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Spontaneous polarization close to phase transitions in NaNO2
Materials Chemistry and Physics 71 (2001) 206–209
Materials science communication
Spontaneous polarization close to phase transitions in NaNO2 H. Yurtseven∗ , S. Saliho˘glu, Ö. Tari Department of Physics, Istanbul Technical University, Maslak, Istanbul 80626, Turkey Received 18 May 2000; received in revised form 4 October 2000; accepted 23 October 2000
Abstract We give here the calculation of the order parameter and the spontaneous polarization as functions of temperature using the mean field model close to the phase transitions in NaNO2 . We have fitted our calculated order parameter to the observed one from the literature. From this fitting, we have then calculated the spontaneous polarization as a function of temperature close to the phase transitions in NaNO2 . Our calculated spontaneous polarization is in good agreement with the observed spontaneous polarization reported in the literature. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Polarization; Order parameter; Phase transition; NaNO2
1. Introduction Sodium nitrite (NaNO2 ) exhibits a phase transition from the paraelectric to the antiferroelectric phase at the Neel temperature (TN = 165.3◦ C) which is of a second-order type [1]. As the temperature decreases further, the antiferroelectric phase disappears and the ferroelectric phase occurs at the Curie temperature (TC = 163.9◦ C), which is of a first-order type. These phase transitions have been well established both experimentally and theoretically in the literature since the discovery of NaNO2 as a ferroelectric due to Sawada et al. [2]. Experimentally, there have been various studies on the phase transitions in NaNO2 using different techniques. The existence of an antiferroelectric phase between the ferroelectric and the paraelectric phases has been studied using the X-ray technique [1,3–6]. The dielectric properties of NaNO2 have been investigated [2,7–9]. In order to obtain the thermodynamic data, the calorimetric measurements have been performed for NaNO2 [10,11]. Using the other spectroscopic techniques, NaNO2 has been investigated extensively in the literature. The far infrared absorption of NaNO2 has been obtained [12]. Also, the infrared spectrum of this material has been obtained [13]. Besides, the reflection spectra of NaNO2 have been analysed [14,15]. The Raman studies of NaNO2 have been reported in the literature [16–19]. The phase transitions of NaNO2 have also been investigated ∗ Corresponding author. Tel.: +90-212-285-3285; fax: +90-212-285-6386. E-mail address: [email protected] (H. Yurtseven).
using ultrasonic and Brillouin scattering [20–26]. Those investigations have been reported in the literature using the NMR [27] and neutron scattering [28,29] techniques. Theoretically, various aspects of the phase transitions in NaNO2 have been studied. In earlier studies [1,4] an order–disorder phase transition in NaNO2 has been investigated on the basis of an Ising model. By making the analogy between the two values of the Ising spin and the two possible equilibrium orientations of the NO2 − dipole in the paraelectric phase of NaNO2 , a pseudospin model for this material has been proposed by Yamada and Yamada [30]. On this basis, a microscopic model has been given [31–33]. In an another study, a thermodynamic potential for the phase transitions in NaNO2 has been given [34,35]. By means of the molecular dynamics calculations, the critical fluctuations in the paraelectric phase of NaNO2 have been studied [36–38]. In the meantime, the incommensurate (IC) transition in NaNO2 has been treated using the mean field theory [39]. Regarding the spontaneous polarization and dielectric constant, a number of studies have been reported in the literature [1,40]. Later, the order parameter for NaNO2 was measured experimentally using X-ray powder diffractometry [41]. In this study we have calculated the order parameter ψ as a function of temperature close to the phase transitions in NaNO2 using the mean field model containing the ψ 8 term in the free energy. By fitting our calculated order parameter to the experimentally measured order parameter from the literature, we have been able to calculate the spontaneous polarization as a function of temperature close to phase transitions
0254-0584/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 1 ) 0 0 2 7 4 - 7
H. Yurtseven et al. / Materials Chemistry and Physics 71 (2001) 206–209
in NaNO2 . This then enables us to compare our calculated spontaneous polarization with the observed one for NaNO2 . In Section 2 we introduce the mean field model from which we derive the order parameter and the spontaneous polarization. In Section 3 we give our results and discussion. Conclusions are given in Section 4.
