Application of the spectroscopic modifications of Pippard relations to NaNO2 in the ferroelectric phase

Journal of Molecular Structure 560 (2001) 189±196

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Application of the spectroscopic modi®cations of Pippard relations to NaNO2 in the ferroelectric phase H. Yurtseven*, A. Aydogdu Department of Physics, Istanbul Technical University, Maslak, Istanbul, Turkey Received 19 June 2000; accepted 8 September 2000

Abstract This study examines a linear variation of the speci®c heat CP with the frequency shifts 1/n (2n /2T ) for the Brillouin frequencies of the L-mode [010], [001] and [100] in the ferroelectric phase of NaNO2 according to our spectroscopically modi®ed Pippard relation. We obtain this linear relationship for those modes studied and calculate dTC/dP in the ferroelectric phase of NaNO2. Our calculated values of dTC/dP for the [001] and [100] modes are in good agreement with the values given in the literature. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Pippard relations; Ferroelectric phase; NaNO2

1. Introduction Sodium nitrite undergoes the antiferroelectric phase from the paraelectric phase at the NeeÂl temperature TN ˆ 165:38C: This transition is of a second order. At lower temperatures, the antiferroelectric phase transforms into the ferroelectric phase, which occurs at the Curie temperature TC ˆ 163:98C as a ®rst-order transition. Sodium nitrite was discovered as a ferroelectric by Sawada et al. [1]. Since then, many researchers have studied this crystalline system both experimentally and theoretically. Experimentally, X-ray studies [2± 4], dielectric [1,5,6] and calorimetric [7,8] measurements have been reported in the literature. The measurements for the thermal expansivity [9] and for the elastic compliances of NaNO2 [10] have also been reported. Apart from those experimental studies, * Corresponding author. Tel.: 190-212-285-3285; fax: 190-212285-6386.

the spectroscopic measurements on NaNO2 have been carried out. The far infrared spectra [11], infrared spectra [12], re¯ection spectra [13] and the Raman spectra [14,15] have been obtained for NaNO2. The ultrasonic and Brillouin spectra of NaNO2 have also been obtained [16±19]. Besides, NMR [20] and the neutron spectroscopic study [21] on NaNO2 have been reported in the literature. Theoretically, the order±disorder phase transition in NaNO2 has been studied using some microscopic models. Because of the two possible orientations of the NO2 2 dipole in the paraelectric phase of NaNO2, the spin-up and spin-down orientations of an Ising model has been applied to this crystal [22]. Also, a microscopic model [23] and the thermodynamic potential for the order±disorder phase transition in NaNO2 [24] have been reported in the literature. In our recent study, we have correlated the thermal expansivity to the frequency shifts 2n /2T for the Brillouin frequencies of the L-mode [010], [001] and [100] in the ferroelectric phase of NaNO2 [25].

0022-2860/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0022-286 0(00)00756-0

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H. Yurtseven, A. Aydogdu / Journal of Molecular Structure 560 (2001) 189±196

Fig. 1. The speci®c heat CP data [28] as a function of our calculated frequency shifts [25] 1/n (2n /2T)P for the L-mode [010] in the vicinity of the critical point …TC ˆ 163:158C† in the ferroelectric phase of NaNO2 according to Eq. (5). The experimental Cp data shown here are taken from Hatta et al. [28].

H. Yurtseven, A. Aydogdu / Journal of Molecular Structure 560 (2001) 189±196

By means of this correlation, we have calculated the Brillouin frequencies of this mode studied in the ferroelectric phase of NaNO2 [25]. In this study, we correlate the speci®c heat CP to the frequency shifts 1/n (2n /2T )P for the L-mode [010], [001] and [100] in the ferroelectric phase of NaNO2 according to the spectroscopic modi®cations of the Pippard relations, which we have developed in our earlier studies [27,28]. For this correlation, we use the experimental CP data for NaNO2 [28]. In Section 2, we give an outline of the Pippard relations and also our spectroscopically modi®ed Pippard relations. In Section 3, we present our results. In Section 4, we discuss our results. Finally, conclusions are given in Section 5. 2. Theory The Pippard relations describe the l -type of behaviour of the speci®c heat CP, the thermal expansivity a P and the isothermal compressibility k T in the vicinity of Tl . Those thermodynamic quantities are correlated to each other by the relations     dP dS 1T …1† CP ˆ aP VT dT l dT l and

aP ˆ kT



dP dT

 l

1

1 V



dV dT

 l

…2†

where (dS/dT )l and (dV/dT )l are constants at the l point. From linear plots of CP versus a P and also a P versus k T, one obtains the slope value, which can be compared with the experimentally measured dP/dT at the l -point by means of a plot of pressure versus temperature. As we have given in our earlier studies [27,28], by de®ning the isobaric mode GruÈneisen parameter   1 1 2n gP ˆ 2 …3† aP n 2T P and the isothermal GruÈneisen parameter   1 1 2n gT ˆ kT n 2P T

…4†

we can obtain the spectroscopically modi®ed Pippard relations. By using g P (Eq. (3)) in Eq. (1) and g T (Eq.

