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Quasielastic light scattering in Tl-β ″-gallate
Solid State Communications, Vol. 68, No. 6, pp. 581-585, 1988. Printed in Great Britain.
0038-1098/88 $3.00 + .00 Pergamon Press plc
QUASIELASTIC L I G H T S C A T T E R I N G IN T1-/d"-GALLATE T. Suemoto, Y. Takeda and M. lshigame Research Institute for Scientific Measurements, Tohoku University, Katahira Sendal 980, Japan
(Received 24 June 1988 by G. Burns) Quasielastic light scattering (QELS) due to fast ionic diffusion has been studied in /d"-gallates. Intense scattering has been observed in the TI substituted /d'-gallate at room temperature. From the comparison of the scattering intensity in TI- and Na-fl'-gallates, the observed QELS is ascribed to the polarizability fluctuation of the mobile ions caused by the hopping motion of the ions. The polarization characteristics of the QELS suggest that the vacancy collisions play an important role for the light scattering mechanism in this material. Unexpectedly large intensity of the QELS observed at 155 K suggests that the T1 ~ ions have a large hopping rate even at low temperatures. 1. I N T R O D U C T I O N T H E / d - A L U M I N A type superionic conductors have been receiving a considerable interest for a decade as a candidate for the solid electrolyte material for sodium sulphur battery from a view point of application, and also as a typical 2D ionic conductor from a view point of solid state physics. Within the family of/d-alumina type compounds there are two important variations which have been explored to search for an improved version of/d-alumina. Firstly, /d'-alumina was found to have a conductivity several times larger than that for /d-alumina for sodium ions at 500K [I, 2]. The crystal structure of/d"-alumina is closely related to that of/J-alumina [3]. In both compounds, the conduction planes are separated by spinel-like blocks, and the alkali ions can migrate within the 2D space. The conduction path in this plane has a honeycomb structure and there are two types of cross-points, which are called BR (Beevers-Ross) site and aBR (anti-BeeversRoss) site. The most essential difference between/d and fl" phases is believed to be the conduction mechanism of the ions. In the stoichiometric/d-alumina, all the BR sites are occupied by the mobile ions and in the hyperstoichiometric/d-alumina, the excess ions make pairs occupying the two adjacent m-0 sites and migrate on the honeycomb-structured conduction path [4]. In the /d"-alumina, on the other hand, the BR and the aBR sites become equivalent and both sites will be occupied by the mobilc ions in the hypothetical stoichiometric compound. However, in the realistic compound the number of the mobile ions is less than the stoichiometric value, and the ions are believed to migrate through the vacancies on the BR or aBR site [5]. Another important variation of the/d-and/d"-alumina
is realized by replacing the aluminium atoms in the spinel blocks by gallium atoms, which will enlarge the lattice constant and reduce the interaction between the framework and the ions in the conduction plane to result in higher ionic conductivity [6]. This modification was found particularly effective for larger ions such as potassium [7]. In spite of the importance of these //-alumina related materials for understanding the ion conduction mechanism, only a small number of reports are available up to now. Recently, we have shown that the quasielastic light scattering (QELS) in the Brillouin region ( < 2cm ') gives a detailed intbrmation about the ion hopping rate and the ion-ion interaction mechanism [8, 9]. As for the/d-alumina, the diffusion coefficient of Ag' ions estimated from the temperature dependence of the QELS intensity was in good agreement with the value obtained from N M R measurements [9]. One of the interesting results was the polarization characteristics, which showed an evidence for hopping of ion pairs confined in the 2D space parallel to the conduction plane. In contrast to the fl phase, the ion pairs arc not expected in the /d" phase, and the diffusion process will be very different. Therefore, a measurement of QELS will cast a light on the dynamical behavior of the ions in the fl" phase material. In this paper we study the QELS arising from the mobile ions in the/d"-gallate, which is one of the variation of the fl-alumina type compounds, and discuss the behavior of the mobile ions.
