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Calculation of the Brillouin frequencies close to phase transitions in NaNO2
Spectrochimica Acta Part A 58 (2002) 55 – 65 www.elsevier.com/locate/saa
Calculation of the Brillouin frequencies close to phase transitions in NaNO2 H. Yurtseven *, I: .E. C ¸ aglar Department of Physics, Istanbul Technical Uni6ersity, Maslak, 80626 Istanbul, Turkey Received 27 June 2000; received in revised form 20 April 2001; accepted 20 April 2001
Abstract We calculate here the Brillouin frequencies of the L-mode [010], [001] and [100] of NaNO2 for the phase transitions from the paraelectric phase to the sinusoidal anti-ferroelectric phase near the Neel temperature (TN = 437.7 K) and to the ferroelectric phase near the critical temperature (Tc = 436.3 K) in this crystalline system. For calculating the frequencies, we use the thermal expansivity data for the phase regions considered, under the assumption that the mode Gruneisen parameter determined for each mode remains constant across the phase transitions. Our calculated frequencies agree well with the observed frequencies for the modes studied in NaNO2. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Brillouin frequencies; Phase transitions; NaNO2
1. Introduction Sodium nitrite undergoes phase transitions from the paraelectric phase to the antiferroelectric phase at the Neel temperature TN =437.7 K, as the temperature decreases from high temperatures. As the temperature decreases further, the antiferroelectric phase transforms into the ferroelectric phase at the Curie temperature Tc = 436.3 K. This ferroelectric behaviour of NaNO2 was first discovered by Sawada et al. [1]. Following Sawada, there have been X-ray studies on
* Corresponding author. Tel.: + 90-212-2853209; fax: + 90212-2856386. E-mail address: [email protected] (H. Yurtseven).
NaNO2 due to Tanisaki [2,3], Yamada et al. [4], Hoshino and Motegi [5], Durand et al. [6] and the dielectric studies on NaNO2 due to Sawada et al. [1], Tagaki and Gesi [7], Hamano [8] and Wyncke et al. [9]. Apart from these experimental techniques, calorimetric measurements for NaNO2 have been performed by Hoshino [10] and Sakiyama et al. [11]. Also, the thermal expansion and the elastic compliances of NaNO2 have been measured by Ema et al. [12] and Hamano and Ema [13], respectively. Some spectroscopic studies on NaNO2 have also been reported in the literature. Far infrared absorption of NaNO2 has been obtained by Vogt and Happ [14]. Infrared spectra by Brehat and Wyncke [15], the reflection spectra by Axe [16] and Barnoski and Ballantyne [17] and also the Raman spectra by Chisler and Shur
1386-1425/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 1 4 2 5 ( 0 2 ) 0 0 5 0 5 - 4
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[18,19], Takase and Miyakawa [20], Asawa and Barnoski [21] have been obtained for the NaNO2 system. The ultrasonic and the Brillouin scattering of NaNO2 have also been reported in the literature by Hatta et al. [22,23], Shimizu et al. [24], Yagi et al. [25,26], Esayan et al. [27] and Hatta [28]. The spectroscopic technique NMR by Buchheit et al. [29] was used to investigate 23Na nucleus for the commensurate and incommensurate phases in NaNO2. Also, the neutron spectroscopic technique by Sakurai et al. [30] and Durand et al. [31], has been used to obtain the spectroscopic data for NaNO2. The experimental studies on NaNO2 have been accompanied by some theoretical studies on NaNO2, as reported in the literature. An Ising model has been applied to the NaNO2 system by means of the analogy between the two possible orientations of an Ising spin (spin up and spin down) and the two equilibrium positions of the NO− 2 dipole in the paraelectric phase of NaNO2, due to Tanisaki [2], Yamada et al. [4], Yamada and Yamada [32]. A microscopic model has been developed in the works of Erhardt and Michel [33,34] and Fivez and Michel [35]. The mechanism of phase transitions in NaNO2 has been explained on the basis of the model proposed by Ishibashi and Shiba [36] and Ishibashi et al. [37]. In this study, we calculate the Brillouin frequencies of the L-mode [010], [001] and [100] by relating the thermal expansivity to the frequency for the phase transitions from the paraelectric to the sinusoidal antiferroelectric and ferroelectric phases in NaNO2. We use for this calculation, the thermal expansion data due to Ema et al. [12] and the Brillouin frequency data due to Yagi et al. [25,26]. In Section 2, we give an outline of our method of calculation for the Brillouin frequency. In Section 3, we give our calculations and results. Discussion is presented in Section 4. Finally, conclusions are given in Section 5.
2. Theory The thermodynamic functions can be related to the frequency shifts by means of the Gruneisen parameter defined as
k=
( ln w ( ln V
(1)
Using the definitions of the thermal expansivity hP =
1 (V V (T
P
and the isothermal compressibility iT = −
1 (V V (P
,
T
we then define the isobaric mode Gruneisen parameter kP and the isothermal mode Gruneisen parameter kT, respectively as kP = − and kT =
1 1 (w hP w (T
1 1 (w iT w (P
(2)
P
(3)
T
Close to phase transitions, the frequency shifts and also the frequencies can be calculated using the data for the thermal expansivity and the isothermal compressibility by taking the mode Gruneisen parameters kP and kT given in Eq. (2) and Eq. (3), respectively, as constants across the phase transitions. Here, we are going to calculate the Brillouin frequencies of the longitudinal acoustic mode (Lmode) along the [010], [001] and [100] directions for NaNO2 close to the phase transitions, from the paraelectric phase to the sinusoidal antiferroelectric and to the ferroelectric phase in this crystalline system. For this calculation, we use the temperature dependence of the thermal expansivity for the three phases, namely, the paraelectric, sinusoidal antiferroelectric and the ferroelectric phases, according to Ema et al. [12]. By expressing Eq. (2) in the form
&
6
1 dw= − 60 w
&
T0
hPkP dT+ Dw
(4)
T
we can calculate the frequencies by means of the temperature dependence of the thermal expansivity. In Eq. (4), Dw represents the order– disorder contribution to the frequency and w0 is the value of the frequency at a constant temperature T0.
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2.1. Paraelectric phase (T \ TN)
2.3. Ferroelectric phase (TB Tc)
Since there is no ordering above the Neel temperature in the paraelectric phase, we take the order–disorder contribution to the Brillouin frequency of the LA mode as zero, Dw = 0. By assuming that the mode Gruneisen parameter kP remains constant close to the Neel temperature TN, we have from Eq. (4) the frequency relation
We can calculate the frequencies from Eq. (4) using the relation
ln w(T)=ln w0 − kP
&
TN
hP(T) dT
(5)
T
In Eq. (5), the temperature dependence of the thermal expansivity is given by hP(T) =A1m − h1 +B1
(6)
where the critical exponent is h1, the reduced temperature is m= ((T − TN)/TN), A1 is the amplitude and B1 is a constant. Those constants h1, A1 and B1 of Eq. (6) can be obtained from the temperature dependence of the thermal expansivity which were obtained experimentally [12].