191

(4)) in Eq. (2), we obtain our modi®ed Pippard relations given by       TV dP 1 2n dS CP ˆ 2 1T …5† gP dT l n 2T P dT l and 1 aP ˆ gT



dP dT



    1 2n 1 dV 1 V dT l l n 2P T

…6†

The above relations relate the speci®c heat CP to the frequency shifts with temperature at constant pressure and the thermal expansivity a P to the frequency shifts with pressure at constant temperatures in the vicinity of the l -point. Linear plots of CP versus 1/n (2n /2T )P and also a P versus 1/n (2n /2P)T give us the slope value of (2P/2T )l , which can be compared with the experimentally measured 2P/2T at the l -point. 3. Calculations and results Here we established our spectroscopically modi®ed Pippard relation (Eq. (5)) by relating the speci®c heat CP to the frequency shifts 1/n (2n /2T )P for the Brillouin frequencies of the L-mode [010], [001] and [100] in the ferroelectric phase of NaNO2, which we have calculated in our recent study [25]. For the speci®c heat CP of NaNO2, we used the observed data given in the literature [28]. 3.1. L-mode [010] of NaNO2 We plot the speci®c heat CP [28] against our calculated frequency shifts for the [010] mode in the vicinity of the transition temperature …TC ˆ 163:158C† in the ferroelectric phase in Fig. 1. From this linear plot, we obtained the inverse slope value of dTC =dP ˆ 13:3 mC bar21 according to Eq. (5) where we used V ˆ 33:2 cm3 mol21 at T ˆ 163:98C [9] and our value of gP ˆ 10:1 as the isobaric mode GruÈneisen parameter for the [010] mode [25]. From this linear variation of CP versus 1/n (2n /2T )P, we obtained the intercept value of dS=dT ˆ 25:85 cal mol 21 C 22 according to Eq. (5). 3.2. L-mode [001] of NaNO2 The speci®c heat CP data [28] are plotted as a function of our calculated frequency shifts for the

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Fig. 2. The speci®c heat CP data [28] as a function of our calculated frequency shifts [25] 1/n (2n /2T)P for the L-mode [001] in the vicinity of the critical point …TC ˆ 163:158C† in the ferroelectric phase of NaNO2 according to Eq. (5). The experimental Cp data shown here are taken from Hatta et al. [28].

H. Yurtseven, A. Aydogdu / Journal of Molecular Structure 560 (2001) 189±196

193

Fig. 3. The speci®c heat CP data [28] as a function of our calculated frequency shifts [25] 1/n (2n /2T)P for the L-mode [100] in the vicinity of the critical point …TC ˆ 163:158C† in the ferroelectric phase of NaNO2 according to Eq. (5). The experimental Cp data shown here are taken from Hatta et al. [28].

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H. Yurtseven, A. Aydogdu / Journal of Molecular Structure 560 (2001) 189±196

[001] mode [25] in the vicinity of TC in the ferroelectric phase …T , TC † in Fig. 2. This linear plot gave us the inverse slope value of dTC =dP ˆ 4:4 mC bar21 using our value of gP ˆ 16:6 for the [001] mode [25] and the value of V ˆ 33:2 cm3 mol21 at T ˆ 163:98C; as before in Eq. (5). From this linear plot, we obtained the intercept value of dS=dT ˆ 33:55 cal mol21 C22 : 3.3. L-mode [100] of NaNO2 Fig. 3 gives a linear plot of CP data [28] versus our calculated frequency shifts for the [100] mode [25] in the vicinity of TC in the ferroelectric phase …T , TC †: From this plot, we obtained that dTC =dP ˆ 5:3 mC bar21 by means of Eq. (5) where we used our calculated value of gP ˆ 2:65 for the [100] mode [25] and the value of V ˆ 33:2 cm3 mol21 at T ˆ 163:98C: A linear variation of CP with 1/n (2n / 2T )P gave us the intercept value of dS=dT ˆ 64:25 cal mol21 C22 : We give in Table 1 our values of the inverse slope dTC/dP and of dS/dT for the Lmode [010], [001] and [100], which we obtained in the ferroelectric phase of NaNO2. 4. Discussion We obtained here the speci®c heat CP as a function of the frequency shifts 1/n (2n /2T )P for the Brillouin frequencies of the L-mode [010], [001] and [100] in NaNO2 as shown in Figs. 1±3, respectively. These ®gures give a linear variation of the speci®c heat with the frequency shifts, as expected from the ®rst Pippard relation (Eq. (5)) in the ferroelectric phase of NaNO2. This linear variation is particularly good for the [010] and [001] modes, as shown in Figs. 1 and 2, respectively, whereas for the [100] mode the data Table 1 Our calculated values of dTC/dP and dS/dT for the L-mode of NaNO2 according to Eq. (5). The values of the isobaric mode GruÈneisen parameter g P are taken from our recent study [25] Mode