581
2. E X P E R I M E N T Single crystals of the Na-fl"-gallate were grown by slow evaporation of Na,O from a mixture consisting
582
Q U A S I E L A S T I C L I G H T S C A T T E R I N G IN TI-fl"-GALLATE
of Ga203 and Na20, following the procedure reported by Foster et al. [10]. The charges were prepared by prereacting a mixture of Ga,O3 and Na2CO3 (99.99%) for 12h at 1000°C to convert Na2CO 3 into Na20. This mixture was loaded in a platinum boat and melted at 1320°C. By keeping one end of the boat at 1310°C in an electric furnace with an appropriate temperature gradient for 2 weeks, single crystals with a typical size of 2 × 5 × 5 m m ~ were obtained. The T1 ions were introduced by soaking the Na-/1"-gallate crystals in molten TINO3 at 250°C in an evacuated Pyrex tube for 3 days [I I]. The samples thus obtained had excellent optical quality, which allowed us to study the details of QELS in very low frequency region avoiding the elastic scattering of the incident laser light. The phase of the crystals was found to be of/3"-type by Laue pattern and the powder diffraction pattern. The general formula of the //"-gallate is A,OxGa20~, where A = Na, K, Ag, TI etc., the value of x ranging typically from 5.6 to 6.0. If we neglect the immobile alkali ions in the spinel blocks, the range o f x becomes 6.9-7.7 [6]. In our case the composition of the TI substituted sample was determined to bc x = 0.75 by X-ray fluorescence analysis. The ion exchange procedure replaces only the Na ions in the conduction plane and not the Na ions which are incorporated in the spinel blocks [I i]. Thus the observed value of x = 7.5 is near the upper limit of the possible composition. The completion of ion exchange was checked by Raman scattering of the mobile ion bands. After the ion exchangc procedure, the peak at 69cm ' which is characteristic to Na-/~'-gallate completely disappeared and a peak due to TI ~ ion appeared at 55 cm '. The Raman frequencies of these bands arc in good agreement with the values reported by Burns et al. [I 1]. For the QELS measurement the 5145 ,/k line from a single mode Ar ion laser was used for excitation and the scattered light was analyzed by a tandem F a b r y Perot interfcrometer equipped with a grating premonochromator [8]. The effective free spectral range of the tandem system was 10cm ' and the resolutiola was 0.04cm '. An iodine vapor cell was used to remove the elastic scattering component. 3. RESULTS A N D D I S C U S S I O N 3.1. Ion dependence and polarization The light scattering spectra were measured at room temperature in air with a right angle scattering geometry. The typical spectra in the low frequency region taken with a polarization of X ( Y Y ) Z + X ( Y X ) Z for the TI and Na materials are shown in Fig. I, where the axis Z is parallel to the c-axis and the
Vol. 68, No. 6
(a)
• .-.
TI
,
•,qte "t
Z ta.I I--
° '~
..
• g.
Z
i
i
i
i
i
1
(b)
,
Na
B
-0.4
0
0.4
0.8
SHIFT (crn-')
Fig. 1. Low frequency light scattering spectra taken in a polarization o f X ( Y Y ) Z + X ( Y X ) Z at room temperature for Tl-/~'-gallate (a) and for Na-fl"-gallate (b). The experimental condition and the unit of the ordinate are the same for (a) and (b). The peak labeled B is ascribed to the Brillouin scattering and the smooth curves are the guides to the eye. axes X and Y lie in the conduction plane. The spectra shown are normalized by the structured transmission efficiency of the iodine vapor filter to remove the spurious structures appeared in the raw data. The normalized data are not shown for the regions where the transmission of the iodine cell is too low ( < 20%) to avoid a poor S/N ratio. In our experiment the component below 0.03cm L was completely masked by the iodine absorption. The peak labelled B in the spectrum for Na-fl"-gailate (Fig. I(b)) is ascribed to Brillouin scattering, while the Briliouin peak is not visible for Tl-/J"-gailate (Fig. I(a)) probably due to accidental coincidence of the frequency with the sharp and intense absorption of the iodine ceil. In the spectrum for Tl-/~"-gallate, a broad feature centered at 0cm ' was observed, which is ascribed to the QELS due to ion diffusion as discussed below. Similar QELS spectra have been previously observed in Ag- and Tl-~-alumina and it has been concluded that the QELS is caused by polarizability fluctuation due to diffusive motion of ions, based on polarization, ion dependence and temperature dependence. The most striking feature in Fig. I is the large difference of the intensity of the QELS depending on
Voi. 68, No. 6
QUASIELASTIC L I G H T S C A T T E R I N G IN TI-/3"-GALLATE
Table 1. Squared Raman tensor for the Q E L S in a frequency region between - 0.3 cm- J and + 0.3 cm ' obtained at 300 K
X Y Z
X
Y
Z
100
15 100
4 4 6
