Dynamical structure of Ag_xCu_1−xI (x = 0.99–0.80) in ionic and superionic phases studied by solid ⁶³Cu and ¹²⁷I NMR spin–lattice relaxation time measurements

Abstract

The temperature dependences of the ⁶³Cu and ¹²⁷I NMR spin–lattice relaxation time T₁ and the ⁶³Cu NMR spin–lattice relaxation time in a rotating frame T_1ρ were measured for Ag_xCu_1−xI (x = 0.99–0.80). In the γ phase, T₁ and T_1ρ were dominated by the lattice vibration at low temperatures. The minimum of T_1ρ caused by the diffusion of the Cu ion was observed at ca. 360 K. The temperature dependence of T_1ρ in the range 420–300 K can be explained in terms of a distribution of correlation times which arises from a distribution of activation energies for the ionic diffusion. The mean and the width of the distribution for the activation energies increased with increasing Cu concentration. The rapid T₁ decrease observed above ca. 400 K can be attributed to the motion of the thermally generated defects. The enthalpy for the formation of the defect was estimated as 120 ± 10 kJ mol⁻¹ from ⁶³Cu NMR T₁ and T_1ρ values. In the α phase, the relaxation of the ⁶³Cu nucleus was in the fast motion region and the activation energy of the ionic diffusion was determined as 7 ± 1 kJ mol⁻¹ from the temperature dependence of ⁶³Cu NMR T₁.

---

Introduction

It is well known that AgI and CuI are superionic conductors that achieve a highly conducting state [italic v]ia phase transitions and that an anomalous increase in the defect concentration occurs near the transition temperature to the superionic phase.^1–6 However, the crystal structures and the paths of ionic diffusion in the superionic phases of these compounds are different.^7–9 The superionic phase of AgI is the α-phase where the iodine ions form a body-centered cubic structure, whereas that of CuI is the α′-phase where the iodine ions form a face-centered cubic structure.^7–9 These differences are considered to be attributable to the size and electronic structure of the diffusing ions. In order to understand how the nature of these ions relates to ionic diffusion and the formation of defects, study of the transport properties of the cations in the mixed crystal, Ag_xCu_1−xI, is important. Kusakabe et al. investigated the crystal structure and ionic diffusion for Ag_xCu_1−xI using electrical conductivity, heat capacity and X-ray diffraction methods.^10–14 We analyzed the crystal structure of Ag_xCu_1−xI by using ⁶³Cu NMR and X-ray diffraction methods.^15–17 We clarified the sites of the Cu and Ag ions in Ag_xCu_1−xI by analyzing the observed ⁶³Cu NMR chemical shift using ab initio MO calculations. In the present work, we study the influence of doped Cu ions on ionic diffusion and the formation of defects in Ag_xCu_1−xI by using ⁶³Cu and ¹²⁷I NMR. The ⁶³Cu and ¹²⁷I NMR spin–lattice relaxation time T₁ and the ⁶³Cu NMR spin–lattice relaxation time in the rotation frame T_1ρ were measured for Ag_xCu_1−xI (x = 0.99–0.80). In this range of x, Ag_xCu_1−xI transforms with increasing temperature from the γ phase, which has a zinc-blende structure, into the α phase [italic v]ia a coexistent α + γ phase.¹⁸ For the relaxation of ⁶³Cu NMR in the γ phase, two processes are considered.¹ One is due to the hopping of a Cu atom between the cubic lattice sites and the other is caused by the defect motion.¹⁹ Usually, the former process occurs at a lower frequency than the latter. Therefore, the former and latter processes are considered to affect T_1ρ and T₁, respectively. When the relaxation of nuclei is caused by ionic diffusion, T₁ and T_1ρ sometimes deviate from the Bloembergen, Purcell and Pound (BPP) theory.²⁰ Such deviations can be explained by a distribution of correlation times. In this case, T₁ in a log(T₁) [italic v]s. (1/T) plot and T_1ρ in a log( T_1ρ) [italic v]s. (1/T) plot show an asymmetric temperature dependence with a steeper slope on the high-temperature side of the minimum.^21–23 We discuss the distribution of the barrier height for ionic diffusion in the γ phase calculated from ⁶³Cu NMR T_1ρ values. The ionic diffusion in the α phase is also discussed on the basis of the ⁶³Cu NMR T₁ values.

