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Ac response of polycrystalline sodium yttrium fluoride
*H __ .
__ 44._ t8 Solid State Ionics 69 (1994) 37-41
ELFEVIER
ACresponse of polycrystalline sodium yttrium fluoride Md. Shareefuddin, M. Narasimha Chary Department of Physics, Osmania University, Hyderabad 500 007, India
Received 10 October 1993; accepted for publication 28 February 1994
Abstract AC response of polycrystalline sodium yttrium fluoride (NaYF,) has been studied in the temperature region of 350-950 K and in the frequency range 50 Hz-100 kHz using blocking electrodes. From the admittance plots, the bulk conductivity (a), dielectric constant (c’) and loss angle (tan 6) were determined at different temperatures and frequencies. The conductivity was found to increase with increasing temperature. Large value of E’observed at high temperatures was attributed to space charge polarization effects. The relaxation character of tan 6 was observed. The ac conductivity was found to increase with increasing frequency.
1. Introduction AC measurements in a wide frequency range are widely employed to determine the bulk conductivity of solids, since dc conductivity and mono-frequency studies are often hampered by the occurrence of extensive interlacial polarization at elevated temperatures [ 1,2 1. In addition, ac response studies provide detailed information on bulk polarization, and electrode interfacial polarization phenomena [ 3-6 1. In a recent communication [ 71 we reported thermally stimulated depolarization currents (TSDC) studies of pure polycrystalline sodium yttrium fluoride and samarium-doped sodium yttrium fluoride. In this paper we report ac response studies of pure NaYF, sample.
2. Experimental
studies, cylindrical pellets of NaYF, were cold pressed at 500 kg cmm2 and annealed at 873 K for two days to remove strains. The sample was spring loaded between two silver blocking electrodes in a conductivity cell placed in an electrically heated furnace. Temperature was measured by a chromel-alumel thermocouple. The ac response parameters of the samples were measured using a phase-sensitive detector (PSD) based electronic system [ 8,9], which can be operated in the frequency range 50 Hz-100 kHz. AC pickups were avoided by using perfect shielding and by employing a furnace giving low inductive noise. In addition to this, a thick stainless steel sheet (connected to ground) was inserted between the conductivity set-up and the furnace. The furnace body, input and output terminals of the sample were also shielded perfectly. The thickness (1) of the pellet used in the present study was 0.3 cm with a diameter (d) of 1.25 cm.
Polycrystalline sodium yttrium fluoride (NaYF,) sample was prepared using the solid state reaction method described earlier [ 71. For the ac response 0167-2738/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSD10167-2738(94)00055-W
38
Md. Shareefuddin,M. NarasimhaChary /Solid State Ionics 69 (1994) 37-41
3. Results
723,758 and 803 K. At all the temperatures, the typical dependence of the imaginary part Y” on the real part Y’ of the admittance Y consisted of a depressed circular arc for the low frequency region while a straight line was observed in the high frequency region. The low frequency dependence of Y” (I”) can be
The ac response data of NaYF4 is presented on a complex plane, in which the imaginary part Y” of the admittance Y (YE Y’+jv”) was plotted against the real part Y’. Fig. 1 shows the plots of complex admittance parameters (Y’ and Y”) at temperatures 653,
I
T-723
K
K
T=803
9
. ‘OK
8’ I.5 76’ t r 2” z >
5432I-
3.0
34
3.8 Y’( lo-‘)
4.2
4.8
5.0
I.2
1.4
I.6
I.6
2.0
Y’(lo-‘l-
Admittance
different
I.0
plots
of
NoY
F, for
four
temprrotures
frequencies
Fig. 1. Complex admittance plots of NaYF, at 653,723, 758 and 803 K.
at
Md. Shareefuddin,M. NarasimhaChary /Solid State Ionics 69 (1994) 37-41 2
Table 1 Double layer capacitance (C,) values of NaYF,.
* NOW,
Sample No.