2.2. Sinusoidal antiferroelectric phase (Tc BT B TN) Since there is very little change in the Brillouin frequencies of the L-mode below the Neel temperature TN down to the critical temperature Tc, as observed experimentally in NaNO2 [26], we take Dw =0 in Eq. (4). We then obtain similar to Eq. (5) that ln w(T)= ln w0 + kP
&
TN
hP(T) dT
(7)
T
In Eq. (7), the temperature dependence of the thermal expansivity is given by
ln w(T)=ln w0 + kP
&
Tc
hP(T) dT+ Dw
(9)
T
where we define the order–disorder contribution to the frequency as Dw= wobserved − wcalculated
(10)
which we take its temperature dependence as Dw= aT+bT 2
(11)
Here, a and b are constants which can be determined from the observed [26] and our calculated Brillouin frequencies for the L-mode [010], [001] and [100]. In Eq. (9), the temperature dependence of the thermal expansivity is given by hP(T)= A3 m − h3 + B3
(12)
where the critical exponent is a3, the reduced temperature is m= ((T − Tc)/Tc), A3 is the amplitude and B3 is a constant. Those constants a3, A3 and B3 of Eq. (12) can be obtained from the temperature dependence of the thermal expansivity that was obtained experimentally [12].
3. Calculations and results Here, we first determined the values of the mode Gruneisen parameter kP for the L-mode along the [010], [001] and [100] directions in NaNO2. We then calculated the Brillouin frequencies of those modes for the phases of paraelectric (T\ TN), sinusoidal antiferroelectric (Tc B TBTN) and ferroelectric (TB Tc) in NaNO2.
(8)
3.1. Calculation of the mode Gruneisen parameter and the Brillouin frequencies for the L-mode [010]
where the critical exponent is h2 and m =((T − TN)/TN) as before. A2 is the amplitude and B2 is a constant. Those constants h2, A2 and B2 of Eq. (8) can be obtained from the temperature dependence of the thermal expansivity that was obtained experimentally [12].
We determined the mode Gruneisen parameter kP by means of Eq. (2) using the experimental data for the Brillouin frequencies [26] and the thermal expansivity [12]. For this calculation, we used those experimental data for the temperature region of 441.96 KB TB438.78 K in the
hP(T) =A2 m
− h2
+B2
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paraelectric phase T\ TN, where TN =437.7 K. This temperature region was appropriate to determine the kP value, since the frequency shifts (#w/ #T) varied linearly with the thermal expansivity hP in the paraelectric phase, according to Eq. (2). This then gave us the values between 6.63 and 8.79 for the mode Gruneisen parameter of the [010] mode, from which we took the average value of kP =7.32. We then assumed that this value of kP for the L-mode // [010] remained constant across the phase transitions of paraelectric– sinusoidal antiferroelectric– ferroelectric in NaNO2.
3.1.1. Calculation of the frequencies for the L-mode // [010] in the paraelectric phase (T \ TN) We calculated the Brillouin frequencies for the L-mode along the [010] direction in the paraelectric phase of NaNO2 using Eq. (5), where 60 = 20.32 GHz, which we obtained from the extrapolation of the frequency data [26] at T0 = 444 K. In Eq. (5), we used the literature values of A1 =17.2×10 − 6, B1 =9.1 ×10 − 5 and h1 =0.36 for the thermal expansivity [12], according to Eq. (6). By taking the integral limits T =441.96 K, TN = 437.7 K and kP =7.32, we calculated the Brillouin frequencies as a function of temperature using Eq. (5) for the L-mode along the [010] direction in the paraelectric phase of NaNO2. Our calculated frequencies for the L-mode // [010] are plotted as a function of temperature together with the experimental data [26] for T \ TN in Fig. 1. Our constant values for kP, 60 and the temperature dependence of Dw for the [010] mode are given in Table 1. The values of the coefficients A1, B1 and a1 for the thermal expansivity are tabulated in Table 2. 3.1.2. Calculation of the frequencies for the L-mode // [010] in the sinusoidal antiferroelectric phase (Tc BTB TN) The Brillouin frequencies for the [010] mode were calculated in the sinusoidal antiferroelectric phase by means of Eq. (7), where we used the value of 60 =20.67 GHz which was obtained from the frequency data of Yagi et al. [26] at T0 =434 K. In Eq. (7), the thermal expansivity data were used with the parameter values of A1 =18.1 ×
10 − 5, B2 = 9.1× 10 − 5 and h2 = 0.11 [12] according to Eq. (8). The integral limits for the temperature dependence of thermal expansivity in Eq. (7) were taken as Tc = 436.3 K and TN = 437.7 K. By using our value of kP = 7.32 for the mode Gruneisen parameter of the [010] mode, we were able to predict through Eq. (7) the Brillouin frequencies of this mode for Tc B TB TN in the sinusoidal antiferroelectric phase of NaNO2. We plot our calculated frequencies of the [010] mode, together with the observed frequencies [26] for Tc B TB TN in Fig. 1. We give our values for the parameters (Eq. (7)) and those values for the thermal expansivity (Eq. (8)) in Tables 1 and 2, respectively.
3.1.3. Calculation of the frequencies for the L-mode [010] in the ferroelectric phase (TB Tc) For the ferroelectric phase, we used Eq. (9) to calculate the Brillouin frequencies of the [010]
Fig. 1. Our calculated frequencies as a function of temperature for the L-mode [010] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN =437.7 K, Tc =436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown.
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Table 1 The values of the frequency shifts calculated for the paraelectric phase (T\TN), the values of the mode Gruneisen parameter kP calculated for the temperature intervals indicated, values of the Brillouin frequency 60 extrapolated at the temperatures given and the temperature dependence of the order–disorder contribution to the frequency D6 for the [010], [001] and [100] modes of NaNO2 (see text) Mode
(w (GHz/K) (T
[010] [001] [100]
−3.41×10−2 7.32 −2.42×10−2 7.01 −2.3×10−3 0.62
kP
Temperature Interval (K)
w0(GHz)
Dw(GHz)
441.96BTB438.78 440.57BTB441.64 438.28BTB441.42
20.32 (at 444 K) 20.67 (at 434 K) 17.94 (at 444 K) 18.23 (at 432 K) 15.95 (at 444 K) 15.98 (at 432 K)
0.1006T−2.301×10−4T 2 0.0386T−8.746×10−5T 2 0.0264T−6.014×10−5T 2
mode using the temperature dependence of the thermal expansivity with the integral limits of T =435.21 K and Tc =436.3 K. In Eq. (9), the value of 60 = 20.67 GHz was used from the extrapolation at T0 = 434 K. We also determined the temperature dependence of D6 with the constant values of a= 0.1006 and b = − 2.301 × 10 − 4 according to Eq. (11). We then calculated the Brillouin frequencies of the [010] mode using the temperature dependence of the thermal expansivity hP (Eq. (12)) with the experimental values of A3 =2.098×10 − 4, B3 =9.1 ×10 − 5 and h3 =0.08 due to Ema et al. [12] and our constant value of kP = 7.32. Our constant values for kP, w0 and the temperature dependence of Dw are tabulated in Table 1. The values of A3, B3 and h3 for the thermal expansivity are given in Table 2. Our calculated frequencies for the [010] mode of NaNO2 are plotted as a function of temperature in Fig. 1. Experimental frequencies [26] are also plotted in this figure.
to Eq. (2), which gave kP = 7.01 as the average value. We then assumed that this average value of the mode Gruneisen parameter did not vary across the phase transitions from the paraelectric to the ferroelectric phases in NaNO2 as we also assumed for the [010] mode in this crystalline system.