Experimental

The Ag_xCu_1−xI crystals were prepared by the melt annealing method as described earlier.^15–1763Cu and ¹²⁷I NMR were measured with a CMX-300 spectrometer at 79.12 and 59.73 MHz, respectively. A single π/2 pulse and a (π/2)_x–τ–(π/2)_y–τ–acquisition pulse sequence were used for the measurements of the ⁶³Cu and ¹²⁷I NMR spectra, respectively. The T₁ measurements were made by using the inversion recovery method in the temperature range 523–173 K. The π/2 pulse widths were 3 and 4 μs for ⁶³Cu and ¹²⁷I, respectively. τ was 30 μs. T_1ρ was measured at a spin-locking rf field of 31 G in the γ phase.

Results

⁶³Cu NMR

The temperature dependences of the ⁶³Cu NMR T₁ and T_1ρ are shown in Fig. 1. Below ca. 280 K, T₁ and T_1ρ are considered to be dominated by the lattice vibration, since T₁ and T_1ρ agreed well and were proportional to T⁻². Above ca. 280 K, T_1ρ decreased rapidly with increasing temperature and showed a minimum at ca. 360 K. In this temperature range, T_1ρ is considered to be dominated by the fluctuation of the quadrupole interaction and the dipole interaction due to the hopping of a Cu ion.¹ For all compounds, except x = 0.99, T_1ρ deviated from BPP behavior and the slope of the temperature dependence of T_1ρ became steeper on the high-temperature side of the minimum. In the case of x = 0.85, the activation energies estimated from the slope of the temperature dependence of T_1ρ were 51 kJ mol⁻¹ on the low-temperature side and 82 kJ mol⁻¹ on the high-temperature side. These temperature dependences of T_1ρ can be explained by the distribution of the correlation times which arises from the distribution of the activation energies for the diffusion of the Cu ions.^21–23 Above ca. 400 K, T₁ decreased rapidly with increasing temperature. For AgI and CuI, the defects increase the anomaly at temperatures approaching the transition to a superionic phase.^1–6 Therefore, T₁ is considered to be dominated by the fluctuation of the electric field gradient at the Cu nuclei due to the defect motion.^1,19 The open circles in Fig. 1 show the temperature dependence of T₁ in the α phase. T₁ in the α phase was shorter than that in the γ phase and increased with increasing temperature. These results suggest that T₁ is dominated by fast ionic diffusion in the α phase.

Fig. 1 Temperature dependences of ⁶³Cu NMR T₁ and T_1ρ for Ag_xCu_1−xI (x = 0.99–0.80). The open and the closed circles show T₁ above and below T_c, respectively. The triangles show T_1ρ. The solid and the broken lines show the theoretical fitting to the observed T₁ and T_1ρ, respectively.

¹²⁷I NMR

The ¹²⁷I NMR spectrum could not be observed in the α phase. Fig. 2 shows the temperature dependence of ¹²⁷I NMR T₁ for the x = 0.99 compound. T₁ was proportional to T⁻² below ca. 280 K and decreased rapidly above ca. 300 K. ¹²⁷I NMR T₁ can be explained by fluctuation of the electric field gradient at the ¹²⁷I nuclei due to lattice vibration and defect motion, in analogy with ⁶³Cu NMR T₁. Fig. 3 shows the ratio of T₁ for ⁶³Cu and ¹²⁷I (T₁(⁶³Cu)/T₁(¹²⁷I)). Below ca. 280 K, where the lattice vibration is considered to dominate the relaxation of both nuclei, T₁(⁶³Cu)/T₁(¹²⁷I) was constant. Above ca. 300 K, T₁(⁶³Cu)/T₁(¹²⁷I) increased rapidly with increasing temperature. These results, which indicate that the defect motion affects more strongly the relaxation of ¹²⁷I than that of ⁶³Cu, are connected with the fact that the distance between the Ag or Cu defect and the I atom is shorter than that between the defect and the Cu atom in the regular site.

Fig. 2 Temperature dependence of ¹²⁷I NMR T₁ in the γ phase of Ag_0.99Cu_0.01I. The solid line shows the theoretical fitting to T₁ below 280 K by using T₁ = aT⁻².

