dipolar fluctuation parameters in the calculation of the R1(6Li) (changing only the nuclear gyromagnetic ratio). The obtained results are 2 orders of magnitude higher than the experimental ones, and we ruled out this hypothesis. In a second step we consider isotropic transferred hyperfine interaction. Again, we calculate R1(7Li) with the parameters given in Table 1 and we used these last parameters to analyze the 6Li case. With very weak modifications of these found transferred hyperfine interaction parameters (given in Table 2),

H′(t ) =

S0 = Sz , I0 = Iz

α , β = 0, ± 1

α ,β

(9)

to a site without this interaction. f v is a time function which depends upon the correlation function of the electronic spin and the hyperfine parameters Aαβ. These parameters verify the relation |A−α−β| = |Aαβ|. T1 and T2 relations were established in the case where there is no residual quadrupolar interaction.29 Our contribution was to widen these results to the cases of T1ρ, T1Q as well as in case there is a residual quadrupolar interaction. Thus, eq 8 becomes 1 = RKQ = (1 − ⟨f KQ ⟩)ZcW TKQ Q = 0 for T1(K = 1) and T1Q (K = 2), K = 1 and Q = ± 1 for T2 and T1ρ

(10) −Ea/kT

The hopping rate is supposed activated with W = W0e ⟨f KQ ⟩ = W

a

∫0

−Wτ

∞

dτ e

f KQ (τ )

(11) −Wτ

KQ

A00

A01

A11

A1−1

τc0PAR

τc0PER

EaPAR

EaPER

ND

1.45

?

?

ND

22.

ND

0.13

.

is the average value of f over all time τ with We dτ the probability of the nucleus remaining at given site between τ and τ + dτ before jumping. f KQ is defined from

Table 2. Hyperfine Parameter for T1 of Li 6

∑ Aαβ IαSβ(t )

f KQ (τ ) = exp( −

a

ND (respectively, ?) means not determined because it is not pertinent for T1 calculation (respectively, for NMR parameters because it depends upon the electronic frequency). T1e = 8.5 × 10−8 s. τc0PER = τc021 = τc022.

∫0

τ

du(τ − u)F KQ (u))

(12)

F functions contain the fluctuation characteristics together with the electronic spin parameters and the hyperfine parameters Aαβ.27 It is worth noting that the amplitudes of the relaxations rates are monitored by the parameters Z, c, and Aαβ. The Kubo−Tomita perturbation technique,33 with suitable approximations and averages, allows writing the relation: KQ

we can account for the R1(6Li) relaxation rate. These weak discrepancies between parameters can be understood considering the number of neutrons which modifies the amplitude of the hyperfine interaction and the nucleus mass which modifies the frequencies of oscillations inside the well (which depends on spring constant K and mass M: ω2 = K/M). So, afterward we shall give only the results relative to transferred hyperfine interaction fluctuations. 4.2. Model for Relaxation by Transferred Hyperfine Interaction. The model presented here is a generalization of the model established by Richards for R1 and R2 without residual quadrupolar interaction. 29 The details of the calculation are reported in the Supporting Information. The used method has the advantage to handle simultaneously with spin diffusion, particle diffusion, and interactions. Spin diffusion mechanism was studied in the past.30−32 Our original contribution concerns our method of calculation which allows determining R1, R1Q, R2, and R1ρ in presenceor notof residual quadrupolar interaction (CQ ≠ 0). For nuclei undergoing a random walk governed by a standard rate equation and submitted to hyperfine transferred fluctuations, it was shown that the nuclear relaxation times T1 and T2 are given by29 1 = R v = (1 − ⟨fv ⟩)ZcW v = 0 for T1, v = ±1 for T2sS Tv

⟨TQ(K )⟩A (t + τ ) = ⟨TQ(K )⟩A (t ) − Tr

∫t

t+τ

dt ′

∫t

t′

dt ″{GAKQ (t ′, t ″)ρA (t )}

(13)

ρ(t) is the density operator of the whole system, and Tr is the trace of the matrix of the operator in the parentheses, with the approximation ρA(t′′) = ρA(t) and GAKQ (t ′, t ″) = [HA(t ″), [HA(t ′), TQ(K )]]

(14)

The trace operation concerns all the variables (network, electronic, nuclear), and the index A indicates result of the application of the unitary transformation UA = e−iAt (see eq A20 of Supporting Information). Equation 13 connects the average value of an observed physical quantity T(K) Q at the time t + τ to its average value at the time t. Thus, after we performed the average, eq 13 becomes ⟨TQK ⟩A (t + τ ) = ⟨TQK ⟩A (t ) − Tr