There exist mainly two models to study the phase transitions from the paraelectric to the ferroelectric phases in borasicites [42]. These two models are the Dvorak and Levanyuk–Sannikov models. In both models there are two-order parameters, namely, the transition parameter ψ and the polarization P. In the Dvorak model ψ is the primary order parameter and P the secondary order parameter. In the Levanyuk–Sannikov model P is the primary order parameter whereas ψ the secondary order parameter. Now, we want to study the phase transitions from the paraelectric phase to the ferroelectric phase in NaNO2 , where there also exist two-order parameters, ψ and P. In this study we take the transition parameter ψ as the primary order parameter and the polarization P as the secondary order parameter. Hence, our Gibbs free energy for NaNO2 will be similar to the Dvorak model of borasicites, namely, our Gibbs free energy is given by
(1)
Here, χ 0 is the paraelectric susceptibility, a2 = α(T − Tc ) where Tc is the critical temperature and we take a2 as negative. In Eq. (1) the constants a4 , a6 and a8 are positive since we consider the para–ferroelectric phase transition as a second order. The coupling constants are taken as b < 0 and c > 0. Minimization of the free energy given by Eq. (1) with respect to P gives ∂G P + bψ 2 + 2cψ 2 P = 0 = ∂P χ0
Inserting P given by Eq. (3) into Eq. (4) gives a2 = −a4 ψ 2 − a6 ψ 4 − a8 ψ 6 2b2 cχ02 ψ 4 2b2 χ0 ψ 2 + − 1 + 2cχ0 ψ 2 (1 + 2cχ0 ψ 2 )2
(5)
By taking a2 = α(T − Tc ), from Eq. (5) we get A + Bψ 2 + Cψ 4 + Dψ 6 + Eψ 8 T = Tc A + F ψ 2 + Gψ 4
2. Theory
1 1 1 1 G = a2 ψ 2 + a4 ψ 4 + a 6 ψ 6 + a 8 ψ 8 2 4 6 8 1 P2 + + bψ 2 P + cψ 2 P 2 2 χ0
207
(6)
where A = αTc ,
B = 4αTc cχ0 − a4 + 2b2 χ0 ,
C = 4αTc c2 χ02 − 4a4 cχ0 − a6 + 2b2 cχ02 , D = −4a4 c2 χ02 − 4a6 cχ0 − a8 , E = −4a6 c2 χ02 − 4a8 cχ0 , G=
4αTc c2 χ02
F = 4αTc cχ0 , (7)
In Eq. (6) we have expanded the numerator up to the ψ 8 term because we have included powers of ψ up to the ψ 8 term in the free energy given by Eq. (1).
3. Results and discussion In the literature there exist experimental data for the order parameter ψ as a function of T/Tc [41]. Since there exist many experimental data for the order parameter ψ in the region of T/Tc from 0.8 to 1.0, we fitted Eq. (6) to the experimental data within this region and we obtained the coefficients in the free energy (Eq. (1)). The graph of ψ as a function of T/Tc is shown in Fig. 1. The coefficients obtained from this fitting are given in Table 1. Using these coefficients, we calculated the polarization from Eq. (3). We show our calculated and observed [40] values of the
(2)
By solving P in Eq. (2) we get P =−
bχ0 ψ 2 1 + 2cχ0 ψ 2
(3)
By taking b as negative and c as positive constants, we have a positive P value in Eq. (3). Minimization of the free energy (Eq. (1)) with respect to ψ gives ∂G = ψ(a2 + a4 ψ 2 + a6 ψ 4 + a8 ψ 6 + 2bP + 2cP2 ) = 0 ∂ψ (4)
Fig. 1. The temperature dependence of the order parameter obtained from the fitting of Eq. (6) to the experimental data of da Costa Lamas et al. [41] close to the phase transition in NaNO2 .
208
H. Yurtseven et al. / Materials Chemistry and Physics 71 (2001) 206–209
Table 1 Values of the parameters in the free energy given by Eq. (1); we note that the constant α is defined by a2 = α(T − Tc ) α (J ◦ C−1 ) a4 (J) a6 (J) a8 (J) b (J m2 ◦ C−1 ) c (J m2 ◦ C−1 )
0.924 0.142 0.015 56.170 −3.002×10−2 5.413×10−2
region of 0.8–1.0, we were unable to get the spontaneous polarization adequately in the temperature region of 0.3–0.8.