Fig. 3 Temperature dependence of the ratio of ⁶³Cu and ¹²⁷I NMR T₁ for Ag_0.99Cu_0.01I.

Discussion

Motion of Cu ions and defects in the γ phase

In order to relate the results of ⁶³Cu T_1ρ and T₁ to the dynamical properties of the diffusing ions and defects, we consider two relaxation processes. One is due to the hopping Cu ion. The hopping Cu ion undergoes fluctuations of the electric field gradient and the magnetic dipole field and these fluctuations cause the relaxation of the Cu nuclei. The other is caused by fluctuation of the electric field gradient at the ⁶³Cu nuclei on the regular sites due to motion of the defect. In general, the fluctuation of the local field for the second process is faster than for the first process.¹ Therefore, T_1ρ and T₁ are predicted to be sensitive to the two different processes. For the relaxation due to the hopping motion of the Cu ion, the correlation time τ_a corresponds to the time between hops of a Cu ion and assuming an Arrhenius relation can be expressed as

where τ_a0 and E_a are the correlation time at infinite temperature and the activation energy for the hopping motion of the Cu ion, respectively. When a distribution of τ_a is caused by a distribution of E_a, T_1ρ is represented by^21–23

where C_a is the average change in the dipole and the quadrupole interactions for the Cu nuclei due to the hopping motion and ω₁ is the locking rf frequency. For the present case, we assume a Gaussian distribution of E_a and the distribution function Z is written as^21–23

where E_m and E_b are the mean of the activation energies and the half width at half height of the distribution, respectively.

For the relaxation due to the motion of the defect, the correlation time τ_d is the time between hops of the defect and is connected with τ_a by¹

where n_d is the atomic fraction of defects. The relaxation rate due to the motion of the thermally generated defect, which dominates T₁ on the high-temperature side in the γ phase, is written as^21–23

where C_d and ω₀ are the average change in the quadrupole interaction for the ⁶³Cu nuclei due to defect diffusion and the angular NMR frequency. The atomic fraction of thermally generated defects is represented by

where ΔH is the enthalpy for the formation of the Frenkel defect. Since n_d⪡1 holds in the γ phase,^3–5ω₀τ_d⪡1 is predicted and eqn. (5) is rewritten as^19,21–23

The contribution of the lattice vibration to T₁ is written as

The temperature dependence of T_1ρ and T₁ can be explained by

The fitting calculations of T_1ρ and T₁ were performed using eqn. (1)–(10) with C_a, τ_a0, E_m, E_b, C_dn_d^*2, ΔH and a as parameters. The parameters which give the minimum of ∑[(log(T₁)_obs − log(T₁)_cal)² + (log(T_1ρ)_obs − log(T_1ρ)_cal)²] were obtained by using the software package MultiSimplex for the optimization. The broken and the solid lines in Fig. 1 show the theoretical fitting curves for T_1ρ and T₁, respectively. The parameters obtained are shown in Table 1. The ΔH value obtained, which was almost constant at 120 ± 10 kJ mol⁻¹, is comparable to those of the silver halides [AgCl (140 kJ mol⁻¹), AgBr (110 kJ mol⁻¹), β-AgI (79 kJ mol⁻¹)].^6,24 From a molecular dynamics (MD) study of β-AgI, ΔH for the octahedral site and the tetragonal site were obtained as 53 and 115 kJ mol⁻¹, respectively.^3–5 ΔH obtained by this work agrees with the latter value. In a similar manner to Ag_xCu_1−xI (x = 0.99–0.80), the ⁶³Cu NMR T₁ for CuI decreased rapidly above ca. 400 K.¹ Therefore, it is predicted that the ⁶³Cu NMR T₁ is mainly dominated by the dynamics of the defect in the tetrahedral site and the enthalpy for the formation of this defect is little affected by the different cations.