the ion species. The intensity at 0.05cm ~ is about 130 times larger for Tl-fl"-gallate compared with that for the Na-fl"-gallate. The QELS intensity is proportional to the square of the modulation amplitude in the polarizability caused by the ion motion and the magnitude of the modulation may be proportional to the polarizability ct of the ion itself. Then the QELS intensity will be proportional to the square of the polarizability of the mobile ions. Therefore the ratio of the QELS in T1- and Na-fl"-gallates should be explained by the ratio of ~2, provided the diffusion coefficient is the same for TI and Na. This assumption seems reasonable because the electric conductivity in these two materials was found almost the same at room temperature by our preliminary measurements. The polarizability is ~t = 0.41 for Na t and ~ = 5.2 for TI + in a unit of 10 .-24cm 3, and the squared ratio is (5.2/0.41) 2 = 160. This value corresponds roughly to the observed intensity ratio. This fact shows that the origin of the observed QELS is the polarizability fluctuation of the diffusing ions due to the hopping motion. The squared elements of the Raman tensor for Tl-/3-gallate were determined from polarization measurement of QELS intensity integrated in between - 0 . 3 c m ~and + 0 . 3 c m ~and are shown in Table l. Here, the YY, Z X and YX components are measured in X - Z scattering geometry. The Z Z component is measured in X- Y scattering geometry and normalized by the YXcomponent measured in the same scattering geometry, to match with the former measurement. 3.2. The vacancy pair model
It is generally agreed that the ionic diffusion in the fl" type material occurs through the vacancy on the BR and aBR sites [5], which are equivalent unlike the fl type material. The simplest picture of ion hopping in this system is depicted in Fig. 2(a). We do not have any detailed information about the site occupancy in the/3"-gallate, but the situation will be similar to that in fl"-alumina. In the/3"-alumina, the normal site of the mobile ion is the 6c position of the space group R~m, which corresponds to the BR and aBR site in/3 phase. The ions surrounding the vacancy tend to relax toward the vacant site. As a consequence, these ions occupy the ! 8 h positions, which are close to the mid-
583
(a) i.~/6c-site
_
~
18h-site ~" m- 0 site
(b)
c1
c2
c3
c4
Fig. 2. The figure (a) depicts the ion configuration before and after hopping of a single vacancy in the conduction plane. The figures C I - C 3 in (b) depict three possible ion configurations of a vacancy pair and the figure C4 shows two separated vacancies. oxygen sites. If we consider hopping of a single vacancy as shown in Fig. 3(a), the polarizability tensor of the system is the same before and after the hopping, even if we consider the relaxation of the surrounding ions, because the vacant site has an approximate three-fold axis along the c-axis and the configuration before and after hopping (Fig. 2(a)) are equivalent so far as the polarizability tensor is concerned. Thus, in terms of this picture we can expect no polarizability fluctuation and no QELS due to ion hopping. In order to understand the observed intense QELS in Tl-/Y'-gallate, we take in account the vacancy collision. From the composition determined by the X-ray fluorescence analysis, the vacancy concentration of our samples in the conduction plane was found as high as 25%. Therefore, we expect very frequent collision of the vacancies, which means simultaneous occupation of adjacent 6c sites by two vacancies. The configurations C I - C 3 in Fig. 2(b) show the situation in the vacancy collision and the configuration C4 shows an example of separated vacancies. The ion hopping will cause transitions between these four ion configurations. If we have, for example, another vacancy on the site 4 in the configuration C l, the hopping of a vacancy from site 2 to site 3 leaves an isolated vacancy on site 6 and creates a new vacancy pair (3-4) which is equivalent to C2. By solving rate equations which describe the transition between these states, we can obtain the QELS spectrum and the polarization character. The problem is formally equivalent to the ion pair model, which was applied to fl-alumina [9]. The time dependence of the correlation function
584
Q U A S I E L A S T I C L I G H T S C A T T E R I N G IN T I - f f ' - G A L L A T E ( a ) TI
300
Vol. 68, No. 6
and those for ( ' 2 and C3 can be obtained by rotating the X and Y axes for +120 ° around Z axis. The polarizability for the single vacancy configuration (Fig. 2(a)) is
K
a4
I-tt~ Z i,i
I
~4 =
a~
.
(4)
Cl5
Z
'~;,",~ ,,;,~ ,
" -
-
The intensity of the scattered light at a frequency shift ,~ is given by.
_
(b) TI
A
SO.)) -
155 K
;
2n J (.zt(t)~(O))e""'dt.
(5)
t
By using the solution of the rate equation (I), the correlation function in the integrand is written as. (cc(t):~(0))
~, :,
;:..
O
:
:
R =
1
2 SHIFT
3
of the configuration population N,,(t) is described by rate equations: d o~(N"(t)N'(O))
=
~] C,,(N,(t)N,(O))
(I)
wherc/~ (p = I, 2, 3, 4) denotes the ion configuration and C,,,. denotes the transition rate from p to v. We define C, ~ C'12 = C'23 = (~'31 = ("21 = (-']'32 = (~'13 for rotation, (': - (4,_ = ('a3 = Cat for creation, and ('~ - ( ' u = C:4 = ('~.~ for destruction of the ion pairs, respectively. Wc assume that the average local polarizability is expressed as
~ ~N,,(t).
L (P,,)~ :~,, %(k).
(2)
where ~,, denotes the polarizability of configuration p. Disregarding the orientation of the oxygen tetrahedron around the mobile ion on the 6c site [10], the polarizability of the ion pair CI in Fig. 2 is written as,
a. °
a3
I
(7)
(3)
('1
ZC I - ~-,~
• 2Cl --~'~
('1 r
~
CI
\/"t~'2 ('~
('1
~"C2C'~
• •
Cl
CI \."C.
~"t;'2('x
C ~
2Cl --('3 \,C2C 3
\;'C?C~ ~ 3t2%
(8) p,, is the averaged population of configuration It. By performing thc Fourier transform given by equation (5). we obtain a superposition of Lorcntzians with various polarization characteristics for the QELS spectrum:
S(e~) -¢. ~ IR(k)!: k
F(k) F-'(k) + ~,-"
(9)
The non-zero cigenvalucs and the corresponding Raman tensors are. ( i ) F I = 3(71 + ('~
i R l i 2 -b [R212
=
H I
=
(6)
F(k) is thc eigenvalue and e,(k) is the eigenvcctor of the symmetrized rate matrix.