4. Conclusions We have studied here the order parameter and the spontaneous polarization as a function of temperature using the Dvorak model close to the phase transition in NaNO2 . We have fitted our calculated order parameter to the measured order parameter. This has allowed us to calculate the spontaneous polarization as a function of temperature. Our calculated spontaneous polarization agrees well with those in the literature. This shows that the Dvorak model containing the eighth power of the order parameter in the free energy, which we have introduced here, explains adequately the observed features of the NaNO2 close to its phase transition. References
Fig. 2. The temperature dependence of the spontaneous polarization calculated from Eq. (3) and the experimental data due to Nomura [40] close to the phase transition in NaNO2 .
polarization in Fig. 2. We note that in Eqs. (6) and (3) we took the paraelectric susceptibility χ 0 as 4.2 C2 (J m4 )−1 . 1 We also took the critical temperature Tc as 163◦ C. The graph of polarization as a function of T/Tc shows that agreement between our calculated and the observed values for polarization is better within the temperature region of T/Tc from 0.8 to 1.0. The reason for this good agreement in this region of T/Tc is that the fit of the order parameter was done within the temperature region of 0.8–1.0. Hence, the calculation of spontaneous polarization depends upon the coefficients of the free energy, which were calculated (see Table 1) from the fitting of the order parameter (see Eq. (6)) in this region of T/Tc . Within the temperature region of T/Tc from 0.3 to 0.8, agreement between our calculated and the observed values is not as good as the region from 0.8 to 1.0. The reason for this disagreement in the temperature region of 0.3–0.8 is that there exist very few experimental data for the order parameter in this region. Hence, the fit of the order parameter was not performed sufficiently within this temperature region. Therefore, using those coefficients of the free energy, which we calculated in the temperature 1 This value of χ was obtained from the dielectric constant value 0 of ε = 5.2 C2 (J m4 )−1 for sodium nitrate in SI unit system (see the dielectric constants table in Ref. [43]). In SI units χ 0 is given by ε − 1. By assuming the same value of dielectric constant for sodium nitrite we have χ0 = 4.2 C2 (J m4 )−1 .
[1] S. Tanisaki, J. Phys. Soc. Jpn. 16 (1961) 579. [2] S. Sawada, S. Nomura, S. Fujii, I. Yoshida, Phys. Rev. Lett. 1 (1958) 320. [3] S. Tanisaki, J. Phys. Soc. Jpn. 18 (1963) 1181. [4] Y. Yamada, I. Shibuya, S. Hoshino, J. Phys. Soc. Jpn. 18 (1963) 1594. [5] S. Hoshino, H. Motegi, Jpn. J. Appl. Phys. 6 (1967) 708. [6] D. Durand, F. Denoyer, M. Lampert, L. Bernard, R. Currat, J. Phys. 43 (1982) 149. [7] Y. Takagi, K. Gesi, J. Phys. Soc. Jpn. 19 (1964) 142. [8] K. Hamano, J. Phys. Soc. Jpn. 19 (1964) 945. [9] B. Wyncke, F. Brehat, G.V. Kozlov, Phys. Stat. Sol. (b) 129 (1984) 531. [10] S. Hoshino, J. Phys. Soc. Jpn. 19 (1964) 140. [11] M. Sakiyama, A. Kimoto, S. Seki, J. Phys. Soc. Jpn. 20 (1965) 2180. [12] H. Vogt, H. Happ, Phys. Stat. Sol. 30 (1968) 67. [13] F. Brehat, B. Wyncke, J. Phys. C 18 (1985) 1705. [14] J.D. Axe, Phys. Rev. 167 (1968) 573. [15] M.K. Barnoski, J.M. Ballantyne, Phys. Rev. 174 (1968) 946. [16] E.V. Chisler, M.S. Shur, Phys. Stat. Sol. 17 (1966) 163. [17] E.V. Chisler, M.S. Shur, Sov. Phys. Solid State 9 (1967) 796. [18] A. Takase, K. Miyakawa, J. Phys. C 18 (1985) 5579. [19] C.K. Asawa, M.K. Barnoski, Phys. Rev. B 2 (1970) 205. [20] I. Hatta, Y. Shimizu, K. Hamano, J. Phys. Soc. Jpn. 44 (1978) 1887. [21] I. Hatta, M. Hanami, K. Hamano, J. Phys. Soc. Jpn. 48 (1980) 160. [22] H. Shimizu, Y. Ishibashi, M. Tsukamoto, M. Umeno, J. Phys. Soc. Jpn. 36 (1974) 498. [23] T. Yagi, Y. Hikada, K. Miura, J. Phys. Colloq. C6 (1981) 731. [24] T. Yagi, Y. Hikada, K. Miura, J. Phys. Soc. Jpn. 51 (1982) 3562. [25] S.Kh. Esayan, V.V. Lemanov, N. Mamatkulov, Sov. Phys. Solid State 23 (1981) 1195. [26] I. Hatta, J. Phys. Soc. Jpn. 53 (1984) 635. [27] W. Buchheit, G. Herth, J. Peterson, Solid State Commun. 40 (1981) 411. [28] J. Sakurai, R.A. Cowley, G. Dolling, J. Phys. Soc. Jpn. 28 (1970) 1426. [29] D. Durand, L. Bernard, F. Mezei, R. Currat, F. Denoyer, M. Lambert, Physica B 136 (1986) 325. [30] Y. Yamada, T. Yamada, J. Phys. Soc. Jpn. 21 (1966) 2167. [31] K.D. Ehrhardt, K.H. Michel, Z. Phys. B 41 (1981) 329.