Temp. (K)
Capacitance (Farads)
1 2 3 4
653 723 758 803
6.61 x 2.04x 9.45 x 2.46x
10-‘O lO-9 lo-9 lo-*
connected with the polarization effects in the double layer which is formed at the electrode-sample interface. The capacitance C, of the double layer was found to increase with increasing temperature (Table 1). The high frequency dependence of Y” (Y ‘) can be connected with the bulk conductance (G) and geometric or high frequency capacitance of the sample. The bulk conductance at different temperatures was determined from the complex admittance spectra, which was found to increase with increasing temperature. The conductivity was calculated using the relation:
10
1.e
1.4
1.6 3 1.8 +--
2.0
2.2
Fig. 2. Variation of -log dversus 103/T. -4.0
I
o-=Gi,
39
(1) -4.5
where l/A is the geometrical factor (I is the thickness, and A is the area) of the sample. The variation of -log o with 103/T is shown in Fig. 2. The activation energy was found to be 1.24 eV. The variation of the total ac conductivity ( -log a, versus logfl is shown in Fig. 3. The real part of the dielectric constant t’ was evaluated from the complex admittance plots using the following relation:
I
mYt&
ar653K
-
t
d g-5.0J .. -5.5
*
.- . . . . . . .” * . .
-
I
-6.0.. 2.0
2.5
I
I
3.0
3.5
Log
I 4.0
I 4.5
1
5.0
f-
Fig. 3. Variation of -log uw versus logf:
f( Y) =b+iOeot*,
(2)
where e* in the complex dielectric constant (e*= e’-ie”) and o being the angular frequency. The loss angle tan S was calculated by using the relation: Y’ tan&=-== Y" t”
(3)
The temperature dependance of the dielectric constant e’ at different frequencies of 5,9, 18.2 and 3 1.25 kHz are shown in Fig. 4. e’ was found to increase exponentially with increasing temperature, and at-
tained a value of the order lo4 at a temperature of 770 K. It was also observed that for a given temperature, e’ increases with decreasing frequency. The variation of tan 6 with 1O’/T is shown in Fig. 5 at different frequencies. The loss angle was found to decrease with increasing frequency. 4. Discussion The complex admittance spectra of NaYF4 consisted of a depressed circular arc and a straight line
Md. Shareefuddin, M. Narasimha Chary /Solid State Ionics 69 (1994) 37-41
40
IO5 NoY F.
@9
KHz
@I82 @
t -W
KHz 31.25 KHz
to”-
10*-
1.0
f.2
t.4
2.0
t.8
1.6
2.2
(03 T
_3
Fig. 4. Variation of the dielectric constant e’ versus lO’/T.
NaYF, 251 20-
@
5 KHz
@
31.2 KHz
@9
KHz
@
50.0KHz
@!8.2KHz 1
i5-
a c z
io-
5-
0 I.0
t.2
1.4
I.6
1.8
2.0
103, T
Fig. 5. Variation of tan 6 versus 103/T.
in the low and high frequencies respectively. This is the behaviour of the most ionic solids [ 10,11,12- 16 1. The low frequency dependence of Y” (Y’) may be due to the polarization effects in the double layer which is formed at the electrode-sample interface
[ 17,18 1. The increase in C, value may be attributed to the increase in the dielectric constant with increasing temperature. A similar dependence was observed for Ba,_XLaXF2_Xand PbFz crystals [ 13,15,19]. At high frequencies the response of the system reflected predominantly bulk dispersion. The admittance plots reflect the blocking nature of the silver electrodes. The cell configuration Ag/NaYFJAg may be represented by an equivalent circuit comprising of a parallel network of z,,~ and bulk resistance Rb in series with another constant phase angle element z,,,. This constant phase angle element accounts for the semicircle flattening and spike tilting. It may be thought conceptually as a leaky capacitor. The conductivity of NaYF, calculated from dc studies was found to be less than the values calculated from the admittance studies. This discrepancy may be due to polarization effects at the interface between the silver electrode and the sample. The ac conductivity was found to increase slightly in the high frequency region. This increase may be attributed to the reduction in space charge polarization at high frequencies [ 2 1,221. This increased conductivity may also be the result of an increase in jump frequency of the defects with increasing frequency. The dielectric constant of a material is due to electronic, ionic, dipolar and space charge polarizations. All these factors may be active at lower frequencies. It is reasonable to suggest that there is contribution from all four sources of polarization measured in our experiment, where the maximum value of frequency is only 70 kHz. The high values oft’ at low frequencies are normally due to the presence of space charge polarization effects [ 23-25 1. This is the behaviour of most of the ionic solids [ 15,26,27]. At low frequencies ionic migration takes place which increases the measured dielectric constant by the creation of space charge layers. The observed rise in E’may also be connected with a temperature change of the relaxation time r and dipole moment connected with point defect polarization of the samples. The thermally generated anion Frenkel defects can move in the solid [ 28 1. Hence in the presence of an external electric field oriented displacement of lattice defects will occur. Large values of E’ at high temperature is connected with the polarization effects of space charges which are created by the fluorine interstitials and vacancies. These results are supported by our earlier
hid. Shareefuddin,M. NarasimhaChary /SolidState Ionics 69 (1994) 37-41
conclusions drawn from the TSDC studies on NaYF, [71The relaxation character of tan 6 is observed. The increase in tan 6 in the high temperature region may be due to the increased conductivity [ 20,291.