Table 1 Parameters obtained by the fitting of the temperature variation of T_1ρ and T₁ for Ag_xCu_1−xI

Ag0.99Cu0.01I	Ag0.95Cu0.05I	Ag0.90Cu0.10I	Ag0.85Cu0.15I	Ag0.80Cu0.20I
C a/s^−2	(3.0 ± 0.3) × 10^9	(7.0 ± 0.5) × 10^9	(8.0 ± 0.5) × 10^9	(1.0 ± 0.2) × 10^10	(7.0 ± 0.5) × 10^9
C d n d ^*2/s^−2	(4.0 ± 0.4) × 10^23	(1.0 ± 0.2) × 10^24	(4.0 ± 0.4) × 10^22	(1.2 ± 0.2) × 10^23	(7.5 ± 0.5) × 10^22
E m/kJ mol^−1	70 ± 2	75 ± 2	75 ± 2	80 ± 2	80 ± 2
E b/kJ mol^−1	≃0	1.5 ± 0.2	2.5 ± 0.2	5.0 ± 0.5	5.5 ± 0.5
τ a0/s	(1.5 ± 0.3) × 10^−16	(4.0 ± 0.5) × 10^−17	(5.0 ± 0.4) × 10^−17	(2.5 ± 0.5) × 10^−17	(3.0 ± 0.5) × 10^−17
ΔH/kJ mol^−1	120 ± 5	125 ± 5	115 ± 5	125 ± 5	125 ± 5
a/s^−1K^−2	(5.5 ± 0.3) × 10^−4	(5.5 ± 0.3) × 10^−4	(5.5 ± 0.3) × 10^−4	(5.0 ± 0.3) × 10^−4	(5.0 ± 0.3) × 10^−4

---

Fig. 4 shows the distribution of E_a. Although the distribution of E_a was not seen in the x = 0.99 compound, the mean and width of the distribution of E_a for the other compounds increased gradually with increasing Cu concentration. These phenomena are considered to be closely related to the size of the cations. Since the ionic radius of the Cu ion is less than that of the Ag ion, the lattice constant of Ag_xCu_1−xI decreases with increasing Cu concentration.¹⁵ Therefore, the decrease in the available space for ionic diffusion due to the increase in Cu concentration is predicted to cause the increase in the mean and the width of the activation energy. The diffusion constant D_NMR was derived from τ_a0, E_m and E_b using the equation

Fig. 4 Distribution of the activation energies (E_a) for the hopping of the copper ion in the γ phase.

where 〈r²〉 is the mean square jump distance, which was estimated as 2.1 × 10⁻¹⁹ m² from the distance between the nearest regular Ag ion sites in γ-AgI. Fig. 5 shows the temperature dependence of D_NMR . The diffusion constant D_σ can be obtained from the electrical conductivity, using the Nernst–Einstein relation:

Fig. 5 Temperature dependence of the diffusion constant (D_NMR) in the γ phase of Ag_xCu_1−xI obtained from ⁶³Cu NMR T_1ρ.

where σ, Ze and k are the conductivity, the charge of the diffusing ion and Boltzmann's constant, respectively. The number of diffusing ions per unit volume N was estimated as 1.5 × 10²⁸ m⁻³ using the crystal structure of γ-AgI. From the value of the electrical conductivity for Ag_xCu_1−xI (x = 1.0– 0.6) measured by Kusakabe et al.,¹⁴D_σ is found to be of the order of 10⁻¹⁴ m² s⁻¹ at 370 K. These results are in agreement with the value of D_NMR. D_σ represents the total diffusion of the Ag and Cu ions, whereas D_NMR is a constant for the diffusion of the Cu ion. The agreement of both constants suggests that the mobility of the Cu ion is nearly the same as that of the Ag ion in the γ phase.

Ionic diffusion in the α phase

The activation energy of the ionic diffusion in the α phase E_a^α was obtained from the temperature dependence of T₁ by assuming an Arrhenius relation. E_a^α shows an almost constant value of 7 ± 1 kJ mol⁻¹. The E_a^α value obtained from the ⁶³Cu NMR T₁ was smaller than that (∽10 kJ mol⁻¹) determined from the electrical conductivity.¹⁴ The result from the electrical conductivity measurements suggests that the activation energy of the Cu ion is the same as that of the Ag ion in the α phase.¹⁴ Therefore, the deviation is considered to reflect the difference between the ionic motions which are observed by NMR and by electrical conductivity. The correlation time of NMR is considered to be mainly dominated by short-range hopping in the α phase. On the contrary, the electrical conductivity is predicted to be determined by long-range diffusion of the Ag and Cu ions.