H. Yurtseven et al. / Materials Chemistry and Physics 71 (2001) 206–209 [32] [33] [34] [35]
K.D. Ehrhardt, K.H. Michel, Phys. Rev. Lett. 46 (1981) 291. J. Fivez, K.H. Michell, Z. Phys. B 51 (1983) 127. Y. Ishibashi, H. Shiba, J. Phys. Soc. Jpn. 45 (1978) 409. Y. Ishibashi, W. Buchheit, J. Peterson, Solid State Commun. 38 (1981) 1277. [36] M.L. Klein, I.R. McDonald, Y. Ozaki, Phys. Rev. Lett. 48 (1982) 1197. [37] R.M. Lynden-Bell, M.L. Klein, I.R. McDonald, Z. Phys. B 54 (1984) 325.
209
[38] R.M. Lynden-Bell, R.W. Impey, M.L. Klein, Chem. Phys. 109 (1986) 25. [39] J. Naudts, J. Phys. C 16 (1983) 3457. [40] S. Nomura, J. Phys. Soc. Jpn. 16 (1961) 2440. [41] A. da Costa Lamas, S.L. Chang, S. Caticha-Ellis, Phys. Stat. Sol. (a) 68 (1981) 173. [42] H.E. Arakelian, T.R. Hart, Ferroelectrics 74 (1987) 13. [43] CRC Handbook of Chemistry and Physics, 61st Edition, CRC Press, Boca Raton, FL, p. E-58.
Materials science communication
Spontaneous polarization close to phase transitions in NaNO2 H. Yurtseven∗ , S. Saliho˘glu, Ö. Tari Department of Physics, Istanbul Technical University, Maslak, Istanbul 80626, Turkey Received 18 May 2000; received in revised form 4 October 2000; accepted 23 October 2000
Abstract We give here the calculation of the order parameter and the spontaneous polarization as functions of temperature using the mean field model close to the phase transitions in NaNO2 . We have fitted our calculated order parameter to the observed one from the literature. From this fitting, we have then calculated the spontaneous polarization as a function of temperature close to the phase transitions in NaNO2 . Our calculated spontaneous polarization is in good agreement with the observed spontaneous polarization reported in the literature. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Polarization; Order parameter; Phase transition; NaNO2
1. Introduction Sodium nitrite (NaNO2 ) exhibits a phase transition from the paraelectric to the antiferroelectric phase at the Neel temperature (TN = 165.3◦ C) which is of a second-order type [1]. As the temperature decreases further, the antiferroelectric phase disappears and the ferroelectric phase occurs at the Curie temperature (TC = 163.9◦ C), which is of a first-order type. These phase transitions have been well established both experimentally and theoretically in the literature since the discovery of NaNO2 as a ferroelectric due to Sawada et al. [2]. Experimentally, there have been various studies on the phase transitions in NaNO2 using different techniques. The existence of an antiferroelectric phase between the ferroelectric and the paraelectric phases has been studied using the X-ray technique [1,3–6]. The dielectric properties of NaNO2 have been investigated [2,7–9]. In order to obtain the thermodynamic data, the calorimetric measurements have been performed for NaNO2 [10,11]. Using the other spectroscopic techniques, NaNO2 has been investigated extensively in the literature. The far infrared absorption of NaNO2 has been obtained [12]. Also, the infrared spectrum of this material has been obtained [13]. Besides, the reflection spectra of NaNO2 have been analysed [14,15]. The Raman studies of NaNO2 have been reported in the literature [16–19]. The phase transitions of NaNO2 have also been investigated ∗ Corresponding author. Tel.: +90-212-285-3285; fax: +90-212-285-6386. E-mail address: [email protected] (H. Yurtseven).