3.2. Calculation of the mode Gruneisen parameter and the Brillouin frequencies for the L-mode [001]
3.2.1. Calculation of the frequencies for the L-mode [001] in the paraelectric phase (T\ TN) Using Eq. (5), we were able to predict the Brillouin frequencies of the [001] mode of NaNO2 in the paraelectric phase by means of the experimental data for the thermal expansivity [12]. We first determined the extrapolated value of 60 = 17.94 GHz at T0 = 444 K and we took the limits of the integral given in Eq. (5) as T=441.64 K and TN = 437.7 K. Using the values of A1, B1 and h1 for the thermal expansivity Eq. (5) from Table 2 and our value of kP = 7.01, we predicted the Brillouin frequencies for the [001] mode by means of Eq. (5). We plot our calculated frequencies as a function of temperature in Fig. 2. The experimental data due to Yagi et al. [26] are also plotted in this figure.
As for the [010] mode, we calculated the value of the mode Gruneisen parameter for the [001] mode in the paraelectric phase (T \ TN) of NaNO2. In this region, the frequency shifts (#w/ #T) [26] of this mode varied linearly with the thermal expansivity [12]. From this linear variation, we obtained values of the mode Gruneisen parameter between 6.85 and 7.27 within the temperature interval of 440.57 KBT B441.64 K in the paraelectric phase (TN =437.7 K) according
3.2.2. Calculation of the frequencies for the L-mode [001] in the sinusoidal antiferroelectric phase (Tc B TB TN) We calculated the Brillouin frequencies of the [001] mode in the sinusoidal antiferroelectric phase. For this calculation, we used Eq. (7), where we had w0 = 18.23 GHz from the extrapolation at T0 = 432 K and the integral limits were Tc = 436.3 K and TN = 437.7 K, as before. Using the values of A2, B2 and a2 (Eq. (8)) given in Table 2 and our
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Table 2 The values of the coefficients A1, B1 and h1 (Eq. (6)), A2, B2 and h2 (Eq. (8)), A3, B3 and h3 (Eq. (12)) using the thermal expansivity data of NaNO2 [12] A1(K−1)
A2(K−1)
A3×10−4(K−1)
B1 =B2 =B3(K−1)
h1
h2
h3
1.72×10−5
1.81×10−4
2.098 [010] 1.81 [001] 1.248 [100]
9.1×10−5
0.36
0.11
0.08 [010] 0.17 [001] 0.15 [100]
The values of the coefficients A3 and h3 are given for the modes [010], [001] and [100] as indicated, according to our analysis of the thermal expansivity (see text).
value of kP = 7.01, we calculated the Brillouin frequencies of the [001] mode through Eq. (7) using the thermal expansivity data [12] in the sinusoidal antiferroelectric phase of NaNO2. In Fig. 2, we show our calculated Brillouin frequencies for the [001] mode as a function of temperature, together with the experimental frequencies [26] in the sinusoidal antiferroelectric phase of NaNO2.
3.2.3. Calculation of the frequencies for the L-mode [001] in the ferroelectric phase (T B Tc) In order to calculate the frequencies for the [001] mode in the ferroelectric phase, we first determined the extrapolated value of w0 =18.23 GHz at T0 = 432 K and the temperature dependence of Dw according to Eq. (11), with the values of a =0.0386 and b= −8.746 × 10 − 5, as given in Table 1. We then predicted the Brillouin frequencies for the [001] mode by reanalyzing the thermal expansivity data [12] for the ferroelectric phase, according to Eq. (12) with the values of A3 = 1.181× 10 − 4, B3 = 9.1 × 10 − 5 and h3 =0.17, which we also give in Table 2. For predicting the Brillouin frequencies for this mode by means of Eq. (9), we had our value of kP = 7.01 and the integral limits were T = 434 K and Tc = 436.3 K. Our calculated frequencies for the [001] mode are plotted as a function of temperature for the ferroelectric phase (T B Tc) in NaNO2 in Fig. 2. The experimental data [26] are also plotted in this figure.
3.3. Calculation of the mode Gruneisen parameter and the Brillouin frequencies for the L-mode [100]
Here, we determined the value of the mode Gruneisen parameter for the [100] mode of NaNO2 through Eq. (2) using the frequency data [26] and the thermal expansivity data [12]. As we did for the [010] and [001] modes, for determining the mode Gruneisen parameter of the [100] mode, we used the linear variation of the frequency shifts (#w/#T) and of the thermal expansivity in the
Fig. 2. Our calculated frequencies as a function of temperature for the L-mode [001] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN =437.7 K, Tc =436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown.
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paraelectric phase (T \Tc). According to our analysis, we obtained the values between 0.52 and 0.77 for the mode Gruneisen parameter of the [100] mode, which gave us the average value of kP = 0.62 within the temperature interval of 438.28 KBT B 441.42 K in the paraelectric phase (TN =437.7 K.). We then assumed that this value of kP remains constant through phase transitions from the paraelectric to the sinusoidal antiferroelectric and ferroelectric phases in NaNO2.
3.3.1. Calculation of the frequencies for the L-mode [100] in the paraelectric phase (T \ TN) As we calculated the Brillouin frequencies of the [010] and [001] modes, here we also calculated the Brillouin frequencies of the [100] mode in the paraelectric phase by means of the same analysis given above. By determining the extrapolated value of 60 =15.95 GHz at T0 =444 K and our value of kP =0.62, we predicted the Brillouin frequencies of the [100] mode through Eq. (5), where T = 441.42 K and TN =437.7 K using the thermal expansivity data [12] according to Eq. (6) with the values of A1, B1 and h1, given in Table 2. In Fig. 3, we plot our calculated frequencies for the [100] mode together with the observed frequencies [26] as a function of temperature for the paraelectric phase (T\TN) of NaNO2. 3.3.2. Calculation of the frequencies for the L-mode [100] in the sinusoidal antiferroelectric phase (Tc BTB TN) The Brillouin frequencies were calculated using Eq. (7), where we had the extrapolated value of V0 =15.98 GHz at T0 =432 K, our value of kP = 0.62 and the thermal expansivity data [12] according to Eq. (8) with the values of A2, B2 and h2 given in Table 2. In Eq. (7), the integral limits were Tc =436.3 K and TN =437.7 K, as before. Our calculated frequencies for the [100] mode are given as a function of temperature for the sinusoidal antiferrolectric phase in NaNO2. In this figure, the experimental frequencies [26] are also plotted. 3.3.3. Calculation of the frequencies for the L-mode [100] in the ferroelectric phase (T B Tc)
Fig. 3. Our calculated frequencies as a function of temperature for the L-mode [100] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN =437.7 K, Tc =436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown.
We also calculated here the Brillouin frequencies of the [100] mode in the ferroelectric phase, as we did for the [010] and [001] modes in NaNO2. We obtained the extrapolated value of 60 = 15.95 GHz at T0 = 432 K and the temperature dependence of D6 with the values of a= 0.0264 and b= − 6.014×10 − 5 (Eq. (11)), which are also tabulated in Table 1. We then used our value of kP = 0.62 and the thermal expansivity data [12], which we reanalyzed in the ferroelectric phase according to Eq. (12) with the values of A3 = 1.248× 10 − 4, B3 = 9.1× 10 − 5 and h3 = 0.15, as given in Table 2 and we were then able to calculate the Brillouin frequencies for the [100] mode by means of Eq. (9) where we had the integral limits of T=434.14 K and Tc = 436.3 K. Our calculated frequencies and those observed [26] for the [100] mode are plotted as a function of temperature for the ferroelectric phase (TB Tc) of NaNO2 in Fig. 3.
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Finally, we plot our calculated Brillouin frequencies for the [010], [001] and [100] modes of NaNO2 as a function of temperature in the paraelectric, sinusoidal antiferroelectric and the ferroelectric phases in this crystalline system in Fig. 4. The experimental frequencies [26] are also plotted in this figure.