(cm-')
Fig. 3. The QELS spectra for Tl-fl"-gallate at 300 K (a) and 155K (b) takcn in a polarization o f X ( Y Y ) Z + X ( Y X ) Z. The labels B in thc spectrum (b) denote the Brillouin peaks. The dashed curve in thc spectrum (a) is a Lorentzian fitted to the data in a frequency range over - 0.3 cm ~to + 0.3 cm ~. Both spectra arc plotted in the same intensity units.
i,
JRI z e t,k~,l
where R is the Raman tensor:
B.
,.
B
0
7(1) =
N~ k
.-:.~
• "~
=
i °oI A
0
(10)
0
where A = (3/8)p,(a, - a 2 ) z and p, is the averaged population of one of the pair configurations. (ii) F 3 = 3C2 + ('3
[R~[ e =
0
B
0
0
(11) •
Vol. 68, No. 6
Q U A S I E L A S T I C L I G H T S C A T T E R I N G IN TI-fl"-GALLATE
where B = 3C~p~/4(3C2 + C~)(aj + a, - 2a4) 2 and C = 3C3p~/(3C, + C3)(a3 - as) 2. The observed polarization character in Table 1 shows that the X X and Y Y elements are about 7 times larger than the off-diagonal ones. But from the mode (i) we expect the same intensity for the diagonal and off-diagonal elements. This fact indicates that the main contribution comes from the other mode i.e., the mode (ii), which corresponds to the fluctuation due to creation and destruction of the ion pairs. The observed ratio of the elements can be interpreted by setting A :B" C = 15"85"6. We can not determine the ratio of the original parameters a~-as and Ct-C3, because we have too many adjustable parameters. However, it is evidently possible to find a set of parameters to explain the ratio of A • B' C quoted above. This result shows that the polarizability fluctuation is mainly caused by the creation and destruction of the vacancy pairs. 3.3. Spectral shape
Figure 3 shows the unpolarized QELS spectra corrected for the absorption spectrum of the iodine vapor filter measured at 300 and 155 K, The ordinates of these spectra are in the same unit. The spikes at 0.62 and 1.0 cm-~ in (b) are the Brillouin scattering. These peaks are absent in (a) probably due to accidental coincidence of the Brillouin frequencies with the sharp absorption lines of the iodine vapor. One interesting feature of these spectra is the large deviation of the shape from the Lorentzian; the best fit below 0.3 cm t is indicated by a dashed curve in Fig. 3(a). According to the discussion in the preceding section based on the polarization character, we expect only one Lorentzian component corresponding to the mode (ii) except for a relatively small contribution from mode (i). This argument holds in the frequency range where the polarization was measured i.e., below 0.3cm -t. If the high frequency tail beyond 0.3 cm ~were due to mode (i), the off-diagonal component below 0.3 cm ~would be of comparable magnitude with the diagonal components. But, as readily seen from the figure, this is not the case. Therefore, we have to consider other reasons for this non-Lorentzian spectral shape. We have observed such a feature in some superionic conductors [8] and shown that this can be well explaincd in terms either of the barrier height distribution for the ion hopping or of the many body interaction among the
585
ions as discussed by Ngai et al. Although we can not decide the origin of this non-Lorentzian shape for the T1-//"-gallate at present, it seems reasonable to expect a strong interaction between the ions which can modify the individual hopping rate and gives a distribution in the relaxation time, because the concentration of the mobile ions is very high. Another interesting fact seen in Fig. 3 is the existence of a surprisingly intense QELS even at a temperature as low as 155 K. In Ag- or Tl-//-alumina, for example, the QELS intensity decreases rapidly by lowering the temperature below room temperature [9]. The existence of intense QELS suggests that the ion hopping rate is very high even at low temperatures. Such a situation might be possible, because the intrinsic activation energy for the single ion hopping in the Na-/~"-alumina was calculated to tge 0.02eV [2], which is extremely lower than the observed activation energy of conductivity. This point needs further investigation to bc clarificd. Acknowledgement - This work has been supported by a Grant-in-Aid for Special Project Research from the Ministry of Education, Science and Culture, Japan.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. il.
J.T. Kummer, Prog. in SolidState Chem. 7, 141 (1972). J.C. Wang, Phys. Rev. 26, 5911 (1982). J.P. Boilot, G. Collin, Ph. Colomban & R. Comes, Phys. Rev. B22, 5912 (I 980). J.C. Wang, M. Gaffari & Sang-il Choi, J. Chem. Phys. 63, 772 (1975). W. Hayes & G.F. Hopper, J. Phys. C: SolM State Phys. 16, 2529 (1983). L.M. Foster, Fast hm Transport in Solids, (Ed. Vashishta, Mundy, Shenoy), Elsevier, North Holland, Inc. (1979). G.V. Chandrashekhar & L.M. Foster, Solid State Commun. 27, 269 (1978). T, Suemoto & M. Ishigame, Phys. Rev. B33, 2757 (1986). T. Suemoto & M. Ishigame, Phys. Rev. B32, 4126 (1985). L.M. Foster, G.V. Chandrashekhar, J.E. Scardefield & R.B. Bradford, J. Amer. Ceram. Soc. 63, 509 (1980). Gerald Burns, G.V. Chandrashekhar, F.H. Dacol, L.M. Foster & H.R. Chandrasekhar, Phys. Rev, B22, 1073 (1980).