using ultrasonic and Brillouin scattering [20–26]. Those investigations have been reported in the literature using the NMR [27] and neutron scattering [28,29] techniques. Theoretically, various aspects of the phase transitions in NaNO2 have been studied. In earlier studies [1,4] an order–disorder phase transition in NaNO2 has been investigated on the basis of an Ising model. By making the analogy between the two values of the Ising spin and the two possible equilibrium orientations of the NO2 − dipole in the paraelectric phase of NaNO2 , a pseudospin model for this material has been proposed by Yamada and Yamada [30]. On this basis, a microscopic model has been given [31–33]. In an another study, a thermodynamic potential for the phase transitions in NaNO2 has been given [34,35]. By means of the molecular dynamics calculations, the critical fluctuations in the paraelectric phase of NaNO2 have been studied [36–38]. In the meantime, the incommensurate (IC) transition in NaNO2 has been treated using the mean field theory [39]. Regarding the spontaneous polarization and dielectric constant, a number of studies have been reported in the literature [1,40]. Later, the order parameter for NaNO2 was measured experimentally using X-ray powder diffractometry [41]. In this study we have calculated the order parameter ψ as a function of temperature close to the phase transitions in NaNO2 using the mean field model containing the ψ 8 term in the free energy. By fitting our calculated order parameter to the experimentally measured order parameter from the literature, we have been able to calculate the spontaneous polarization as a function of temperature close to phase transitions
0254-0584/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 1 ) 0 0 2 7 4 - 7
H. Yurtseven et al. / Materials Chemistry and Physics 71 (2001) 206–209
in NaNO2 . This then enables us to compare our calculated spontaneous polarization with the observed one for NaNO2 . In Section 2 we introduce the mean field model from which we derive the order parameter and the spontaneous polarization. In Section 3 we give our results and discussion. Conclusions are given in Section 4.
There exist mainly two models to study the phase transitions from the paraelectric to the ferroelectric phases in borasicites [42]. These two models are the Dvorak and Levanyuk–Sannikov models. In both models there are two-order parameters, namely, the transition parameter ψ and the polarization P. In the Dvorak model ψ is the primary order parameter and P the secondary order parameter. In the Levanyuk–Sannikov model P is the primary order parameter whereas ψ the secondary order parameter. Now, we want to study the phase transitions from the paraelectric phase to the ferroelectric phase in NaNO2 , where there also exist two-order parameters, ψ and P. In this study we take the transition parameter ψ as the primary order parameter and the polarization P as the secondary order parameter. Hence, our Gibbs free energy for NaNO2 will be similar to the Dvorak model of borasicites, namely, our Gibbs free energy is given by
(1)
Here, χ 0 is the paraelectric susceptibility, a2 = α(T − Tc ) where Tc is the critical temperature and we take a2 as negative. In Eq. (1) the constants a4 , a6 and a8 are positive since we consider the para–ferroelectric phase transition as a second order. The coupling constants are taken as b < 0 and c > 0. Minimization of the free energy given by Eq. (1) with respect to P gives ∂G P + bψ 2 + 2cψ 2 P = 0 = ∂P χ0
Inserting P given by Eq. (3) into Eq. (4) gives a2 = −a4 ψ 2 − a6 ψ 4 − a8 ψ 6 2b2 cχ02 ψ 4 2b2 χ0 ψ 2 + − 1 + 2cχ0 ψ 2 (1 + 2cχ0 ψ 2 )2
(5)
By taking a2 = α(T − Tc ), from Eq. (5) we get A + Bψ 2 + Cψ 4 + Dψ 6 + Eψ 8 T = Tc A + F ψ 2 + Gψ 4
2. Theory
1 1 1 1 G = a2 ψ 2 + a4 ψ 4 + a 6 ψ 6 + a 8 ψ 8 2 4 6 8 1 P2 + + bψ 2 P + cψ 2 P 2 2 χ0
207
(6)
where A = αTc ,
B = 4αTc cχ0 − a4 + 2b2 χ0 ,
C = 4αTc c2 χ02 − 4a4 cχ0 − a6 + 2b2 cχ02 , D = −4a4 c2 χ02 − 4a6 cχ0 − a8 , E = −4a6 c2 χ02 − 4a8 cχ0 , G=
4αTc c2 χ02
F = 4αTc cχ0 , (7)
In Eq. (6) we have expanded the numerator up to the ψ 8 term because we have included powers of ψ up to the ψ 8 term in the free energy given by Eq. (1).