4. Discussion We predicted here the Brillouin frequencies for the L-mode [010], [001] and [100] of NaNO2 for the paraelectric (T \TN), sinusoidal antiferroelectric (Tc BTB TN) and ferroelectric (T B Tc) phases in this crystalline system. For this calculation, we used the frequency data [26] and the thermal expansivity data [12] by means of the functional forms for each phase transition region. For the paraelectric phase, we had our frequency relation (Eq. (5)) with the thermal expansivity (Eq. (6)). For the sinusoidal antiferroelectric and ferroelectric phases we used our frequency relations given by Eq. (7) and Eq. (9), respectively. Those frequency relations accompanied with the thermal expansivity expressions are given by Eq. (8) and Eq. (12), respectively. The expressions for the thermal expansivity (Eq. (6), Eq. (8) and Eq. (12)) were taken from the work of Ema et al. [12]. In the frequency relation for the ferroelectric phase (Eq. (9)) we introduced a nonzero Dw as an order –disorder contribution to the frequency, which we take zero in the paraelectric and the sinusoidal antiferroelectric phases. We obtain Dw as a function of temperature for each mode, as given in Table 1. In order to calculate the Brillouin frequencies for the modes studied, we first determined the values of the mode Gruneisen parameter, as given in Table 1. For this determination, we assumed that the mode Gruneisen parameter does not vary with the temperature across the phase transitions. This assumption has been examined for ammonium halides [38,39]. Since the frequency shifts (#w/#T) and the thermal expansivity vary linearly with the temperature away from the phase transitions in the paraelectric phase, we determined the value
of each mode Gruneisen parameter within the temperature intervals given in Table 1. As we see from Table 1, the values of the mode Gruneisen parameter for the [010] and [001] modes, are close to each other (nearly 7), whereas for the [100] mode this value is very small (0.6) because of the smallness of the frequency shift ((6/(T), which is about one order of magnitude smaller than that for the [010] and [001] modes. We plotted our predicted frequencies with the observed data [26] as a function of temperature in Figs. 1– 3 for the [010], [001] and [100] modes of NaNO2, respectively. As shown in these figures, there are some discrepancies between our calculated and those observed in some temperature regions. In particular, for the [010] mode in the paraelectric phase nearly above 440 K and for the temperature interval of 437.5 KBTB 438.5 K, whereas in the ferroelectric phase there is a good agreement between our calculated frequencies and the observed frequencies, as shown in Fig. 1. For the [001] mode, there is discrepancy between calculated and observed frequencies for the temperature interval of nearly 437 KB TB 438 K, whereas in the ferroelectric and paraelectric phases our calculated frequencies agree reasonably well with the observed data, as given in Fig. 2. On the other hand, for the [100] mode, as seen in Fig. 3, our calculated frequencies seem to follow the experimental data points, which are scattered within the temperature region of 436.5 KBTB 438 K and also just below Tc. Because the experimental frequency data are scattered within the temperature regions indicated, this may be one reason for this discrepancy between our calculated frequencies and the observed data. We also note that since we calculated the Brillouin frequencies for those modes studied using the thermal expansivity data for NaNO2 and we compared our calculated frequencies with the observed frequencies, there are also discrepancies about the locations of the Neel temperature TN and the critical temperature Tc for two sets of experimental measurements. Those locations are, namely, TN = 438.2 K and Tc = 436.9 K due to
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Fig. 4. Our calculated frequencies as a function of temperature for the L-mode [010], [001] and [100] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN = 437.7 K, Tc = 436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown. Vertical dots represent the three phases considered.
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Ema et al. [12], whereas they are TN =437.7 K and Tc = 436.3 K from the work of Yagi et al. [26]. Considering the fact that we did our analysis within the temperature interval of nearly 8 K, this difference of DT $ 0.5 K can be important to determine the location of the transition points between the phases of the paraelectric, sinusoidal antiferroelectric and ferroelectric in NaNO2. This discrepancy between our calculated frequencies and those observed within the temperature regions indicated, may then be, namely, TN =438.2 K and Tc = 436.9 K due to Ema et al. [12], whereas they are TN =437.7 K and Tc =436.3 K from the work of Yagi et al. [26]. Considering the fact that we performed our analysis within the temperature interval of nearly 8 K, this difference of DT $ 0.5 K can be important in determining the location of the transition points between the phases of the paraelectric, sinusoidal antiferroelectric and ferroelectric in NaNO2. This discrepancy between our calculated frequencies and those observed within the temperature regions indicated, may then be decreased. Although these discrepancies occur, agreement between our predicted frequencies and the experimental frequencies is good for the phase transitions from the paraelectric phase to the sinusoidal antiferroelectric and ferroelectric phases for the [010], [001] and [100] modes of NaNO2, as also given in Fig. 4.
5. Conclusions We calculated in this study the Brillouin frequencies for the L-mode [010], [001] and [100] through the phase transitions from the paraelectric phase to the sinusoidal antiferroelectric and to the ferroelectric phases in NaNO2. The frequencies were calculated by means of the thermal expansivity and they were compared with those measured experimentally. Our calculated frequencies are in good agreement with the observed data for those modes throughout the transition regions studied in NaNO2. This shows that the method employed here for calculating the frequencies is applicable to NaNO2 to predict its observed behavior close to phase transitions.
References [1] S. Sawada, S. Nomura, S. Fujii, I. Yoshida, Phys. Rev. Lett. 1 (1958) 320. [2] S. Tanisaki, J. Phys. Soc. Jap. 16 (1961) 579. [3] S. Tanisaki, J. Phys. Soc. Jap. 18 (1963) 1181. [4] Y. Yamada, I. Shibuya, S. Hoshino, J. Phys. Soc. Jap. 18 (1963) 1594. [5] S. Hoshino, H. Motegi, Jap. 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. Jap. 19 (1964) 142. [8] K. Hamano, J. Phys. Soc. Jap. 19 (1964) 945. [9] B. Wyncke, F. Brehat, O.V. Kozlov, Phys. Stat. Sol. (b) 129 (1984) 531. [10] S. Hoshino, J. Phys. Soc. Jap. 19 (1964) 140. [11] M. Sakiyama, A. Kimoto, S. Seki, J. Phys. Soc. Jap. 20 (1965) 2180. [12] K. Ema, K. Hamano, I. Hatta, J. Phys. Soc. Jap. 39 (1975) 726. [13] K. Hamano, K. Ema, J. Phys. Soc. Jap. 45 (1978) 923. [14] H. Vogt, H. Happ, Phys. Stat. Sol. 30 (1968) 67. [15] F. Brehat, B. Wyncke, J. Phys. C: Sol. State Phys. 18 (1985) 1705. [16] J.D. Axe, Phys. Rev. 167 (1968) 573. [17] M.K. Barnoski, J.M. Ballantyne, Phys. Rev. 174 (1968) 946. [18] E.V. Chisler, M.S. Shur, Phys. Stat. Sol. 17 (1966) 163. [19] E.V. Chisler, M.S. Shur, Sov. Phys. Sol. State 9 (1967) 796. [20] A. Takase, K. Miyakawa, J. Phys. C: Sol. State Phys. 18 (1985) 5579. [21] C.K. Asawa, M.K. Barnoski, Phys. Rev. B 2 (1970) 205. [22] I. Hatta, Y. Shimizu, K. Hamano, J. Phys. Soc. Jap. 44 (1978) 1887. [23] I. Hatta, M. Hanami, K. Hamano, J. Phys. Soc. Jap. 48 (1980) 160. [24] H. Shimizu, Y. Ishibashi, M. Tsukamoto, M. Umeno, J. Phys. Soc. Jap. 36 (1974) 498. [25] T. Yagi, Y. Hikada, K. Miura, J. Phys. Colloq. C6 (1981) 731. [26] T. Yagi, Y. Hikada, K. Miura, J. Phys. Soc. Jap. 51 (1982) 3562. [27] S.Kh. Esayan, V.V. Lemanov, N. Mamatkulov, Sov. Phys. Sol. State 23 (1981) 1195. [28] I. Hatta, J. Phys. Soc. Jap. 53 (1984) 635. [29] W. Buchheit, G. Herth, J. Peterson, Sol. State Comm. 40 (1981) 411. [30] J. Sakurai, R.A. Cowley, G. Dolling, J. Phys. Soc. Jap. 28 (1970) 1426. [31] D. Durand, L. Bernard, F. Mezei, R. Currat, F. Denoyer, M. Lambert, J. Phys. 136B (1986) 325. [32] Y. Yamada, T. Yamada, J. Phys. Soc. Jap. 21 (1966) 2167. [33] K.D. Ehrhardt, K.H. Michel, Z. Phys. B. Cond. Mat. 41 (1981) 329.