∫t

⟨TQK ⟩A (t + τ ) = ⟨TQK ⟩A (t ) − Tr

∫t

t+τ

dt ′

∫t

dt ′

∫t

t+τ

t′

t′

dt ″⟨GAKQ (t ′, t ″)⟩ dt ″⟨GAKQ (t ′, t ″)⟩ (15)

In Supporting Information (paragraph III-1 part A), it is shown

(8) F


Article

The Journal of Physical Chemistry C ⟨GAKQ (t ′, t ″)⟩ = FAKQ (t ′, t ″)⟨TQ(KQ )⟩(t )

The transformation UA is defined by UA(t) = UQ(t)U0(t) where U0 is given by eq A59 and UQ is defined with eq A60. The theoretical values of R1, R1Q, R2 with residual quadrupolar coupling are obtained with the function

(16)

Then supposing that the fluctuations are stationary we obtain f AKQ (τ ) = =

∫t ∫0

t+τ

dt ′ τ

∫t

t′

dt ″{FAKQ (t ′, t ″)}

du(τ − u)FAKQ (u)

u = t′ − t″

3 ⎧ ⎪∑ CS(α , u)CL(α , β , u) ∑ ⎪ n , n ′= 1 ⎪ α ,β FωKQ (τ ) = ⎨ f 1n ′ f 1n a a Q ⎪ −β , −ν β , ν β −β ⎪ ⎪ ...........ΨQαβ , ν(τ , ωs , ωI , ωQ , τce) ⎩

(17)

We introduce the average values CS(α , u) =

(21)

S(S + 1) −u/ τcα e and CL(α , β , u) = ⟨⟨|Aαβ |2 ⟩(u)⟩ 3 (18)

Z(n,n′,K,L,β,Q) is defined in eq A63, and f1n β,v parameters are defined in paragraph III-2 part B of Supporting Information. The function ΨQαβ,v(τ,ωs,ωI,ωQ,τce) is defined in eq A61.

The double bracket means a statistical average over the lattice and electronic spin variables followed by an average on the paramagnetic defects. Details of the calculations in different cases are given in Supporting Information (paragraph III-1 part A), but we can summarize the various results which we shall need to analyze the experimental results of 7Li: ωQ = 0, ω1 = 0

ωQ ≠ 0, ω1 ≠ 0

∑ α ,β

(a)

KQ + + F AQx = FAx + FQQ

+ FQQ (τ , ω1, τce) =

⎫ ⎧ +2 [Ω0 ]2 π ⎞⎟ (2) ⎛⎜ π ⎞⎟ Q (2) ⎛ ⎪ ∑ ⎪ ⎜ Γ d d (1, 2, m ) ,0 ,0 spin q m m q ⎝ ⎠ q ⎝ ⎠ ⎪ ⎪ 2 2 a1 ⎬ − ∑ ⎨ mq =−2 ⎪ ⎪ λ =|k − 2| QQ ⎪ Λ(2, 2, λ , − mq + 1, mq + 1)Ψ mq (τ , ω1, τce) ⎪ ⎩ ⎭ 3

(19)

Γspin(β,K,Q) are given by eq A16 of Supporting Information, and the functions ΨKQ αβ (τ,ωs,ωI) are defined in eq A48 of Supporting Information. These results contain the ones of ref 27: K = 1 Q = 0 for T1 and K = 1 Q = 1 for T2. The ratio R1Q/R1 is an indication to verify whether the model of the hyperfine fluctuations is at the origin of the relaxation. In this case, there is no residual quadrupolar interaction and in absence of radio frequency field this ratio is easy to calculate. Relation 19 shows that R1Q and R1 are only differentiated by the K parameter in the Γ coefficient. In Supporting Information (paragraph III-1 part A) we show R1Q/R1 = 3. ωQ = 0, ω1 ≠ 0

(23)

(2) dm,n (β) are the Wigner matrix, Γspin(k1,k2,m) are defined in eq A16, and Λ(k1,k2,λ,m,m′) are parameters defined in eq A15. The function ΨQQ mq (τ,ω1,τce) is defined in eq A71. In this case some restraints were applied as explained in paragraph III-1 part A of Supporting Information. It is important to notice that if the quadrupolar fluctuations are negligible we obtain the same results as with ωQ = 0 and ω1 ≠ 0. In our model, anisotropic relaxation mechanism is introduced by differentiating the correlation time of the hyperfine parameter A00 of that of A01. The introduction of different correlation times in the direction of the I−S bond and its perpendicular plane is justified by the observation of anisotropy on the ESR spectrum line width.

