Effect of vacancy diffusion on NQR parameters in LaF3

1. Phys.

Chem.

F’rinted in Great

Sofids Vol. 48. No

9, pp. 83%336,

1987

OOZZ-3697/87 S3.00 + 0 00 Pergamon Journals Ltd.

Britain.

EFFECT OF VACANCY DIFFUSION ON NQR PARAMETERS IN LaF, Department

NOBUO NAKAMURA and HIDEAKIC~nrm of Chemistry, Faculty of Science, Osaka University, Toyonaka 560, Japan (Received 24 December

1986; accepted 13 February 1987)

Abstract-A simple model calculation of the electric field gradient (EFG) at the La site in crystalline LaF, was carried out using Bertaut’s method in order to elucidate a possible relation between the anomalous temperature dependence of the asymmetry parameter of the EFG found in the ‘:%a nuclear q~d~pole resonance and the chancy-m~iated difIusion of the fluorine anions. R~ist~bution of some particular fluorine species among the four crystallographically inequivalent positions between their original sites and the Schottky-type vacancy sites via rapid diffusion contributes to the reduction of the asymmetry of the EFG, this being consistent with the experimental fact that the asymmetry parameter decreases sharply above room temperature. Keywords: Superionic conductor,

diffusion, electric field gradient, nuclear quadrupole resonance.

~RODU~ION In 1966 Lee et al. reported on the temperature resonance dependence of 139La nuclear quadrupole (NQR) frequencies in LaF, [l]. They found that its nuclear quadrupole coupling constant, e2Qq/h, de-

This paper aims to interpret such anomalous behavior of the NQR parameters of ‘39Lain terms of the results of a model calculation of the EFG at the La site in the LaF, crystal in the presence of the vacancymediated diffusion of the fluoride anions.

creases monotonically on heating and, although its temperature coefficient becomes larger above 300 K, the gross feature can be accounted for by the vibrational averaging effect on the electric field gradient (EFG) at the ‘39La site. Gn the other hand, the asymmetry parameter, q, of the EFG increases with an increase in temperature up to about 300 K, and then suddenly begins to decrease on further heating. They attempted to interpret this anomalous decreasing behavior of q as an anisotropic lattice vibrational effect but their analyses led to unrealistic vibrational parameters such as too large vibrational amplitudes. LaF, is known by now as a typical superionic conductor in which the fluoride anions transport the electric charges. Many experimental works such as electrical conductivity [2-51, nuclear magnetic resonance [2,4,6-91, thermal expansion [2], and lattice vibrational spectra [IO] have been carried out on this material to elucidate the m~h~isrn of the anionic conduction: it has been considered that there are two (or more) crystallographically inequivalent fluoride anion sites with their rates of diffusion being considerably different from each other and that the fluorines diffuse via the Scottky-type vacancies. Since the above-mentioned works indicate that fluorine diffusion becomes predominant above room temperature, it is reasonable to consider that the anomalous behavior of tl of the EFG at the ‘3%a site above 3OOK is brought about by some kind of averaging effect of the fluorine diffusion on the EFG at nearby La sites.

A number of X-ray diffraction studies have been carried out on LaF,, leading to different proposed space groups for this substance; these include P6,/mcm [2, 111, P6,/mmc [12], P%l [13, 141, and P63cm [ 151.Recent single crystal neutron diffraction experiments by Gregson and others [ 161 showed that the most probable space group is P6,cm with 6 formula units per unit cell. They also showed that two of the four inequivalent ftuorines have very large and highly anisotropic temperature factors. Sher and others measured the lattice and bulk thermal expansion coefficients, fluorine NMR, and electrical conductivity on single crystals of LaF, above room temperature [2]. Their experimental data could be described by a model in which neutral Schottky defects are produced with a small activation energy but their diffusion requires higher energy. On the other hand, 19F NMR relaxation measur~ents [6,8,9] led to models of two or more sublattices in which two inequivalent fluorine lattice sites exist and the rates of the vacancy-mediated diffusion of these fluorines are significantly different. On the basis of the results of these previous studies we set up a model for the static and dynamic structure of the LaF, crystal and calculated the EFG at the La site: (1) The LaF3 crystal assumes the P6,cm structure 1161and its unit cell contains 6 formula units. There are four crystallographically different fluorines, the numbering of which is the same as in Ref. [16];

THE MODEL STRUCTURE OF LaF,

833

834

NOBUO NAKAMIJRA and HIDEAKI CHIHARA

(2) La and F carry formal charges + 3 and - 1, respectively, in the perfect crystal. A polarization effect was not taken into account in the EFG calculation; (3) There exist neutral Schottky-type anion vacancies at a concentration of 1.21 x lo-’ at room temperature, which increases in number on heating and becomes twice as many at 400 K [2]. These neutral vacancy sites can be used to accommodate diffusing fluoride anions; (4) F- ions diffuse via anion vacancies and their rate of diffusion is higher than the NQR frequencies of “‘La so that the F- ions are distributed at their accessible sites uniformly on the time scale of La NQR. Recent 19F NMR results indicate that this situation holds well above room temperature [8,9]. Under such conditions, the amount of effective charge on a fluorine which each La sees is reduced as the number of vacancies increases. If the number of the fluorine atoms engaging the diffusion is denoted by Nr and the number of the anion vacancies by Nv, the effective charge on an average fluorine may then be represented by (- 1). Nr/(Nr + Nv). The computation of the EFG was carried out by Bertaut’s method [17]. This method takes the lattice sum in the reciprocal lattice space and essentially corresponds to the point charge model. However, contrary to the usual point charge model, it assumes a special form of the charge distribution on each ion to cause very rapid convergence.