gP

dTC/dP (mC/bar)

2 dS/dT (cal mol 21 C 2)

[010] [001] [100]

10.1 16.6 2.65

13.3 4.4 5.3

25.85 35.55 64.25

seem to be scattered (Fig. 3). From these ®gures, we obtained the values of dT/dP for the [011], [001] and [100] modes as given in Table 1. Our values of dT/dP can be compared with the value of 5.7 mdeg/bar due to Ema et al. [9], which was obtained from a plot of the thermal expansivity a P against the speci®c heat CP for NaNO2 in the ferroelectric phase (below TC). Our value of 5.3 mC/bar for the [100] mode is the closest to the value of 5.7 mdeg/bar [9], which is also close to the value of 4.9 mdeg/bar determined by Gesi et al. [29] as indicated in the work of Ema et al. [9]. Our value of 4.4 mC/bar for the [001] mode can be compared with the value of 4.9 mdeg/bar [29]. However, our value of 13.3 mC/bar for the [010] mode is far from those values for the [001] and [100] modes. Considering the ®rst-order transition from the ferroelectric to the antiferroelectric phase in NaNO2, the Clasius±Clapeyron equation dTC T …DV=V† ˆ C L=V dP

…7†

can be written as given in Ema et al. [9]. By taking the values of the latent heat L ˆ 56 cal mol21 ; the discontinuous volume change DV/V is between 9:0 £ 1024 and 12:1 £ 1024 and V ˆ 33:2 cm3 mol21 at TC ˆ 163:158C; Ema et al. [12] have obtained the value of dTC =dP ˆ 5:6±7:5 mdeg bar21 : This value of dTC/dP can also be compared with our dTC/dP values obtained for the [010], [001] and [100] modes of NaNO2 as given in Table 1. On the other hand, for a secondorder transition from the antiferroelectric to the paraelectric phase near the NeeÂl temperature TN, the generalized Pippard relation [30,31]

bi ˆ gi

CP 1 bti VT

…8†

has been applied to NaNO2 [9]. In this relation, b i represents the thermal expansivity and t the neighbourhood temperature de®ned as t ˆ T 2 TN …Xi † where Xi is the stress. bti is the thermal expansivity along the transition line, which is assumed to be constant and the stress dependence of the NeeÂl temperature is given by   2TN gi ˆ 2 …9† 2Xi which is also assumed to be constant. Thus the pressure dependence of the NeeÂl temperature is