3. Results and discussion In the literature there exist experimental data for the order parameter ψ as a function of T/Tc [41]. Since there exist many experimental data for the order parameter ψ in the region of T/Tc from 0.8 to 1.0, we fitted Eq. (6) to the experimental data within this region and we obtained the coefficients in the free energy (Eq. (1)). The graph of ψ as a function of T/Tc is shown in Fig. 1. The coefficients obtained from this fitting are given in Table 1. Using these coefficients, we calculated the polarization from Eq. (3). We show our calculated and observed [40] values of the
(2)
By solving P in Eq. (2) we get P =−
bχ0 ψ 2 1 + 2cχ0 ψ 2
(3)
By taking b as negative and c as positive constants, we have a positive P value in Eq. (3). Minimization of the free energy (Eq. (1)) with respect to ψ gives ∂G = ψ(a2 + a4 ψ 2 + a6 ψ 4 + a8 ψ 6 + 2bP + 2cP2 ) = 0 ∂ψ (4)
Fig. 1. The temperature dependence of the order parameter obtained from the fitting of Eq. (6) to the experimental data of da Costa Lamas et al. [41] close to the phase transition in NaNO2 .
208
H. Yurtseven et al. / Materials Chemistry and Physics 71 (2001) 206–209
Table 1 Values of the parameters in the free energy given by Eq. (1); we note that the constant α is defined by a2 = α(T − Tc ) α (J ◦ C−1 ) a4 (J) a6 (J) a8 (J) b (J m2 ◦ C−1 ) c (J m2 ◦ C−1 )
0.924 0.142 0.015 56.170 −3.002×10−2 5.413×10−2
region of 0.8–1.0, we were unable to get the spontaneous polarization adequately in the temperature region of 0.3–0.8.
4. Conclusions We have studied here the order parameter and the spontaneous polarization as a function of temperature using the Dvorak model close to the phase transition in NaNO2 . We have fitted our calculated order parameter to the measured order parameter. This has allowed us to calculate the spontaneous polarization as a function of temperature. Our calculated spontaneous polarization agrees well with those in the literature. This shows that the Dvorak model containing the eighth power of the order parameter in the free energy, which we have introduced here, explains adequately the observed features of the NaNO2 close to its phase transition. References
Fig. 2. The temperature dependence of the spontaneous polarization calculated from Eq. (3) and the experimental data due to Nomura [40] close to the phase transition in NaNO2 .
polarization in Fig. 2. We note that in Eqs. (6) and (3) we took the paraelectric susceptibility χ 0 as 4.2 C2 (J m4 )−1 . 1 We also took the critical temperature Tc as 163◦ C. The graph of polarization as a function of T/Tc shows that agreement between our calculated and the observed values for polarization is better within the temperature region of T/Tc from 0.8 to 1.0. The reason for this good agreement in this region of T/Tc is that the fit of the order parameter was done within the temperature region of 0.8–1.0. Hence, the calculation of spontaneous polarization depends upon the coefficients of the free energy, which were calculated (see Table 1) from the fitting of the order parameter (see Eq. (6)) in this region of T/Tc . Within the temperature region of T/Tc from 0.3 to 0.8, agreement between our calculated and the observed values is not as good as the region from 0.8 to 1.0. The reason for this disagreement in the temperature region of 0.3–0.8 is that there exist very few experimental data for the order parameter in this region. Hence, the fit of the order parameter was not performed sufficiently within this temperature region. Therefore, using those coefficients of the free energy, which we calculated in the temperature 1 This value of χ was obtained from the dielectric constant value 0 of ε = 5.2 C2 (J m4 )−1 for sodium nitrate in SI unit system (see the dielectric constants table in Ref. [43]). In SI units χ 0 is given by ε − 1. By assuming the same value of dielectric constant for sodium nitrite we have χ0 = 4.2 C2 (J m4 )−1 .