Conclusions

(1) The distribution of the activation energy for diffusion of the Cu ion in the γ phase was studied by ⁶³Cu NMR T_1ρ. The mean and the width of the distribution for the activation energy increased with increasing Cu concentration.

(2) The diffusion constant of the Cu ion in the γ phase was estimated from ⁶³Cu NMR T_1ρ. The agreement of the diffusion constants from ⁶³Cu NMR and electrical conductivity measurement shows that the mobilities of the Cu and Ag ions are nearly the same in the γ phase.

(3) The rapid increase in the number of defects near the transition temperature to a superionic phase was confirmed by ⁶³Cu NMR T₁. The enthalpy of formation of the Frenkel defect in the γ phase of Ag_xCu_1−xI (x = 0.99–0.80) was almost independent of the Cu concentration.

(4) The activation energy estimated from ⁶³Cu NMR T₁ in the α phase shows the barrier height for the short-range hopping of the cations.

References

1. J. B. Boyce and B. A. Huberman, Solid State Commun., 1977, 21, 31.
2. S. Miyake, S. Hoshino and T. Takenaka, J. Phys. Soc. Jpn., 1952, 7, 19.
3. J.-S. Lee, S. Adams and J. Maier, J. Phys. Chem. Solids, 2000, 61, 1607.
4. F. Zimmer, P. Ballone, J. Maier and M. Parrinello, J. Chem. Phys., 2000, 112, 6416.
5. N. Hainovsky and J. Maier, Phys. Re[italic v]. B, 1995, 51, 15789.
6. R. J. Cava and E. A. Rietman, Phys. Re[italic v]. B, 1984, 30, 6896.
7. S. Hoshino, J. Phys. Soc. Jpn., 1957, 12, 315.
8. S. Hoshino, T. Sakuma and Y. Fujii, Solid State Commun., 1977, 22, 763.
9. W. Bührer and W. Hälg, Electrochim. Acta, 1977, 22, 701.
10. M. Kusakabe, Y. Ito and S. Tamaki, J. Phys. Condens. Matter, 1996, 8, 6851.
11. M. Kusakabe, Y. Ito, M. Arai, Y. Shirakawa and S. Tamaki, Solid State Ionics, 1996, 86, 231.
12. M. Kusakabe, Y. Ito, M. Arai, Y. Shirakawa and S. Tamaki, Solid State Ionics, 1996, 92, 135.
13. M. Kusakabe, Y. Ito and S. Tamaki, J. Phys. Soc. Jpn., 1996, 65, 3220.
14. M. Kusakabe, Y. Shirakawa, S. Tamaki and Y. Ito, J. Phys. Soc. Jpn., 1995, 64, 170.
15. T. Ida, K. Endo, M. Suhara, M. Kenmotsu, K. Honda, S. Kitagawa and H. Kawabe, Bull. Chem. Soc. Jpn., 1999, 72, 2061.
16. K. Endo, T. Ida, J. Kimura, M. Mizuno, M. Suhara and K. Kihara, Chem. Phys. Lett., 1999, 308, 390.
17. J. Kimura, T. Ida, M. Mizuno, K. Endo, M. Suhara and K. Kihara, J. Mol. Struct., 2000, 522, 61.
18. J. Nölting, Ber. Bunsen-Ges. Phys. Chem., 1964, 68, 932.
19. M. H. Cohen and F. Reif, in Solid State Physics, ed. F. Seitz and D. Turnbull, Academic Press, New York, 1957, vol. 5, p. 321..
20. N. Bloembergen, E. M. Purcell and R. V. Pound, Phys. Re[italic v]., 1948, 73, 679.
21. I. Svare, F. Borsa, D. R. Torgeson and S. W. Martin, Phys. Re[italic v]. B, 1993, 48, 9336.
22. S. Sen, A. M. George and J. F. Stebbins, J. Non-Cryst. Solids, 1996, 197, 53.
23. S. Sen and J. F. Stebbins, Phys. Re[italic v]. B, 1997, 55, 3512.
24. J. K. Aboagye and R. J. Friauf, Phys. Re[italic v]., 1975, 11, 1654.

---