compared 38.78 nm-r,

with the corresponding calculated using old X-ray

quantity,

structural data [18]. The electric potentials at the different fluorine sites differ from each other. This result supports the result of recent NMR studies that there are energetically different fluorine sites. The potential difference between them is 11.5 kJ.mol-’ [9], although the present calculation leads to much larger potential differences. In the course of the present work, two papers on the crystal structure analyses of LaF, were published [19,20], which propose the space group P3c 1 for the LaF, crystal and claim that the previous P6, cm structure [16] had been er-

roneously derived without recognizing a twin structure of the specimen. We then did the same calculation as above with the new crystal data [20]; the results are included in Tables 1 and 2. Table 2 records the EFG tensor components and the NQR parameters for the perfect crystal. The principal axes of the EFG tensor, x, y, and z coincide with the crystallographic a, Q*, and c-axes, respectively. The nuclear quadrupole coupling constant e*Qq/h in this Table was calculated using the nuclear quadrupole moment eQ = 8.01 x 10e4*C m* [21] and the Sternheimer antishielding factor (1 - y) = 107 [22]. An excellent agreement was obtained with the experimental value, 16.14 MHz, at 300 K [l]. The asymmetry parameter, q, for the P6,cm structure

RESULTS AND DISCUSSION

Table 1. The Madelung constant, the electrostatic potentials at various atomic sites, and the electric field components at the La site for the P6,cm and Pkl structures at 300K

We first calculated the Madelung energy, the electrostatic potential at the La and all the F sites, the electric field, and the electric field gradient at the La site in a hypothetical perfect crystal of LaF, using the crystal data [16]. The lattice sum was taken in the range - 15 < h, k, 1~ 15 and the convergence proved to be sufficient; the Madelung constant, for example, was reliable to the 4th significant figure. The Madelung constant corresponding to a lattice parameter of u = 0.7185 nm, the electric potential at each one of the non-equivalent atomic sites, and the electric field at the La site are listed in Table 1. The Madelung energy coefficient which is the quantity, (Madelung constant/a) = 38.6135 nm-‘, may be

27.7354 27.7438 Madelung constant? 5363.28 5364.91 Madelung energy (kJ .mol-‘) Electrostatic potential (10e9 C m-‘) -3.881 -3.888 La 1.356 1.458 F(1) 1.425 1.429 F(2) 1.440 1.369 F(3) 1.346 F(4) Electric field at La site (lo-’ Cm-*) -17.193 5.789 E, 10.092 -3.164 EY 0.032 -0.811 EZ t Referred to the lattice constant a = 0.7185 nm 1161.

P6,cm

Table 2. The principal components of EFG tensors at the La site and ra9La NQR parameters P6,cm

P3c 1

Experimentalt

EFG (lo9 Cm-‘) V xx

v, V*,

e2Qq .h -’ (MHz)$

n

1.3012 0.1333 - 1.4259 16.59 0.8252

-0.4783 1.5528 - 1.0744 18.07 0.3839

16.14 0.8054

t Ref. [I]. $ The antishielding factor (1 - y) = 107 was assumed (Ref. [22]).

P3c 1

NQR parameters in LaF,

agrees very well with the experimental value, 0.8054, but the P&l structure leads to a much smaller q value than the experimental one. Since our simple point charge model does not take into account other important factors such as the polarization effect of the ions, it will not be approp~ate to draw a conclusion about the correct structure of the LaF, crystal at the present stage. We adopt, however, the P6, c1)1 structure in the following computation for assessing the effect of the anion diffusion on the EFG at the La site because the computation based on this structure gave a value of the asymmetry parameter q close to that obtained by experiment. We now apply our method of computation to the LaF3 crystal in which vacancy-mediated fluorine diffusion takes place. According to the neutron structure analysis [16], there is a possibility that the fluoride anions, F(1) and F(2), at the 2a and/or the 4fpositions move through channels which run in the crystallographic c-direction. Thus, EFG was first calculated as a function of the effective formal charge on the two F( 1)‘s. As was mentioned in condition (3) in the preceding section, decreasing the negative formal charge on a fluorine atom corresponds to increasing the number of vacancies, which in turn has the same effect as increasing the temperature of the specimen. The result of the calculation is shown in Fig. 1. This figure predicts that the absolute values of both e*Qq/h and q decrease rapidly when the vacancy concentration is increased on heating the sample, the tendency being consistent with the experimental finding [ 11.