(b)

This case corresponds to T1ρ without residual quadrupolar coupling. The transformation UAx is given by UAx = UxUA with UA defined in eq A20 of Supporting Information and Ux given by eq A52 of Supporting Information. We obtain

5. DISCUSSION Parts a and b of Figure 2 give evidence, at low temperature, to an abrupt variation of R2(7Li), explained by an activated motion, which is not observed in the cases of 31P and 27Al (see part III). This behavior is characteristic of the appearance of the motion of the Li+ ions at increasing temperature. In paragraph 4 we showed that the main contribution to the relaxation resulted from the hyperfine interaction. Figures 2 and 3 summarize the comparison between experimental and calculated data obtained with the parameters reported in Table 1. A00 (respectively, A01) is the main contribution to R2 and R1ρ (respectively, R1 and R1Q). Thus, the value of A00 (respectively, A01) is obtained by adjusting this parameter to give the amplitude of R2 (respectively, R1). It is worth noting that the so determined values also fit R1ρ (respectively, R1Q). In the first step (Figure 2a) we use isotropic diffusion. In this case the correlation times τc0PAR = τc0PERP and the activation

+1 3 ⎧ ⎫ ⎪∑ CS(α , u)CL(α , β , u) ∑ ∑ Z(n , K , n′, β , Q ). ⎪ ⎪ ⎪ n , n ′= 1 v , v ′=−1 ⎪ ⎪ α ,β ⎬ FωKQ (τ ) = ⎨ f 1n ′ f 1n a a Q − β β ⎪ ⎪ −β , −ν β , ν ⎪ ⎪ ⎪ ⎪ ...........ΨQαβ , ν(τ , ωs , ωI , ω1, τce) ⎭ ⎩

(20)

f βk S, ν )

The coefficients aβ (respectively, are given by eq A13 (respectively, paragraph III-2) of Supporting Information, Γspin(β,K,Q) are given by eq A16, and the functions ΨQαβ(τ,ωS,ωI,ωQ,τce) are defined in eq A65. ωQ ≠ 0, ω1 = 0

(22)

The first contribution is given by eq A57, and the second one (quadrupolar fluctuations) is given by eq A72:

S(S + 1) ⟨⟨|Aαβ |2 ⟩⟩Γspin(− β , K , Q + β)Γspin(β , K , Q ) 3

KQ Ψαβ (τ , ωS , ωI)

(d)

This case corresponds to R1ρ with residual quadrupolar interaction. If there is no correlation between hyperfine fluctuations and quadrupolar fluctuations, two contributions characterize the relaxation

The transformation UA is given by eq A35 of Supporting Information. In this case there is no residual quadrupolar coupling and no radio frequency field. This case was already studied for T1 and T2, but it was a test for our method.27 We develop the trace FAKQ (τ ) =

⎫ Ζ(n , n′, K, L, β, Q ). ⎪ ⎪ ν , ν ′=−1 ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ +1

∑

(c) G


Article

The Journal of Physical Chemistry C EaPAR = EaPERP are identical along the z axis of the hyperfine interaction and in the plane perpendicular to this axis (R1, R1Q). τc0PERP value is determined from the R1 maximum. The isotropic case allows us to obtain the shift in the right direction of R1ρ and R2 with regard to the R1 position and to obtain correct amplitudes of relaxation rates. Such a shift had also been observed in ref 29. Furthermore, we obtain also the right shift of R1ρ with regards to R2. The found concentration of defects c = 6.5 × 10−4 is in agreement with that detected by ESR, c ≈ 5.5 × 10−4, and finally, we found a relaxation time of paramagnetic defects in agreement with the one obtained from the line width of the ESR spectrum of the paramagnetic defects. So, we are rather confident in this model to continue along this way. In a second step we introduce anisotropic diffusion (Figure 2b) by differentiating the correlation time of the hyperfine interaction along the z axis of the hyperfine interaction and in the plane perpendicular to this axis (τc0PAR ≠ τc0PERP, EaPAR ≠ EaPERP). This is suggested by the equation 18 which allows differentiating the correlation times of the fluctuations of the A00 and A01 parameters. Physically, this correction has its origin in the CQ parameter behavior, which is due to anisotropic distortion (see part I) and also by the anisotropy of the ESR line widths. We keep the parameters EaPERP and τc0PERP previously obtained, and we adjust the parameters of diffusion EaPAR and τc0PAR to obtain the positions of R2 (Figure 2b). The amplitude and the position of R1ρ are correct. It is worth noting that using this model with NMR frequency ν0(7Li) = 11 MHz that corresponds to the NMR experiment presented in ref 3 we are able to reproduce results in agreement with the experimental data of this paper (see Figure S3 of the Supporting Information). We are also able to reproduce results at 27 MHz (see Figure S4 of the Supporting Information).32 So, differentiating these two directions we obtained results sketched in Figure 2b. Within this model we account for all 7Li NMR results. Let us notice that the temperature of the interaction change (Tc1), detected from the abrupt variation of the ratio R1Q/R1 (inset of Figure 3b), corresponds to the relative minimum of R2. Thus, this relative minimum in R2 occurs when the impurity rate ZcW = ZcW0e−U/kBT in the slow regime catches up, as temperature increases, with the motionally narrowed dipolar contribution⟨Δω2⟩/W. W0 = l/ τc0PAR is the attempt frequency, and U = Ea0PAR is the activation energy, both given in Table 1. Roughly we obtain kBTc1 ≈ 2U/ ln(ZcW02/⟨Δω2⟩) which gives Tc1 = 189 K which is very close to the experimental value Tc1 = 192 K.29 Finally, the experimental ratio R1Q/R1 (inset of Figure 3b) is found close to the theoretical value R1Q/R1 = 3 given by eq A51 of Supporting Information. Thus, we showed that the relaxation of the lithium is mainly due to transferred hyperfine fluctuations. The hyperfine interaction fluctuations are due to lithium which jumps from the neighborhood of an oxygen where there is a defect toward oxygen without defect in its neighborhood. The motion acts as reservoir of energy allowing standardization of the magnetization leading to unique relaxation times. This is an important conclusion because it means that the lithium, in its motion, performs bonds with oxygen on which is a paramagnetic defect. This defect is an unpaired electron on oxygen on the PO4 group. (The hyperfine interaction with phosphorus is a characteristic of the ESR spectrum.) The transferred hyperfine interaction requires that the ion lithium establishes chemical bonds with the oxygen. Nevertheless, these bonds do not have any static feature because no such effect was observed on ESR