Acknowledgement The authors are grateful to Prof. K. Rama Reddy, Head, Department of Physics, Osmania University, for providing laboratory facilities. The authors thank Council of Scientific and Industrial Research (CSIR), New Delhi for providing financial assistance.
References [ 1] V.M. Carr, A.V. Chadwick and D.R. Figueroa, J. Phys. (Paris) 37 (1976) C7-337. [2] J. Olxrschmidt and D. Lazarus, Phys. Rev. B. 21 ( 1980) 5823. [3] AK. Jonscher and J.M. Reau, J. Mat. Sci. 13 (1978) 563. [4] I.D. Raistrick, C.H.O.Y. Wenttu and R.A. Huggins, J. Elextroanal. Chem. 77 (1977) 319. [5] J. Schoonman, L.J. Stil, J.R. Macdonald and D.R. Franceschetti, Solid State Ionics 3/4 ( 1981) 365. [6] K.E.D. Wapcnaar and J. Schoonman, J. Solid State Chem. 25 (1978) 31. [ 71 Md. Shareefuddin and M. Narasimha Chary, Cryst. Res. Tech. 28 (1993) 257.
41
[ 81 Md. Shareefbddin, K. Narasimha Reddy and M. Narasimha Chary, Mat. Sci. Eng. B 22 ( 1994) 159. [ 91 P. Balaya and C.S. Sunandana, Pramana 33 (1989) 627. [lo] J. Schoonman, L.B. Ebert, C.H. Hsieh and R.A. Huggins, J. Appl. Phys. 46 (1975) 2873. 111 A.D. Reed and D. Lazarus, Phys. Rev. B27 ( 1983) 6504. [ 121 R.W. Bonne and J. Schoonman, Solid State Commun. 18 (1976) 1005. [ 131 K.E.D. Wapenaar and J. Schoonman, Solid State Ionics 2 (1981) 253. [ 141 H. Geiger, G. Schon and H. Stork, Solid State Ionics 15 (1985) 155. 151 I. Kosacki and K. Zalibowski, Phys. Status Solidi (a) 108 (1988) 765. [ 161 M. Natsuhara, T. Uchida and M. Wakihara, Solid State Ionics 40 (1990) 874. [ 171 J. Bauerle, Phys. Chem. Solids 30 (1967) 2657. [ 181 K.E.D. Wapenaar and J. Schoonman, Solid State Ionics 5 (1981) 637. [ 191 A. Roos, D.R. Franceschetti and J. Schoonman, J. Phys. Chems. Solids 46 (1985) 645. [ 201 H. Yamane, S. Kikkawa and M. Koizumi, Solid State Ionics 15 (1985) 51. [ 211 B. Tareev, Physics of Dielectric Materials (Mir, Moscow, 1979). [22] K. Jonscher, J. Mater Sci. 24 (1989) 372. [23] K.V. Rao and A. Samakula, J. Appl. Phys. 36 ( 1965) 2031. [24] K.V. Rao and A. Samakula, J. Appl. Phys. 37 (1966) 319. [25] K.V.S. Badrinath and S. Subramanyam, Solid State Commun. 58 (1986) 137. [ 261 S. Palchoudhuri and G.K. Bichile, Solid State Commun. 69 (1989) 271. [27] G.A. Samara, Phys. Rev. B 13 (1976) 4529. [28] A. Azimi, V.M. Carr, A.V. Chadwick, F.G. Kirkwood and R. Saghafian, J. Phys. Chems. Solids 45 (1984) 23. [29] E. Mariani, J. Eckstein and E. Rubinova, Czech. J. Phys. B. 17 (1967) 552.