0038-1098/88 $3.00 + .00 Pergamon Press plc
QUASIELASTIC L I G H T S C A T T E R I N G IN T1-/d"-GALLATE T. Suemoto, Y. Takeda and M. lshigame Research Institute for Scientific Measurements, Tohoku University, Katahira Sendal 980, Japan
(Received 24 June 1988 by G. Burns) Quasielastic light scattering (QELS) due to fast ionic diffusion has been studied in /d"-gallates. Intense scattering has been observed in the TI substituted /d'-gallate at room temperature. From the comparison of the scattering intensity in TI- and Na-fl'-gallates, the observed QELS is ascribed to the polarizability fluctuation of the mobile ions caused by the hopping motion of the ions. The polarization characteristics of the QELS suggest that the vacancy collisions play an important role for the light scattering mechanism in this material. Unexpectedly large intensity of the QELS observed at 155 K suggests that the T1 ~ ions have a large hopping rate even at low temperatures. 1. I N T R O D U C T I O N T H E / d - A L U M I N A type superionic conductors have been receiving a considerable interest for a decade as a candidate for the solid electrolyte material for sodium sulphur battery from a view point of application, and also as a typical 2D ionic conductor from a view point of solid state physics. Within the family of/d-alumina type compounds there are two important variations which have been explored to search for an improved version of/d-alumina. Firstly, /d'-alumina was found to have a conductivity several times larger than that for /d-alumina for sodium ions at 500K [I, 2]. The crystal structure of/d"-alumina is closely related to that of/J-alumina [3]. In both compounds, the conduction planes are separated by spinel-like blocks, and the alkali ions can migrate within the 2D space. The conduction path in this plane has a honeycomb structure and there are two types of cross-points, which are called BR (Beevers-Ross) site and aBR (anti-BeeversRoss) site. The most essential difference between/d and fl" phases is believed to be the conduction mechanism of the ions. In the stoichiometric/d-alumina, all the BR sites are occupied by the mobile ions and in the hyperstoichiometric/d-alumina, the excess ions make pairs occupying the two adjacent m-0 sites and migrate on the honeycomb-structured conduction path [4]. In the /d"-alumina, on the other hand, the BR and the aBR sites become equivalent and both sites will be occupied by the mobilc ions in the hypothetical stoichiometric compound. However, in the realistic compound the number of the mobile ions is less than the stoichiometric value, and the ions are believed to migrate through the vacancies on the BR or aBR site [5]. Another important variation of the/d-and/d"-alumina
is realized by replacing the aluminium atoms in the spinel blocks by gallium atoms, which will enlarge the lattice constant and reduce the interaction between the framework and the ions in the conduction plane to result in higher ionic conductivity [6]. This modification was found particularly effective for larger ions such as potassium [7]. In spite of the importance of these //-alumina related materials for understanding the ion conduction mechanism, only a small number of reports are available up to now. Recently, we have shown that the quasielastic light scattering (QELS) in the Brillouin region ( < 2cm ') gives a detailed intbrmation about the ion hopping rate and the ion-ion interaction mechanism [8, 9]. As for the/d-alumina, the diffusion coefficient of Ag' ions estimated from the temperature dependence of the QELS intensity was in good agreement with the value obtained from N M R measurements [9]. One of the interesting results was the polarization characteristics, which showed an evidence for hopping of ion pairs confined in the 2D space parallel to the conduction plane. In contrast to the fl phase, the ion pairs arc not expected in the /d" phase, and the diffusion process will be very different. Therefore, a measurement of QELS will cast a light on the dynamical behavior of the ions in the fl" phase material. In this paper we study the QELS arising from the mobile ions in the/d"-gallate, which is one of the variation of the fl-alumina type compounds, and discuss the behavior of the mobile ions.