spectrum. We can immediately deduce that the lithium travels in the structure by making bonds with oxygens and the hyperfine interaction serves to highlight these bonds. This conclusion confirms what the analysis of the 7Li NMR spectra versus temperature had allowed us to establish in the part I: the positioning of lithium in planes formed by three oxygen ions. The motion in the plane confers an axial symmetry to the quadrupolar tensor of the lithium, and the jumps between planes (diffusion in the material) explain the difference τc0PAR and τc0PERP. As will be seen in part III, dynamical properties bring information about structural ones.

6. CONCLUSION This work was devoted to the Li+ local dynamics in Li1.3Al0.3Ti1.7(PO4)3 studied by NMR of 6Li and 7Li. Our experimental data concern the relaxation times T1, T1Q, T2, T1ρ for 7Li, T1 for 6Li. They all show a single contribution that led us to analyze these results within a diffusion mechanism (spin and/or particle). We then developed a model of relaxation by hyperfine (transferred and dipolar) fluctuations with residual quadrupolar interaction for these four relaxation times. With this model we were able to account for all the 7Li experimental data and in particular for the shift of the R2 maximum with regard to the one of R1 on one hand, and the shift of the R1ρ maximum with regard to that of R2 on another hand. Nevertheless, to adjust the shift of the maximum R2 with regards to the maximum of R1 we needed to introduce anisotropy in the correlation times of the hyperfine fluctuations. This was suggested by the anisotropy of the ESR line widths. Moreover, within this model we obtain a concentration of paramagnetic defects, responsible for the relaxation by transferred hyperfine fluctuations, consistent with the ESR results. This model accounts also for results obtained at other frequencies (see Supporting Information). The random walk with uncorrelated jumps seems to be sufficient in the temperature range under consideration. Finally, the shift of the maximum of R1ρ with regard to that of R2 would seem to be a characteristic of the relaxation by the hyperfine fluctuations. The results obtained on the dynamics of the lithium also bring us an overview of the position of this ion in the structure. By other techniques (neutron and X-ray diffractions) it is difficult to obtain the position of this ion. Our results show that lithium ions travel through the structure passing near oxygens with which they establishor nottransferred hyperfine couplings as the oxide ion possessesor not transferred electronic hole. It is worth noting that actually, as we shall also see it in the part III, the dynamic studies of three nuclei (7Li, 31 P, 27Al) also allow establishing results concerning the structure and the way it favors lithium motion. Finally we would like to point out that, in a recent article, Vinod Chandran et al. presented some interesting results concerning, in particular T1(7Li) and T1ρ(7Li).35 Nevertheless, the samples on which they carried out their measurements presented some aluminum in a tetrahedral site, and the formulation x = 0.3,34 we report in this present paper, was not studied.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b10392. H


Article


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Experiments with variable temperature, data processing, 7 Li line width and line position vs temperature, model of relaxation by hyperfine transferred fluctuations, ESR experiments, and application of the model to other systems (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: +33 0 2 43 83 33 53. *E-mail: [email protected]. Phone: +33 0 2 43 83 33 53. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

One of us, J.E., would like to thank Pr. Kassiba “Institut des Molécules et Matériaux du Mans”, for his help during EPR experiment.

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