[1] S. Tanisaki, J. Phys. Soc. Jpn. 16 (1961) 579. [2] S. Sawada, S. Nomura, S. Fujii, I. Yoshida, Phys. Rev. Lett. 1 (1958) 320. [3] S. Tanisaki, J. Phys. Soc. Jpn. 18 (1963) 1181. [4] Y. Yamada, I. Shibuya, S. Hoshino, J. Phys. Soc. Jpn. 18 (1963) 1594. [5] S. Hoshino, H. Motegi, Jpn. J. Appl. Phys. 6 (1967) 708. [6] D. Durand, F. Denoyer, M. Lampert, L. Bernard, R. Currat, J. Phys. 43 (1982) 149. [7] Y. Takagi, K. Gesi, J. Phys. Soc. Jpn. 19 (1964) 142. [8] K. Hamano, J. Phys. Soc. Jpn. 19 (1964) 945. [9] B. Wyncke, F. Brehat, G.V. Kozlov, Phys. Stat. Sol. (b) 129 (1984) 531. [10] S. Hoshino, J. Phys. Soc. Jpn. 19 (1964) 140. [11] M. Sakiyama, A. Kimoto, S. Seki, J. Phys. Soc. Jpn. 20 (1965) 2180. [12] H. Vogt, H. Happ, Phys. Stat. Sol. 30 (1968) 67. [13] F. Brehat, B. Wyncke, J. Phys. C 18 (1985) 1705. [14] J.D. Axe, Phys. Rev. 167 (1968) 573. [15] M.K. Barnoski, J.M. Ballantyne, Phys. Rev. 174 (1968) 946. [16] E.V. Chisler, M.S. Shur, Phys. Stat. Sol. 17 (1966) 163. [17] E.V. Chisler, M.S. Shur, Sov. Phys. Solid State 9 (1967) 796. [18] A. Takase, K. Miyakawa, J. Phys. C 18 (1985) 5579. [19] C.K. Asawa, M.K. Barnoski, Phys. Rev. B 2 (1970) 205. [20] I. Hatta, Y. Shimizu, K. Hamano, J. Phys. Soc. Jpn. 44 (1978) 1887. [21] I. Hatta, M. Hanami, K. Hamano, J. Phys. Soc. Jpn. 48 (1980) 160. [22] H. Shimizu, Y. Ishibashi, M. Tsukamoto, M. Umeno, J. Phys. Soc. Jpn. 36 (1974) 498. [23] T. Yagi, Y. Hikada, K. Miura, J. Phys. Colloq. C6 (1981) 731. [24] T. Yagi, Y. Hikada, K. Miura, J. Phys. Soc. Jpn. 51 (1982) 3562. [25] S.Kh. Esayan, V.V. Lemanov, N. Mamatkulov, Sov. Phys. Solid State 23 (1981) 1195. [26] I. Hatta, J. Phys. Soc. Jpn. 53 (1984) 635. [27] W. Buchheit, G. Herth, J. Peterson, Solid State Commun. 40 (1981) 411. [28] J. Sakurai, R.A. Cowley, G. Dolling, J. Phys. Soc. Jpn. 28 (1970) 1426. [29] D. Durand, L. Bernard, F. Mezei, R. Currat, F. Denoyer, M. Lambert, Physica B 136 (1986) 325. [30] Y. Yamada, T. Yamada, J. Phys. Soc. Jpn. 21 (1966) 2167. [31] K.D. Ehrhardt, K.H. Michel, Z. Phys. B 41 (1981) 329.
H. Yurtseven et al. / Materials Chemistry and Physics 71 (2001) 206–209 [32] [33] [34] [35]
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209
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