H. Yurtse6en, I: .E. C ¸ aglar / Spectrochimica Acta Part A 58 (2002) 55–65 [34] K.D. Ehrhardt, K.H. Michel, Phys. Rev. Lett. 46 (1981) 291. [35] J. Fivez, K.H. Michell, Z. Phys. B Cond. Mat. 51 (1983) 127. [36] Y. Ishibashi, H. Shiba, J. Phys. Soc. Jap. 45 (1978) 409.
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[37] Y. Ishibashi, W. Buchheit, J. Peterson, Sol. State Comm. 38 (1981) 1277. [38] H. Yurtseven, W.F. Sherman, Phase Trans. 54 (1995) 1. [39] H. Yurtseven, M. Gulec, W.F. Sherman, Phase Trans. 56 (1996) 137.
Calculation of the Brillouin frequencies close to phase transitions in NaNO2 H. Yurtseven *, I: .E. C ¸ aglar Department of Physics, Istanbul Technical Uni6ersity, Maslak, 80626 Istanbul, Turkey Received 27 June 2000; received in revised form 20 April 2001; accepted 20 April 2001
Abstract We calculate here the Brillouin frequencies of the L-mode [010], [001] and [100] of NaNO2 for the phase transitions from the paraelectric phase to the sinusoidal anti-ferroelectric phase near the Neel temperature (TN = 437.7 K) and to the ferroelectric phase near the critical temperature (Tc = 436.3 K) in this crystalline system. For calculating the frequencies, we use the thermal expansivity data for the phase regions considered, under the assumption that the mode Gruneisen parameter determined for each mode remains constant across the phase transitions. Our calculated frequencies agree well with the observed frequencies for the modes studied in NaNO2. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Brillouin frequencies; Phase transitions; NaNO2
1. Introduction Sodium nitrite undergoes phase transitions from the paraelectric phase to the antiferroelectric phase at the Neel temperature TN =437.7 K, as the temperature decreases from high temperatures. As the temperature decreases further, the antiferroelectric phase transforms into the ferroelectric phase at the Curie temperature Tc = 436.3 K. This ferroelectric behaviour of NaNO2 was first discovered by Sawada et al. [1]. Following Sawada, there have been X-ray studies on
* Corresponding author. Tel.: + 90-212-2853209; fax: + 90212-2856386. E-mail address: [email protected] (H. Yurtseven).
NaNO2 due to Tanisaki [2,3], Yamada et al. [4], Hoshino and Motegi [5], Durand et al. [6] and the dielectric studies on NaNO2 due to Sawada et al. [1], Tagaki and Gesi [7], Hamano [8] and Wyncke et al. [9]. Apart from these experimental techniques, calorimetric measurements for NaNO2 have been performed by Hoshino [10] and Sakiyama et al. [11]. Also, the thermal expansion and the elastic compliances of NaNO2 have been measured by Ema et al. [12] and Hamano and Ema [13], respectively. Some spectroscopic studies on NaNO2 have also been reported in the literature. Far infrared absorption of NaNO2 has been obtained by Vogt and Happ [14]. Infrared spectra by Brehat and Wyncke [15], the reflection spectra by Axe [16] and Barnoski and Ballantyne [17] and also the Raman spectra by Chisler and Shur
1386-1425/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 1 4 2 5 ( 0 2 ) 0 0 5 0 5 - 4
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[18,19], Takase and Miyakawa [20], Asawa and Barnoski [21] have been obtained for the NaNO2 system. The ultrasonic and the Brillouin scattering of NaNO2 have also been reported in the literature by Hatta et al. [22,23], Shimizu et al. [24], Yagi et al. [25,26], Esayan et al. [27] and Hatta [28]. The spectroscopic technique NMR by Buchheit et al. [29] was used to investigate 23Na nucleus for the commensurate and incommensurate phases in NaNO2. Also, the neutron spectroscopic technique by Sakurai et al. [30] and Durand et al. [31], has been used to obtain the spectroscopic data for NaNO2. The experimental studies on NaNO2 have been accompanied by some theoretical studies on NaNO2, as reported in the literature. An Ising model has been applied to the NaNO2 system by means of the analogy between the two possible orientations of an Ising spin (spin up and spin down) and the two equilibrium positions of the NO− 2 dipole in the paraelectric phase of NaNO2, due to Tanisaki [2], Yamada et al. [4], Yamada and Yamada [32]. A microscopic model has been developed in the works of Erhardt and Michel [33,34] and Fivez and Michel [35]. The mechanism of phase transitions in NaNO2 has been explained on the basis of the model proposed by Ishibashi and Shiba [36] and Ishibashi et al. [37]. In this study, we calculate the Brillouin frequencies of the L-mode [010], [001] and [100] by relating the thermal expansivity to the frequency for the phase transitions from the paraelectric to the sinusoidal antiferroelectric and ferroelectric phases in NaNO2. We use for this calculation, the thermal expansion data due to Ema et al. [12] and the Brillouin frequency data due to Yagi et al. [25,26]. In Section 2, we give an outline of our method of calculation for the Brillouin frequency. In Section 3, we give our calculations and results. Discussion is presented in Section 4. Finally, conclusions are given in Section 5.
2. Theory The thermodynamic functions can be related to the frequency shifts by means of the Gruneisen parameter defined as
k=
( ln w ( ln V
(1)
Using the definitions of the thermal expansivity hP =
1 (V V (T
P
and the isothermal compressibility iT = −
1 (V V (P
,
T
we then define the isobaric mode Gruneisen parameter kP and the isothermal mode Gruneisen parameter kT, respectively as kP = − and kT =
1 1 (w hP w (T
1 1 (w iT w (P
(2)
P
(3)
T
Close to phase transitions, the frequency shifts and also the frequencies can be calculated using the data for the thermal expansivity and the isothermal compressibility by taking the mode Gruneisen parameters kP and kT given in Eq. (2) and Eq. (3), respectively, as constants across the phase transitions. Here, we are going to calculate the Brillouin frequencies of the longitudinal acoustic mode (Lmode) along the [010], [001] and [100] directions for NaNO2 close to the phase transitions, from the paraelectric phase to the sinusoidal antiferroelectric and to the ferroelectric phase in this crystalline system. For this calculation, we use the temperature dependence of the thermal expansivity for the three phases, namely, the paraelectric, sinusoidal antiferroelectric and the ferroelectric phases, according to Ema et al. [12]. By expressing Eq. (2) in the form
&
6
1 dw= − 60 w
&
T0
hPkP dT+ Dw
(4)
T
we can calculate the frequencies by means of the temperature dependence of the thermal expansivity. In Eq. (4), Dw represents the order– disorder contribution to the frequency and w0 is the value of the frequency at a constant temperature T0.