581
2. E X P E R I M E N T Single crystals of the Na-fl"-gallate were grown by slow evaporation of Na,O from a mixture consisting
582
Q U A S I E L A S T I C L I G H T S C A T T E R I N G IN TI-fl"-GALLATE
of Ga203 and Na20, following the procedure reported by Foster et al. [10]. The charges were prepared by prereacting a mixture of Ga,O3 and Na2CO3 (99.99%) for 12h at 1000°C to convert Na2CO 3 into Na20. This mixture was loaded in a platinum boat and melted at 1320°C. By keeping one end of the boat at 1310°C in an electric furnace with an appropriate temperature gradient for 2 weeks, single crystals with a typical size of 2 × 5 × 5 m m ~ were obtained. The T1 ions were introduced by soaking the Na-/1"-gallate crystals in molten TINO3 at 250°C in an evacuated Pyrex tube for 3 days [I I]. The samples thus obtained had excellent optical quality, which allowed us to study the details of QELS in very low frequency region avoiding the elastic scattering of the incident laser light. The phase of the crystals was found to be of/3"-type by Laue pattern and the powder diffraction pattern. The general formula of the //"-gallate is A,OxGa20~, where A = Na, K, Ag, TI etc., the value of x ranging typically from 5.6 to 6.0. If we neglect the immobile alkali ions in the spinel blocks, the range o f x becomes 6.9-7.7 [6]. In our case the composition of the TI substituted sample was determined to bc x = 0.75 by X-ray fluorescence analysis. The ion exchange procedure replaces only the Na ions in the conduction plane and not the Na ions which are incorporated in the spinel blocks [I i]. Thus the observed value of x = 7.5 is near the upper limit of the possible composition. The completion of ion exchange was checked by Raman scattering of the mobile ion bands. After the ion exchangc procedure, the peak at 69cm ' which is characteristic to Na-/~'-gallate completely disappeared and a peak due to TI ~ ion appeared at 55 cm '. The Raman frequencies of these bands arc in good agreement with the values reported by Burns et al. [I 1]. For the QELS measurement the 5145 ,/k line from a single mode Ar ion laser was used for excitation and the scattered light was analyzed by a tandem F a b r y Perot interfcrometer equipped with a grating premonochromator [8]. The effective free spectral range of the tandem system was 10cm ' and the resolutiola was 0.04cm '. An iodine vapor cell was used to remove the elastic scattering component. 3. RESULTS A N D D I S C U S S I O N 3.1. Ion dependence and polarization The light scattering spectra were measured at room temperature in air with a right angle scattering geometry. The typical spectra in the low frequency region taken with a polarization of X ( Y Y ) Z + X ( Y X ) Z for the TI and Na materials are shown in Fig. I, where the axis Z is parallel to the c-axis and the
Vol. 68, No. 6
(a)
• .-.
TI
,
•,qte "t
Z ta.I I--
° '~
..
• g.
Z
i
i
i
i
i
1
(b)
,
Na
B
-0.4
0
0.4
0.8
SHIFT (crn-')
Fig. 1. Low frequency light scattering spectra taken in a polarization o f X ( Y Y ) Z + X ( Y X ) Z at room temperature for Tl-/~'-gallate (a) and for Na-fl"-gallate (b). The experimental condition and the unit of the ordinate are the same for (a) and (b). The peak labeled B is ascribed to the Brillouin scattering and the smooth curves are the guides to the eye. axes X and Y lie in the conduction plane. The spectra shown are normalized by the structured transmission efficiency of the iodine vapor filter to remove the spurious structures appeared in the raw data. The normalized data are not shown for the regions where the transmission of the iodine cell is too low ( < 20%) to avoid a poor S/N ratio. In our experiment the component below 0.03cm L was completely masked by the iodine absorption. The peak labelled B in the spectrum for Na-fl"-gailate (Fig. I(b)) is ascribed to Brillouin scattering, while the Briliouin peak is not visible for Tl-/J"-gailate (Fig. I(a)) probably due to accidental coincidence of the frequency with the sharp and intense absorption of the iodine ceil. In the spectrum for Tl-/~"-gallate, a broad feature centered at 0cm ' was observed, which is ascribed to the QELS due to ion diffusion as discussed below. Similar QELS spectra have been previously observed in Ag- and Tl-~-alumina and it has been concluded that the QELS is caused by polarizability fluctuation due to diffusive motion of ions, based on polarization, ion dependence and temperature dependence. The most striking feature in Fig. I is the large difference of the intensity of the QELS depending on
Voi. 68, No. 6
QUASIELASTIC L I G H T S C A T T E R I N G IN TI-/3"-GALLATE
Table 1. Squared Raman tensor for the Q E L S in a frequency region between - 0.3 cm- J and + 0.3 cm ' obtained at 300 K
X Y Z
X
Y
Z
100
15 100
4 4 6