__ 44._ t8 Solid State Ionics 69 (1994) 37-41
ELFEVIER
ACresponse of polycrystalline sodium yttrium fluoride Md. Shareefuddin, M. Narasimha Chary Department of Physics, Osmania University, Hyderabad 500 007, India
Received 10 October 1993; accepted for publication 28 February 1994
Abstract AC response of polycrystalline sodium yttrium fluoride (NaYF,) has been studied in the temperature region of 350-950 K and in the frequency range 50 Hz-100 kHz using blocking electrodes. From the admittance plots, the bulk conductivity (a), dielectric constant (c’) and loss angle (tan 6) were determined at different temperatures and frequencies. The conductivity was found to increase with increasing temperature. Large value of E’observed at high temperatures was attributed to space charge polarization effects. The relaxation character of tan 6 was observed. The ac conductivity was found to increase with increasing frequency.
1. Introduction AC measurements in a wide frequency range are widely employed to determine the bulk conductivity of solids, since dc conductivity and mono-frequency studies are often hampered by the occurrence of extensive interlacial polarization at elevated temperatures [ 1,2 1. In addition, ac response studies provide detailed information on bulk polarization, and electrode interfacial polarization phenomena [ 3-6 1. In a recent communication [ 71 we reported thermally stimulated depolarization currents (TSDC) studies of pure polycrystalline sodium yttrium fluoride and samarium-doped sodium yttrium fluoride. In this paper we report ac response studies of pure NaYF, sample.
2. Experimental
studies, cylindrical pellets of NaYF, were cold pressed at 500 kg cmm2 and annealed at 873 K for two days to remove strains. The sample was spring loaded between two silver blocking electrodes in a conductivity cell placed in an electrically heated furnace. Temperature was measured by a chromel-alumel thermocouple. The ac response parameters of the samples were measured using a phase-sensitive detector (PSD) based electronic system [ 8,9], which can be operated in the frequency range 50 Hz-100 kHz. AC pickups were avoided by using perfect shielding and by employing a furnace giving low inductive noise. In addition to this, a thick stainless steel sheet (connected to ground) was inserted between the conductivity set-up and the furnace. The furnace body, input and output terminals of the sample were also shielded perfectly. The thickness (1) of the pellet used in the present study was 0.3 cm with a diameter (d) of 1.25 cm.
Polycrystalline sodium yttrium fluoride (NaYF,) sample was prepared using the solid state reaction method described earlier [ 71. For the ac response 0167-2738/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSD10167-2738(94)00055-W
38
Md. Shareefuddin,M. NarasimhaChary /Solid State Ionics 69 (1994) 37-41
3. Results
723,758 and 803 K. At all the temperatures, the typical dependence of the imaginary part Y” on the real part Y’ of the admittance Y consisted of a depressed circular arc for the low frequency region while a straight line was observed in the high frequency region. The low frequency dependence of Y” (I”) can be
The ac response data of NaYF4 is presented on a complex plane, in which the imaginary part Y” of the admittance Y (YE Y’+jv”) was plotted against the real part Y’. Fig. 1 shows the plots of complex admittance parameters (Y’ and Y”) at temperatures 653,
I
T-723
K
K
T=803
9
. ‘OK
8’ I.5 76’ t r 2” z >
5432I-
3.0
34
3.8 Y’( lo-‘)
4.2
4.8
5.0
I.2
1.4
I.6
I.6
2.0
Y’(lo-‘l-
Admittance
different
I.0
plots
of
NoY
F, for
four
temprrotures
frequencies
Fig. 1. Complex admittance plots of NaYF, at 653,723, 758 and 803 K.
at
Md. Shareefuddin,M. NarasimhaChary /Solid State Ionics 69 (1994) 37-41 2
Table 1 Double layer capacitance (C,) values of NaYF,.
* NOW,
Sample No.