H. Yurtseven, A. Aydogdu / Journal of Molecular Structure 560 (2001) 189±196

5. Conclusions

given by X dTN ˆ gi dP i

195

…10†

By using the experimental data, Ema et al. [9] have obtained dTN =dP ˆ 8:9 mdeg bar21 for NaNO2, which can be compared with our values of dTC/dP for the [010], [001] and [100] modes in this crystalline system as given in Table 1. Our values of dTC/dP for the modes studied can also be compared with our dP/dT values for ammonium halides [26,27]. Our dTC/dP values or alternatively, dP/dTC values are 75.2 bar/deg for the [010] mode, 190 bar/deg for the [100] mode and 226.4 bar/deg for the [001] mode of NaNO2. Those values can be compared with our dP/dT values of 62.1 bar K 21 for the 134 cm 21 and 90.1 bar K 21 for the 177 cm 21 Raman modes of NH4Br [27]. They can also be compared with our dP/dT values of 105.5 bar K 21 …P ˆ 0† for the 174 cm 21 and 94.9 bar K 21 …P ˆ 0† for the 1708 cm 21 Raman modes of NH4Cl [26,27]. Our intercept values of T(dS/dT ) or alternatively by taking T ˆ TC ˆ 163:158C; our dS/dT values for the [010], [001] and [100] modes of NaNO2 can be compared with the value of 15.67 cal mol 21 C 22, due to Garland and Jones [32] and our value of 13.82 cal mol 21 C 22 for NH4Cl as given in our earlier study [26]. The second Pippard relation, which we have modi®ed spectroscopically (Eq. (6)) can also be tested using the experimental data for the thermal expansivity and for the frequency shifts 1/n (2n / 2P)T. These frequency shift data can be deduced from the Brillouin frequencies measured as a function of pressure at constant temperatures for the L-mode [010], [001] and [100] in the ferroelectric phase of NaNO2. When the frequency shift data are available in the literature, Eq. (6) can be examined for the ferroelectric phase in NaNO2. Eq. (6) can also be examined for the phase transitions from the ferroelectric phase to the antiferroelectric phases in NaNO2 when the frequency shift data are available for these phases as well. Finally, the validity of Eq. (6) can be tested for NaNO2 near TN by comparing with Eq. (2) through the isothermal compressibility k T or the isothermal elastic compliances, which have been measured and tested near TN in this crystalline system [10].

We examined in this study the ®rst Pippard relation, which we have modi®ed spectroscopically for the NaNO2 crystalline system. For this analysis, we used our calculated Brillouin frequencies for the L-mode [010], [001] and [100] in the ferroelectric phase of NaNO2 …T , TC †: Our results show that the speci®c heat CP varies linearly with the frequency shifts 1/ n (2n /2T )P for those modes studied in the ferroelectric phase of NaNO2. Our calculated dTC/dP values obtained for the [001] and [100] modes agree with the literature values. It is suggested that our second Pippard relation (Eq. (6)) can also be tested using the Brillouin frequency data. References [1] S. Sawada, S. Nomura, S. Fujii, I. Yoshida, Phys. Rev. Lett. 1 (1958) 320. [2] S. Tanisaki, J. Phys. Soc. Jpn 18 (1963) 1181. [3] S. Hoshino, H. Motegi, Jap. J. App. Phys. 6 (1967) 708. [4] D. Durand, F. Denoyer, M. Lampert, L. Bernard, R. Currat, J. Phys. 43 (1982) 149. [5] Y. Takagi, K. Gesi, J. Phys. Soc. Jpn 19 (1964) 142. [6] B. Wyncke, F. Brehat, G.V. Kozlov, Phys. Status Solidi B 129 (1984) 531. [7] S. Hoshino, J. Phys. Soc. Jpn 19 (1964) 140. [8] M. Sakiyama, A. Kimoto, S. Seki, J. Phys. Soc. Jpn 20 (1965) 2180. [9] K. Ema, K. Hamano, I. Hatta, J. Phys. Soc. Jpn 39 (1975) 726. [10] K. Hamano, K. Ema, J. Phys. Soc. Jpn 45 (1978) 923. [11] H. Vogt, H. Happ, Phys. Status Solidi 30 (1968) 67. [12] F. Brehat, B. Wyncke, J. Phys. C: Solid State Phys. 18 (1985) 1705. [13] M.K. Barnoski, J.M. Ballantyne, Phys. Rev. 174 (1968) 946. [14] E.V. Chisler, M.S. Shur, Sov. Phys. Solid State 9 (1967) 796. [15] A. Takase, K. Miyakawa, J. Phys. C: Solid State Phys. 18 (1985) 5579. [16] H. Shimizu, Y. Ishibashi, M. Tsukamoto, M. Umeno, J. Phys. Soc. Jpn 36 (1974) 498. [17] S.Kh. Esayan, V.V. Lemanov, N. Mamatkulov, Sov. Phys. Solid State 23 (1981) 1195. [18] T. Yagi, Y. Hikada, K. Miura, J. Phys. Soc. Jpn 51 (1982) 3562. [19] I. Hatta, J. Phys. Soc. Jpn 53 (1984) 635. [20] W. Buchheit, G. Herth, J. Peterson, Solid State Commun. 40 (1981) 411. [21] D. Durand, L. Bernard, F. Mezei, R. Currat, F. Denoyer, M. Lambert, Physica B 136 (1986) 325. [22] Y. Yamada, T. Yamada, J. Phys. Soc. Jpn 21 (1966) 2167. [23] J. Fivez, K.H. Michell, Z. Phys. B: Condens. Matter 51 (1983) 127.

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