the ion species. The intensity at 0.05cm ~ is about 130 times larger for Tl-fl"-gallate compared with that for the Na-fl"-gallate. The QELS intensity is proportional to the square of the modulation amplitude in the polarizability caused by the ion motion and the magnitude of the modulation may be proportional to the polarizability ct of the ion itself. Then the QELS intensity will be proportional to the square of the polarizability of the mobile ions. Therefore the ratio of the QELS in T1- and Na-fl"-gallates should be explained by the ratio of ~2, provided the diffusion coefficient is the same for TI and Na. This assumption seems reasonable because the electric conductivity in these two materials was found almost the same at room temperature by our preliminary measurements. The polarizability is ~t = 0.41 for Na t and ~ = 5.2 for TI + in a unit of 10 .-24cm 3, and the squared ratio is (5.2/0.41) 2 = 160. This value corresponds roughly to the observed intensity ratio. This fact shows that the origin of the observed QELS is the polarizability fluctuation of the diffusing ions due to the hopping motion. The squared elements of the Raman tensor for Tl-/3-gallate were determined from polarization measurement of QELS intensity integrated in between - 0 . 3 c m ~and + 0 . 3 c m ~and are shown in Table l. Here, the YY, Z X and YX components are measured in X - Z scattering geometry. The Z Z component is measured in X- Y scattering geometry and normalized by the YXcomponent measured in the same scattering geometry, to match with the former measurement. 3.2. The vacancy pair model
It is generally agreed that the ionic diffusion in the fl" type material occurs through the vacancy on the BR and aBR sites [5], which are equivalent unlike the fl type material. The simplest picture of ion hopping in this system is depicted in Fig. 2(a). We do not have any detailed information about the site occupancy in the/3"-gallate, but the situation will be similar to that in fl"-alumina. In the/3"-alumina, the normal site of the mobile ion is the 6c position of the space group R~m, which corresponds to the BR and aBR site in/3 phase. The ions surrounding the vacancy tend to relax toward the vacant site. As a consequence, these ions occupy the ! 8 h positions, which are close to the mid-
583
(a) i.~/6c-site
_
~
18h-site ~" m- 0 site
(b)
c1
c2
c3
c4
Fig. 2. The figure (a) depicts the ion configuration before and after hopping of a single vacancy in the conduction plane. The figures C I - C 3 in (b) depict three possible ion configurations of a vacancy pair and the figure C4 shows two separated vacancies. oxygen sites. If we consider hopping of a single vacancy as shown in Fig. 3(a), the polarizability tensor of the system is the same before and after the hopping, even if we consider the relaxation of the surrounding ions, because the vacant site has an approximate three-fold axis along the c-axis and the configuration before and after hopping (Fig. 2(a)) are equivalent so far as the polarizability tensor is concerned. Thus, in terms of this picture we can expect no polarizability fluctuation and no QELS due to ion hopping. In order to understand the observed intense QELS in Tl-/Y'-gallate, we take in account the vacancy collision. From the composition determined by the X-ray fluorescence analysis, the vacancy concentration of our samples in the conduction plane was found as high as 25%. Therefore, we expect very frequent collision of the vacancies, which means simultaneous occupation of adjacent 6c sites by two vacancies. The configurations C I - C 3 in Fig. 2(b) show the situation in the vacancy collision and the configuration C4 shows an example of separated vacancies. The ion hopping will cause transitions between these four ion configurations. If we have, for example, another vacancy on the site 4 in the configuration C l, the hopping of a vacancy from site 2 to site 3 leaves an isolated vacancy on site 6 and creates a new vacancy pair (3-4) which is equivalent to C2. By solving rate equations which describe the transition between these states, we can obtain the QELS spectrum and the polarization character. The problem is formally equivalent to the ion pair model, which was applied to fl-alumina [9]. The time dependence of the correlation function
584
Q U A S I E L A S T I C L I G H T S C A T T E R I N G IN T I - f f ' - G A L L A T E ( a ) TI
300
Vol. 68, No. 6
and those for ( ' 2 and C3 can be obtained by rotating the X and Y axes for +120 ° around Z axis. The polarizability for the single vacancy configuration (Fig. 2(a)) is
K
a4
I-tt~ Z i,i
I
~4 =
a~
.
(4)
Cl5
Z
'~;,",~ ,,;,~ ,
" -
-
The intensity of the scattered light at a frequency shift ,~ is given by.
_
(b) TI
A
SO.)) -
155 K
;
2n J (.zt(t)~(O))e""'dt.
(5)
t
By using the solution of the rate equation (I), the correlation function in the integrand is written as. (cc(t):~(0))
~, :,
;:..
O
:
:
R =
1
2 SHIFT
3
of the configuration population N,,(t) is described by rate equations: d o~(N"(t)N'(O))
=
~] C,,(N,(t)N,(O))
(I)
wherc/~ (p = I, 2, 3, 4) denotes the ion configuration and C,,,. denotes the transition rate from p to v. We define C, ~ C'12 = C'23 = (~'31 = ("21 = (-']'32 = (~'13 for rotation, (': - (4,_ = ('a3 = Cat for creation, and ('~ - ( ' u = C:4 = ('~.~ for destruction of the ion pairs, respectively. Wc assume that the average local polarizability is expressed as
~ ~N,,(t).
L (P,,)~ :~,, %(k).
(2)
where ~,, denotes the polarizability of configuration p. Disregarding the orientation of the oxygen tetrahedron around the mobile ion on the 6c site [10], the polarizability of the ion pair CI in Fig. 2 is written as,
a. °
a3
I
(7)
(3)
('1
ZC I - ~-,~
• 2Cl --~'~
('1 r
~
CI
\/"t~'2 ('~
('1
~"C2C'~
• •
Cl
CI \."C.
~"t;'2('x
C ~
2Cl --('3 \,C2C 3
\;'C?C~ ~ 3t2%
(8) p,, is the averaged population of configuration It. By performing thc Fourier transform given by equation (5). we obtain a superposition of Lorcntzians with various polarization characteristics for the QELS spectrum:
S(e~) -¢. ~ IR(k)!: k
F(k) F-'(k) + ~,-"
(9)
The non-zero cigenvalucs and the corresponding Raman tensors are. ( i ) F I = 3(71 + ('~
i R l i 2 -b [R212
=
H I
=
(6)
F(k) is thc eigenvalue and e,(k) is the eigenvcctor of the symmetrized rate matrix.