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2.1. Paraelectric phase (T \ TN)
2.3. Ferroelectric phase (TB Tc)
Since there is no ordering above the Neel temperature in the paraelectric phase, we take the order–disorder contribution to the Brillouin frequency of the LA mode as zero, Dw = 0. By assuming that the mode Gruneisen parameter kP remains constant close to the Neel temperature TN, we have from Eq. (4) the frequency relation
We can calculate the frequencies from Eq. (4) using the relation
ln w(T)=ln w0 − kP
&
TN
hP(T) dT
(5)
T
In Eq. (5), the temperature dependence of the thermal expansivity is given by hP(T) =A1m − h1 +B1
(6)
where the critical exponent is h1, the reduced temperature is m= ((T − TN)/TN), A1 is the amplitude and B1 is a constant. Those constants h1, A1 and B1 of Eq. (6) can be obtained from the temperature dependence of the thermal expansivity which were obtained experimentally [12].
2.2. Sinusoidal antiferroelectric phase (Tc BT B TN) Since there is very little change in the Brillouin frequencies of the L-mode below the Neel temperature TN down to the critical temperature Tc, as observed experimentally in NaNO2 [26], we take Dw =0 in Eq. (4). We then obtain similar to Eq. (5) that ln w(T)= ln w0 + kP
&
TN
hP(T) dT
(7)
T
In Eq. (7), the temperature dependence of the thermal expansivity is given by
ln w(T)=ln w0 + kP
&
Tc
hP(T) dT+ Dw
(9)
T
where we define the order–disorder contribution to the frequency as Dw= wobserved − wcalculated
(10)
which we take its temperature dependence as Dw= aT+bT 2
(11)
Here, a and b are constants which can be determined from the observed [26] and our calculated Brillouin frequencies for the L-mode [010], [001] and [100]. In Eq. (9), the temperature dependence of the thermal expansivity is given by hP(T)= A3 m − h3 + B3
(12)
where the critical exponent is a3, the reduced temperature is m= ((T − Tc)/Tc), A3 is the amplitude and B3 is a constant. Those constants a3, A3 and B3 of Eq. (12) can be obtained from the temperature dependence of the thermal expansivity that was obtained experimentally [12].
3. Calculations and results Here, we first determined the values of the mode Gruneisen parameter kP for the L-mode along the [010], [001] and [100] directions in NaNO2. We then calculated the Brillouin frequencies of those modes for the phases of paraelectric (T\ TN), sinusoidal antiferroelectric (Tc B TBTN) and ferroelectric (TB Tc) in NaNO2.
(8)
3.1. Calculation of the mode Gruneisen parameter and the Brillouin frequencies for the L-mode [010]
where the critical exponent is h2 and m =((T − TN)/TN) as before. A2 is the amplitude and B2 is a constant. Those constants h2, A2 and B2 of Eq. (8) can be obtained from the temperature dependence of the thermal expansivity that was obtained experimentally [12].
We determined the mode Gruneisen parameter kP by means of Eq. (2) using the experimental data for the Brillouin frequencies [26] and the thermal expansivity [12]. For this calculation, we used those experimental data for the temperature region of 441.96 KB TB438.78 K in the
hP(T) =A2 m
− h2
+B2
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paraelectric phase T\ TN, where TN =437.7 K. This temperature region was appropriate to determine the kP value, since the frequency shifts (#w/ #T) varied linearly with the thermal expansivity hP in the paraelectric phase, according to Eq. (2). This then gave us the values between 6.63 and 8.79 for the mode Gruneisen parameter of the [010] mode, from which we took the average value of kP =7.32. We then assumed that this value of kP for the L-mode // [010] remained constant across the phase transitions of paraelectric– sinusoidal antiferroelectric– ferroelectric in NaNO2.
3.1.1. Calculation of the frequencies for the L-mode // [010] in the paraelectric phase (T \ TN) We calculated the Brillouin frequencies for the L-mode along the [010] direction in the paraelectric phase of NaNO2 using Eq. (5), where 60 = 20.32 GHz, which we obtained from the extrapolation of the frequency data [26] at T0 = 444 K. In Eq. (5), we used the literature values of A1 =17.2×10 − 6, B1 =9.1 ×10 − 5 and h1 =0.36 for the thermal expansivity [12], according to Eq. (6). By taking the integral limits T =441.96 K, TN = 437.7 K and kP =7.32, we calculated the Brillouin frequencies as a function of temperature using Eq. (5) for the L-mode along the [010] direction in the paraelectric phase of NaNO2. Our calculated frequencies for the L-mode // [010] are plotted as a function of temperature together with the experimental data [26] for T \ TN in Fig. 1. Our constant values for kP, 60 and the temperature dependence of Dw for the [010] mode are given in Table 1. The values of the coefficients A1, B1 and a1 for the thermal expansivity are tabulated in Table 2. 3.1.2. Calculation of the frequencies for the L-mode // [010] in the sinusoidal antiferroelectric phase (Tc BTB TN) The Brillouin frequencies for the [010] mode were calculated in the sinusoidal antiferroelectric phase by means of Eq. (7), where we used the value of 60 =20.67 GHz which was obtained from the frequency data of Yagi et al. [26] at T0 =434 K. In Eq. (7), the thermal expansivity data were used with the parameter values of A1 =18.1 ×
10 − 5, B2 = 9.1× 10 − 5 and h2 = 0.11 [12] according to Eq. (8). The integral limits for the temperature dependence of thermal expansivity in Eq. (7) were taken as Tc = 436.3 K and TN = 437.7 K. By using our value of kP = 7.32 for the mode Gruneisen parameter of the [010] mode, we were able to predict through Eq. (7) the Brillouin frequencies of this mode for Tc B TB TN in the sinusoidal antiferroelectric phase of NaNO2. We plot our calculated frequencies of the [010] mode, together with the observed frequencies [26] for Tc B TB TN in Fig. 1. We give our values for the parameters (Eq. (7)) and those values for the thermal expansivity (Eq. (8)) in Tables 1 and 2, respectively.
3.1.3. Calculation of the frequencies for the L-mode [010] in the ferroelectric phase (TB Tc) For the ferroelectric phase, we used Eq. (9) to calculate the Brillouin frequencies of the [010]
Fig. 1. Our calculated frequencies as a function of temperature for the L-mode [010] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN =437.7 K, Tc =436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown.