I

-I

Charge on

F(I)

Fig. 1. The variation of the EFG tensor components at the La site in LaF, as functions of the effective charge on F(1).

835

Table 4. Dependence of the NQR parameters vacancy concentration, s/N Diffusing species

F(1) F(2)

@e2Qq/~)%rlN -

F(3) F(4)

+ +

(~/N)/10-3~

300 400

1.21 2.35

t Ref. [2]. $ Ref. [4].

~,flO-~sf 1000 5.6

The effective charges cr at a temperature T can be related to the experimental Schottky defect concentration s/N = exp[ -4.05 - (6657/RT)] [2] using the relation given in condition (3) above as 1 - lcrl = s/N = NV/(& + NV). Hence, we can estimate the asymmetry parameter q and its shift from the value in the perfect crystal, Aq, at the temperature T. The results are summarized in Table 3. The calculated Au is larger than the experimental one by about a factor of 10 at 400 K, due probably to the crudeness of the simple point charge approximation. The same kind of calculation was carried out by changing the formal charges on four F(2)%, or six F(3)% or F(4)% (the latter two species are at the c-positions) and the gross features of the results are recorded in Table 4. They indicate that the model assuming the diffusion of either of these fluorine species leads to a positive temperature dependence of e*Qq, contrary to the exerimental results. Only F(1) diffusion gives the correct signs that agree with vacancy-mediated Therefore, the experiment. effusion model in which the diffusion of the F(1) is predominant can describe the experimental temperature dependence of both e2Qq and q in LaF, above room temperature. A more sophisticated model, in which the diffusion of F(1) is predominant but the diffusion of F(2) or other fluoride anions is simultaneously excited to some extent, may lead to a more quantitative interpretation of the experimental NQR parameters. It should be noted that the experimental q decreases monotonically on cooling from room temperature [ 11.In order to confirm if such a behavior of the r~ is realized as a mere thermal expansion effect, we computed the electrostatic interaction parameters for the P6,cm structure with a = 0.7180 nm and E = 0.7342 mn estimated by extrapolation of the experimental lattice parameters down to 200 K [2], and obtained results that the Madelung constant = 27.7487, the principal value of the EFG, vzz = - 1.2595 x lOI J/C m*, and q = 0.8452; the calculation predicts that q assumes a negative tem-

Charge on F(1) -0.989 -0.979

+/a (09 + -

Table 3. The Schottky defect concentration and the NQR parameters when the F(1) anions undergo vacancy-mediated diffusion T(K)

on the

eq/109Cm-’ 1.313 1.199

n

An

0.721 0.138 0.567 0.291

An,,r ~0.02

Noeuo

836 perature fore,

tile

NMMURA

coefficient between 200 and 300 K. Thereexperimental temperature dependence of q

cannot be accounted for by a simple thermal expansion ef%ct. A more rigor~s treatment, including the anisotropic vibrational effect, would be necessary to interpret the temperature dependence of the NQR parameters at low temperatures. Acknowledgement--The present work was partially supported by Grant-in-Aid for Scientific Research No. 4i4~ from the Ministry of Education, Science and CuHure.

and

HIDEAN CNIHARA

6. Goldman M. and Shen L., Phys. Rev. 144,321 (1966). 7. Sher L., Phys. I&. 177, 259 (1968). 8. Jaroszkiewicz 0. A. and Strange J. H., J. Pirys. Cl& 233 f ($985). 9. Aalders A. F., Arts A. F. M. and de Wijn H. W., P&s. Reu. BX, Ml2 (1985). 10. Dixon G. S. and Nicklow R. M., Solid St. Commun. 47, 877 (X983). 11. Oftedal I., Z. phys. Chem., Abstr. B, 5, 272 (1929); Z. phys. Chem., k&r. B, 13, 190 (1931). 12. Schlyter K., Ark. Kemi 5, 73 (1952). 13. Mansmann M., Z. Kkitaflogr, 122, 375 (1965). 14. Zalkin A., Tempieton D. H. and Hopkins T, E., fnorg. C&Y?&S, I466 (f966). 15. de Rango C,, Tsoucaris G. and Zelwer C, CR. Acad. sci., ser. C 263, 64 (I966). 16. Gregson D., Catlow C. R. A., Chadwick A. V., Lander G. H., Cormack A. N. and Fender B. E. F., Acta Cryst.

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P. Vashishta. J. N. Mundv and G. K. Sheno~). a. 683. North-Hollahd, Amsterdam (1979). _‘. . 5. Nagel L. E. and O’Keeffe M., Fast Ion Transport in Solids (Edited by W. van Gool), p. 165. North-Holland, Amsterdam (1973).

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(1969). 19. Maximov B. and Schulz H., AC@ C’~_Y~U. B41,88 (t985). 20. Zalkin A. and Templeton D. H., Acta Cryst. B41, 91 (1985). 21. See Ref. [I].

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P. I”., Adu. NQR 1, 277 ’