Temp. (K)
Capacitance (Farads)
1 2 3 4
653 723 758 803
6.61 x 2.04x 9.45 x 2.46x
10-‘O lO-9 lo-9 lo-*
connected with the polarization effects in the double layer which is formed at the electrode-sample interface. The capacitance C, of the double layer was found to increase with increasing temperature (Table 1). The high frequency dependence of Y” (Y ‘) can be connected with the bulk conductance (G) and geometric or high frequency capacitance of the sample. The bulk conductance at different temperatures was determined from the complex admittance spectra, which was found to increase with increasing temperature. The conductivity was calculated using the relation:
10
1.e
1.4
1.6 3 1.8 +--
2.0
2.2
Fig. 2. Variation of -log dversus 103/T. -4.0
I
o-=Gi,
39
(1) -4.5
where l/A is the geometrical factor (I is the thickness, and A is the area) of the sample. The variation of -log o with 103/T is shown in Fig. 2. The activation energy was found to be 1.24 eV. The variation of the total ac conductivity ( -log a, versus logfl is shown in Fig. 3. The real part of the dielectric constant t’ was evaluated from the complex admittance plots using the following relation:
I
mYt&
ar653K
-
t
d g-5.0J .. -5.5
*
.- . . . . . . .” * . .
-
I
-6.0.. 2.0
2.5
I
I
3.0
3.5
Log
I 4.0
I 4.5
1
5.0
f-
Fig. 3. Variation of -log uw versus logf:
f( Y) =b+iOeot*,
(2)
where e* in the complex dielectric constant (e*= e’-ie”) and o being the angular frequency. The loss angle tan S was calculated by using the relation: Y’ tan&=-== Y" t”
(3)
The temperature dependance of the dielectric constant e’ at different frequencies of 5,9, 18.2 and 3 1.25 kHz are shown in Fig. 4. e’ was found to increase exponentially with increasing temperature, and at-
tained a value of the order lo4 at a temperature of 770 K. It was also observed that for a given temperature, e’ increases with decreasing frequency. The variation of tan 6 with 1O’/T is shown in Fig. 5 at different frequencies. The loss angle was found to decrease with increasing frequency. 4. Discussion The complex admittance spectra of NaYF4 consisted of a depressed circular arc and a straight line
Md. Shareefuddin, M. Narasimha Chary /Solid State Ionics 69 (1994) 37-41
40
IO5 NoY F.
@9
KHz
@I82 @
t -W
KHz 31.25 KHz
to”-
10*-
1.0
f.2
t.4
2.0
t.8
1.6
2.2
(03 T
_3
Fig. 4. Variation of the dielectric constant e’ versus lO’/T.
NaYF, 251 20-
@
5 KHz
@
31.2 KHz
@9
KHz
@
50.0KHz
@!8.2KHz 1
i5-
a c z
io-
5-
0 I.0
t.2
1.4
I.6
1.8
2.0
103, T
Fig. 5. Variation of tan 6 versus 103/T.
in the low and high frequencies respectively. This is the behaviour of the most ionic solids [ 10,11,12- 16 1. The low frequency dependence of Y” (Y’) may be due to the polarization effects in the double layer which is formed at the electrode-sample interface
[ 17,18 1. The increase in C, value may be attributed to the increase in the dielectric constant with increasing temperature. A similar dependence was observed for Ba,_XLaXF2_Xand PbFz crystals [ 13,15,19]. At high frequencies the response of the system reflected predominantly bulk dispersion. The admittance plots reflect the blocking nature of the silver electrodes. The cell configuration Ag/NaYFJAg may be represented by an equivalent circuit comprising of a parallel network of z,,~ and bulk resistance Rb in series with another constant phase angle element z,,,. This constant phase angle element accounts for the semicircle flattening and spike tilting. It may be thought conceptually as a leaky capacitor. The conductivity of NaYF, calculated from dc studies was found to be less than the values calculated from the admittance studies. This discrepancy may be due to polarization effects at the interface between the silver electrode and the sample. The ac conductivity was found to increase slightly in the high frequency region. This increase may be attributed to the reduction in space charge polarization at high frequencies [ 2 1,221. This increased conductivity may also be the result of an increase in jump frequency of the defects with increasing frequency. The dielectric constant of a material is due to electronic, ionic, dipolar and space charge polarizations. All these factors may be active at lower frequencies. It is reasonable to suggest that there is contribution from all four sources of polarization measured in our experiment, where the maximum value of frequency is only 70 kHz. The high values oft’ at low frequencies are normally due to the presence of space charge polarization effects [ 23-25 1. This is the behaviour of most of the ionic solids [ 15,26,27]. At low frequencies ionic migration takes place which increases the measured dielectric constant by the creation of space charge layers. The observed rise in E’may also be connected with a temperature change of the relaxation time r and dipole moment connected with point defect polarization of the samples. The thermally generated anion Frenkel defects can move in the solid [ 28 1. Hence in the presence of an external electric field oriented displacement of lattice defects will occur. Large values of E’ at high temperature is connected with the polarization effects of space charges which are created by the fluorine interstitials and vacancies. These results are supported by our earlier
hid. Shareefuddin,M. NarasimhaChary /SolidState Ionics 69 (1994) 37-41
conclusions drawn from the TSDC studies on NaYF, [71The relaxation character of tan 6 is observed. The increase in tan 6 in the high temperature region may be due to the increased conductivity [ 20,291.