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Table 1 The values of the frequency shifts calculated for the paraelectric phase (T\TN), the values of the mode Gruneisen parameter kP calculated for the temperature intervals indicated, values of the Brillouin frequency 60 extrapolated at the temperatures given and the temperature dependence of the order–disorder contribution to the frequency D6 for the [010], [001] and [100] modes of NaNO2 (see text) Mode
(w (GHz/K) (T
[010] [001] [100]
−3.41×10−2 7.32 −2.42×10−2 7.01 −2.3×10−3 0.62
kP
Temperature Interval (K)
w0(GHz)
Dw(GHz)
441.96BTB438.78 440.57BTB441.64 438.28BTB441.42
20.32 (at 444 K) 20.67 (at 434 K) 17.94 (at 444 K) 18.23 (at 432 K) 15.95 (at 444 K) 15.98 (at 432 K)
0.1006T−2.301×10−4T 2 0.0386T−8.746×10−5T 2 0.0264T−6.014×10−5T 2
mode using the temperature dependence of the thermal expansivity with the integral limits of T =435.21 K and Tc =436.3 K. In Eq. (9), the value of 60 = 20.67 GHz was used from the extrapolation at T0 = 434 K. We also determined the temperature dependence of D6 with the constant values of a= 0.1006 and b = − 2.301 × 10 − 4 according to Eq. (11). We then calculated the Brillouin frequencies of the [010] mode using the temperature dependence of the thermal expansivity hP (Eq. (12)) with the experimental values of A3 =2.098×10 − 4, B3 =9.1 ×10 − 5 and h3 =0.08 due to Ema et al. [12] and our constant value of kP = 7.32. Our constant values for kP, w0 and the temperature dependence of Dw are tabulated in Table 1. The values of A3, B3 and h3 for the thermal expansivity are given in Table 2. Our calculated frequencies for the [010] mode of NaNO2 are plotted as a function of temperature in Fig. 1. Experimental frequencies [26] are also plotted in this figure.
to Eq. (2), which gave kP = 7.01 as the average value. We then assumed that this average value of the mode Gruneisen parameter did not vary across the phase transitions from the paraelectric to the ferroelectric phases in NaNO2 as we also assumed for the [010] mode in this crystalline system.
3.2. Calculation of the mode Gruneisen parameter and the Brillouin frequencies for the L-mode [001]
3.2.1. Calculation of the frequencies for the L-mode [001] in the paraelectric phase (T\ TN) Using Eq. (5), we were able to predict the Brillouin frequencies of the [001] mode of NaNO2 in the paraelectric phase by means of the experimental data for the thermal expansivity [12]. We first determined the extrapolated value of 60 = 17.94 GHz at T0 = 444 K and we took the limits of the integral given in Eq. (5) as T=441.64 K and TN = 437.7 K. Using the values of A1, B1 and h1 for the thermal expansivity Eq. (5) from Table 2 and our value of kP = 7.01, we predicted the Brillouin frequencies for the [001] mode by means of Eq. (5). We plot our calculated frequencies as a function of temperature in Fig. 2. The experimental data due to Yagi et al. [26] are also plotted in this figure.
As for the [010] mode, we calculated the value of the mode Gruneisen parameter for the [001] mode in the paraelectric phase (T \ TN) of NaNO2. In this region, the frequency shifts (#w/ #T) [26] of this mode varied linearly with the thermal expansivity [12]. From this linear variation, we obtained values of the mode Gruneisen parameter between 6.85 and 7.27 within the temperature interval of 440.57 KBT B441.64 K in the paraelectric phase (TN =437.7 K) according
3.2.2. Calculation of the frequencies for the L-mode [001] in the sinusoidal antiferroelectric phase (Tc B TB TN) We calculated the Brillouin frequencies of the [001] mode in the sinusoidal antiferroelectric phase. For this calculation, we used Eq. (7), where we had w0 = 18.23 GHz from the extrapolation at T0 = 432 K and the integral limits were Tc = 436.3 K and TN = 437.7 K, as before. Using the values of A2, B2 and a2 (Eq. (8)) given in Table 2 and our
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60
Table 2 The values of the coefficients A1, B1 and h1 (Eq. (6)), A2, B2 and h2 (Eq. (8)), A3, B3 and h3 (Eq. (12)) using the thermal expansivity data of NaNO2 [12] A1(K−1)
A2(K−1)
A3×10−4(K−1)
B1 =B2 =B3(K−1)
h1
h2
h3
1.72×10−5
1.81×10−4
2.098 [010] 1.81 [001] 1.248 [100]
9.1×10−5
0.36
0.11
0.08 [010] 0.17 [001] 0.15 [100]
The values of the coefficients A3 and h3 are given for the modes [010], [001] and [100] as indicated, according to our analysis of the thermal expansivity (see text).
value of kP = 7.01, we calculated the Brillouin frequencies of the [001] mode through Eq. (7) using the thermal expansivity data [12] in the sinusoidal antiferroelectric phase of NaNO2. In Fig. 2, we show our calculated Brillouin frequencies for the [001] mode as a function of temperature, together with the experimental frequencies [26] in the sinusoidal antiferroelectric phase of NaNO2.
3.2.3. Calculation of the frequencies for the L-mode [001] in the ferroelectric phase (T B Tc) In order to calculate the frequencies for the [001] mode in the ferroelectric phase, we first determined the extrapolated value of w0 =18.23 GHz at T0 = 432 K and the temperature dependence of Dw according to Eq. (11), with the values of a =0.0386 and b= −8.746 × 10 − 5, as given in Table 1. We then predicted the Brillouin frequencies for the [001] mode by reanalyzing the thermal expansivity data [12] for the ferroelectric phase, according to Eq. (12) with the values of A3 = 1.181× 10 − 4, B3 = 9.1 × 10 − 5 and h3 =0.17, which we also give in Table 2. For predicting the Brillouin frequencies for this mode by means of Eq. (9), we had our value of kP = 7.01 and the integral limits were T = 434 K and Tc = 436.3 K. Our calculated frequencies for the [001] mode are plotted as a function of temperature for the ferroelectric phase (T B Tc) in NaNO2 in Fig. 2. The experimental data [26] are also plotted in this figure.
3.3. Calculation of the mode Gruneisen parameter and the Brillouin frequencies for the L-mode [100]
Here, we determined the value of the mode Gruneisen parameter for the [100] mode of NaNO2 through Eq. (2) using the frequency data [26] and the thermal expansivity data [12]. As we did for the [010] and [001] modes, for determining the mode Gruneisen parameter of the [100] mode, we used the linear variation of the frequency shifts (#w/#T) and of the thermal expansivity in the
Fig. 2. Our calculated frequencies as a function of temperature for the L-mode [001] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN =437.7 K, Tc =436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown.
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paraelectric phase (T \Tc). According to our analysis, we obtained the values between 0.52 and 0.77 for the mode Gruneisen parameter of the [100] mode, which gave us the average value of kP = 0.62 within the temperature interval of 438.28 KBT B 441.42 K in the paraelectric phase (TN =437.7 K.). We then assumed that this value of kP remains constant through phase transitions from the paraelectric to the sinusoidal antiferroelectric and ferroelectric phases in NaNO2.
3.3.1. Calculation of the frequencies for the L-mode [100] in the paraelectric phase (T \ TN) As we calculated the Brillouin frequencies of the [010] and [001] modes, here we also calculated the Brillouin frequencies of the [100] mode in the paraelectric phase by means of the same analysis given above. By determining the extrapolated value of 60 =15.95 GHz at T0 =444 K and our value of kP =0.62, we predicted the Brillouin frequencies of the [100] mode through Eq. (5), where T = 441.42 K and TN =437.7 K using the thermal expansivity data [12] according to Eq. (6) with the values of A1, B1 and h1, given in Table 2. In Fig. 3, we plot our calculated frequencies for the [100] mode together with the observed frequencies [26] as a function of temperature for the paraelectric phase (T\TN) of NaNO2. 3.3.2. Calculation of the frequencies for the L-mode [100] in the sinusoidal antiferroelectric phase (Tc BTB TN) The Brillouin frequencies were calculated using Eq. (7), where we had the extrapolated value of V0 =15.98 GHz at T0 =432 K, our value of kP = 0.62 and the thermal expansivity data [12] according to Eq. (8) with the values of A2, B2 and h2 given in Table 2. In Eq. (7), the integral limits were Tc =436.3 K and TN =437.7 K, as before. Our calculated frequencies for the [100] mode are given as a function of temperature for the sinusoidal antiferrolectric phase in NaNO2. In this figure, the experimental frequencies [26] are also plotted. 3.3.3. Calculation of the frequencies for the L-mode [100] in the ferroelectric phase (T B Tc)
Fig. 3. Our calculated frequencies as a function of temperature for the L-mode [100] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN =437.7 K, Tc =436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown.