(cm-')
Fig. 3. The QELS spectra for Tl-fl"-gallate at 300 K (a) and 155K (b) takcn in a polarization o f X ( Y Y ) Z + X ( Y X ) Z. The labels B in thc spectrum (b) denote the Brillouin peaks. The dashed curve in thc spectrum (a) is a Lorentzian fitted to the data in a frequency range over - 0.3 cm ~to + 0.3 cm ~. Both spectra arc plotted in the same intensity units.
i,
JRI z e t,k~,l
where R is the Raman tensor:
B.
,.
B
0
7(1) =
N~ k
.-:.~
• "~
=
i °oI A
0
(10)
0
where A = (3/8)p,(a, - a 2 ) z and p, is the averaged population of one of the pair configurations. (ii) F 3 = 3C2 + ('3
[R~[ e =
0
B
0
0
(11) •
Vol. 68, No. 6
Q U A S I E L A S T I C L I G H T S C A T T E R I N G IN TI-fl"-GALLATE
where B = 3C~p~/4(3C2 + C~)(aj + a, - 2a4) 2 and C = 3C3p~/(3C, + C3)(a3 - as) 2. The observed polarization character in Table 1 shows that the X X and Y Y elements are about 7 times larger than the off-diagonal ones. But from the mode (i) we expect the same intensity for the diagonal and off-diagonal elements. This fact indicates that the main contribution comes from the other mode i.e., the mode (ii), which corresponds to the fluctuation due to creation and destruction of the ion pairs. The observed ratio of the elements can be interpreted by setting A :B" C = 15"85"6. We can not determine the ratio of the original parameters a~-as and Ct-C3, because we have too many adjustable parameters. However, it is evidently possible to find a set of parameters to explain the ratio of A • B' C quoted above. This result shows that the polarizability fluctuation is mainly caused by the creation and destruction of the vacancy pairs. 3.3. Spectral shape
Figure 3 shows the unpolarized QELS spectra corrected for the absorption spectrum of the iodine vapor filter measured at 300 and 155 K, The ordinates of these spectra are in the same unit. The spikes at 0.62 and 1.0 cm-~ in (b) are the Brillouin scattering. These peaks are absent in (a) probably due to accidental coincidence of the Brillouin frequencies with the sharp absorption lines of the iodine vapor. One interesting feature of these spectra is the large deviation of the shape from the Lorentzian; the best fit below 0.3 cm t is indicated by a dashed curve in Fig. 3(a). According to the discussion in the preceding section based on the polarization character, we expect only one Lorentzian component corresponding to the mode (ii) except for a relatively small contribution from mode (i). This argument holds in the frequency range where the polarization was measured i.e., below 0.3cm -t. If the high frequency tail beyond 0.3 cm ~were due to mode (i), the off-diagonal component below 0.3 cm ~would be of comparable magnitude with the diagonal components. But, as readily seen from the figure, this is not the case. Therefore, we have to consider other reasons for this non-Lorentzian spectral shape. We have observed such a feature in some superionic conductors [8] and shown that this can be well explaincd in terms either of the barrier height distribution for the ion hopping or of the many body interaction among the
585
ions as discussed by Ngai et al. Although we can not decide the origin of this non-Lorentzian shape for the T1-//"-gallate at present, it seems reasonable to expect a strong interaction between the ions which can modify the individual hopping rate and gives a distribution in the relaxation time, because the concentration of the mobile ions is very high. Another interesting fact seen in Fig. 3 is the existence of a surprisingly intense QELS even at a temperature as low as 155 K. In Ag- or Tl-//-alumina, for example, the QELS intensity decreases rapidly by lowering the temperature below room temperature [9]. The existence of intense QELS suggests that the ion hopping rate is very high even at low temperatures. Such a situation might be possible, because the intrinsic activation energy for the single ion hopping in the Na-/~"-alumina was calculated to tge 0.02eV [2], which is extremely lower than the observed activation energy of conductivity. This point needs further investigation to bc clarificd. Acknowledgement - This work has been supported by a Grant-in-Aid for Special Project Research from the Ministry of Education, Science and Culture, Japan.
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J.T. Kummer, Prog. in SolidState Chem. 7, 141 (1972). J.C. Wang, Phys. Rev. 26, 5911 (1982). J.P. Boilot, G. Collin, Ph. Colomban & R. Comes, Phys. Rev. B22, 5912 (I 980). J.C. Wang, M. Gaffari & Sang-il Choi, J. Chem. Phys. 63, 772 (1975). W. Hayes & G.F. Hopper, J. Phys. C: SolM State Phys. 16, 2529 (1983). L.M. Foster, Fast hm Transport in Solids, (Ed. Vashishta, Mundy, Shenoy), Elsevier, North Holland, Inc. (1979). G.V. Chandrashekhar & L.M. Foster, Solid State Commun. 27, 269 (1978). T, Suemoto & M. Ishigame, Phys. Rev. B33, 2757 (1986). T. Suemoto & M. Ishigame, Phys. Rev. B32, 4126 (1985). L.M. Foster, G.V. Chandrashekhar, J.E. Scardefield & R.B. Bradford, J. Amer. Ceram. Soc. 63, 509 (1980). Gerald Burns, G.V. Chandrashekhar, F.H. Dacol, L.M. Foster & H.R. Chandrasekhar, Phys. Rev, B22, 1073 (1980).