Acknowledgement The authors are grateful to Prof. K. Rama Reddy, Head, Department of Physics, Osmania University, for providing laboratory facilities. The authors thank Council of Scientific and Industrial Research (CSIR), New Delhi for providing financial assistance.
References [ 1] V.M. Carr, A.V. Chadwick and D.R. Figueroa, J. Phys. (Paris) 37 (1976) C7-337. [2] J. Olxrschmidt and D. Lazarus, Phys. Rev. B. 21 ( 1980) 5823. [3] AK. Jonscher and J.M. Reau, J. Mat. Sci. 13 (1978) 563. [4] I.D. Raistrick, C.H.O.Y. Wenttu and R.A. Huggins, J. Elextroanal. Chem. 77 (1977) 319. [5] J. Schoonman, L.J. Stil, J.R. Macdonald and D.R. Franceschetti, Solid State Ionics 3/4 ( 1981) 365. [6] K.E.D. Wapcnaar and J. Schoonman, J. Solid State Chem. 25 (1978) 31. [ 71 Md. Shareefuddin and M. Narasimha Chary, Cryst. Res. Tech. 28 (1993) 257.
41
[ 81 Md. Shareefbddin, K. Narasimha Reddy and M. Narasimha Chary, Mat. Sci. Eng. B 22 ( 1994) 159. [ 91 P. Balaya and C.S. Sunandana, Pramana 33 (1989) 627. [lo] J. Schoonman, L.B. Ebert, C.H. Hsieh and R.A. Huggins, J. Appl. Phys. 46 (1975) 2873. 111 A.D. Reed and D. Lazarus, Phys. Rev. B27 ( 1983) 6504. [ 121 R.W. Bonne and J. Schoonman, Solid State Commun. 18 (1976) 1005. [ 131 K.E.D. Wapenaar and J. Schoonman, Solid State Ionics 2 (1981) 253. [ 141 H. Geiger, G. Schon and H. Stork, Solid State Ionics 15 (1985) 155. 151 I. Kosacki and K. Zalibowski, Phys. Status Solidi (a) 108 (1988) 765. [ 161 M. Natsuhara, T. Uchida and M. Wakihara, Solid State Ionics 40 (1990) 874. [ 171 J. Bauerle, Phys. Chem. Solids 30 (1967) 2657. [ 181 K.E.D. Wapenaar and J. Schoonman, Solid State Ionics 5 (1981) 637. [ 191 A. Roos, D.R. Franceschetti and J. Schoonman, J. Phys. Chems. Solids 46 (1985) 645. [ 201 H. Yamane, S. Kikkawa and M. Koizumi, Solid State Ionics 15 (1985) 51. [ 211 B. Tareev, Physics of Dielectric Materials (Mir, Moscow, 1979). [22] K. Jonscher, J. Mater Sci. 24 (1989) 372. [23] K.V. Rao and A. Samakula, J. Appl. Phys. 36 ( 1965) 2031. [24] K.V. Rao and A. Samakula, J. Appl. Phys. 37 (1966) 319. [25] K.V.S. Badrinath and S. Subramanyam, Solid State Commun. 58 (1986) 137. [ 261 S. Palchoudhuri and G.K. Bichile, Solid State Commun. 69 (1989) 271. [27] G.A. Samara, Phys. Rev. B 13 (1976) 4529. [28] A. Azimi, V.M. Carr, A.V. Chadwick, F.G. Kirkwood and R. Saghafian, J. Phys. Chems. Solids 45 (1984) 23. [29] E. Mariani, J. Eckstein and E. Rubinova, Czech. J. Phys. B. 17 (1967) 552.