We also calculated here the Brillouin frequencies of the [100] mode in the ferroelectric phase, as we did for the [010] and [001] modes in NaNO2. We obtained the extrapolated value of 60 = 15.95 GHz at T0 = 432 K and the temperature dependence of D6 with the values of a= 0.0264 and b= − 6.014×10 − 5 (Eq. (11)), which are also tabulated in Table 1. We then used our value of kP = 0.62 and the thermal expansivity data [12], which we reanalyzed in the ferroelectric phase according to Eq. (12) with the values of A3 = 1.248× 10 − 4, B3 = 9.1× 10 − 5 and h3 = 0.15, as given in Table 2 and we were then able to calculate the Brillouin frequencies for the [100] mode by means of Eq. (9) where we had the integral limits of T=434.14 K and Tc = 436.3 K. Our calculated frequencies and those observed [26] for the [100] mode are plotted as a function of temperature for the ferroelectric phase (TB Tc) of NaNO2 in Fig. 3.
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Finally, we plot our calculated Brillouin frequencies for the [010], [001] and [100] modes of NaNO2 as a function of temperature in the paraelectric, sinusoidal antiferroelectric and the ferroelectric phases in this crystalline system in Fig. 4. The experimental frequencies [26] are also plotted in this figure.
4. Discussion We predicted here the Brillouin frequencies for the L-mode [010], [001] and [100] of NaNO2 for the paraelectric (T \TN), sinusoidal antiferroelectric (Tc BTB TN) and ferroelectric (T B Tc) phases in this crystalline system. For this calculation, we used the frequency data [26] and the thermal expansivity data [12] by means of the functional forms for each phase transition region. For the paraelectric phase, we had our frequency relation (Eq. (5)) with the thermal expansivity (Eq. (6)). For the sinusoidal antiferroelectric and ferroelectric phases we used our frequency relations given by Eq. (7) and Eq. (9), respectively. Those frequency relations accompanied with the thermal expansivity expressions are given by Eq. (8) and Eq. (12), respectively. The expressions for the thermal expansivity (Eq. (6), Eq. (8) and Eq. (12)) were taken from the work of Ema et al. [12]. In the frequency relation for the ferroelectric phase (Eq. (9)) we introduced a nonzero Dw as an order –disorder contribution to the frequency, which we take zero in the paraelectric and the sinusoidal antiferroelectric phases. We obtain Dw as a function of temperature for each mode, as given in Table 1. In order to calculate the Brillouin frequencies for the modes studied, we first determined the values of the mode Gruneisen parameter, as given in Table 1. For this determination, we assumed that the mode Gruneisen parameter does not vary with the temperature across the phase transitions. This assumption has been examined for ammonium halides [38,39]. Since the frequency shifts (#w/#T) and the thermal expansivity vary linearly with the temperature away from the phase transitions in the paraelectric phase, we determined the value
of each mode Gruneisen parameter within the temperature intervals given in Table 1. As we see from Table 1, the values of the mode Gruneisen parameter for the [010] and [001] modes, are close to each other (nearly 7), whereas for the [100] mode this value is very small (0.6) because of the smallness of the frequency shift ((6/(T), which is about one order of magnitude smaller than that for the [010] and [001] modes. We plotted our predicted frequencies with the observed data [26] as a function of temperature in Figs. 1– 3 for the [010], [001] and [100] modes of NaNO2, respectively. As shown in these figures, there are some discrepancies between our calculated and those observed in some temperature regions. In particular, for the [010] mode in the paraelectric phase nearly above 440 K and for the temperature interval of 437.5 KBTB 438.5 K, whereas in the ferroelectric phase there is a good agreement between our calculated frequencies and the observed frequencies, as shown in Fig. 1. For the [001] mode, there is discrepancy between calculated and observed frequencies for the temperature interval of nearly 437 KB TB 438 K, whereas in the ferroelectric and paraelectric phases our calculated frequencies agree reasonably well with the observed data, as given in Fig. 2. On the other hand, for the [100] mode, as seen in Fig. 3, our calculated frequencies seem to follow the experimental data points, which are scattered within the temperature region of 436.5 KBTB 438 K and also just below Tc. Because the experimental frequency data are scattered within the temperature regions indicated, this may be one reason for this discrepancy between our calculated frequencies and the observed data. We also note that since we calculated the Brillouin frequencies for those modes studied using the thermal expansivity data for NaNO2 and we compared our calculated frequencies with the observed frequencies, there are also discrepancies about the locations of the Neel temperature TN and the critical temperature Tc for two sets of experimental measurements. Those locations are, namely, TN = 438.2 K and Tc = 436.9 K due to
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Fig. 4. Our calculated frequencies as a function of temperature for the L-mode [010], [001] and [100] in the paraelectric, sinusoidal antiferroelectric and ferroelectric phases (TN = 437.7 K, Tc = 436.3 K) for NaNO2. The experimental data points taken from Yagi et al. [26] are also shown. Vertical dots represent the three phases considered.
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Ema et al. [12], whereas they are TN =437.7 K and Tc = 436.3 K from the work of Yagi et al. [26]. Considering the fact that we did our analysis within the temperature interval of nearly 8 K, this difference of DT $ 0.5 K can be important to determine the location of the transition points between the phases of the paraelectric, sinusoidal antiferroelectric and ferroelectric in NaNO2. This discrepancy between our calculated frequencies and those observed within the temperature regions indicated, may then be, namely, TN =438.2 K and Tc = 436.9 K due to Ema et al. [12], whereas they are TN =437.7 K and Tc =436.3 K from the work of Yagi et al. [26]. Considering the fact that we performed our analysis within the temperature interval of nearly 8 K, this difference of DT $ 0.5 K can be important in determining the location of the transition points between the phases of the paraelectric, sinusoidal antiferroelectric and ferroelectric in NaNO2. This discrepancy between our calculated frequencies and those observed within the temperature regions indicated, may then be decreased. Although these discrepancies occur, agreement between our predicted frequencies and the experimental frequencies is good for the phase transitions from the paraelectric phase to the sinusoidal antiferroelectric and ferroelectric phases for the [010], [001] and [100] modes of NaNO2, as also given in Fig. 4.
5. Conclusions We calculated in this study the Brillouin frequencies for the L-mode [010], [001] and [100] through the phase transitions from the paraelectric phase to the sinusoidal antiferroelectric and to the ferroelectric phases in NaNO2. The frequencies were calculated by means of the thermal expansivity and they were compared with those measured experimentally. Our calculated frequencies are in good agreement with the observed data for those modes throughout the transition regions studied in NaNO2. This shows that the method employed here for calculating the frequencies is applicable to NaNO2 to predict its observed behavior close to phase transitions.
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