Ultrasonic frequency shifts close to the first order and tricritical phase transitions in NH4Cl

Journal of Molecular Structure 598 (2001) 109±116

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Ultrasonic frequency shifts close to the ®rst order and tricritical phase transitions in NH4Cl È . Tari, M. BasË H. Yurtseven*, O Department of Physics, Istanbul Technical University, 80626 Maslak, Istanbul, Turkey Received 31 January 2001; accepted 7 March 2001

Abstract This study examines our spectroscopic modi®cations of the Pippard relations using the ultrasonic frequencies of the q[110]  modes for the ®rst order phase transition in NH4Cl. We also examine our relations using the ultrasonic frequencies of and q‰110Š the q[110] mode for the tricritical phase transition in this crystal. From our plots of the speci®c heat Cp as a function of the frequency shifts …1=n†…2n=2T†P ; we obtain that our spectroscopically modi®ed Pippard relation considered here is satis®ed for those phonon modes studied close to the ®rst order and tricritical phase transitions in NH4Cl. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Pippard relations; Ultrasonic frequencies; NH4Cl

1. Introduction Ammonium chloride undergoes the l-type of phase transition at T ù 242 K …P ˆ 0†: As a ®rst order transition, this transition transforms into the tricritical phase transition under the pressure of P ù 1:6 kbar; as has been observed experimentally [1] and when the pressure increases further, it then transforms into a second order transition at P ù 2:8 kbar: The mechanism of the l-phase transitions in the ammonium halides, in particular, in the ammonium chloride is attributed to the two orientations of the NH41 tetrahedra [2]. Over the years, the l-phase transitions in the ammonium halides have been studied both experimentally and theoretically in the literature, * Corresponding author. Tel.: 190-212-285-3258; fax: 190-212285-6386. E-mail address: [email protected] (H. Yurtseven).

as we have reviewed in our earlier studies for NH4Cl [3] and for NH4Br [4]. Among the experimental techniques, the spectroscopic technique has been largely used to study the phase transitions in the ammonium halides. In particular, using the Raman spectroscopic technique, the l-phase transitions in NH4Cl have been investigated [5±13]. We have also reported some of our Raman studies on NH4Cl [3,14]. As we have given in our earlier study [3], we have correlated the volume changes to the Raman frequencies of some phonon modes in NH4Cl. We have reported our correlations for the NH4Br crystal in an another study [4]. We have also established this relationship between the volume change and the frequencies of the acoustic modes in NH4Cl [15]. Very recently, we have correlated the volume changes to the ultrasonic frequencies of the q[110] mode (®rst order and tricritical) [16], the  mode (®rst order) [17] and the q[100] mode q‰110Š

0022-2860/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0022-286 0(01)00628-7

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(®rst order, tricritical and second order) [18] phase transitions in NH4Cl. Once, the frequencies (Raman, acoustic and ultrasonic) of various modes are correlated to the volume, their variations with temperature or pressure can also be correlated with each other close to the phase transitions. Therefore, the frequency shifts …1=n†…2n=2T†P are correlated to …1=V†…2V=2T†P or the thermal expansivity aP through the isobaric mode GruÈneisen parameter gP : Similarly, the frequency shifts …1=n†…2n=2P†T are correlated to 2…1=V†…2V=2P†T or the isothermal compressibility kT by means of the isothermal mode GruÈneisen parameter gT close to the phase transitions. On the other hand, the thermal expansivity aP can be related to the isothermal compressibility kT and also to the speci®c heat Cp close to phase transitions. In fact, Pippard established a linear variation of Cp with the aP and also a linear variation of aP with the kT in the vicinity of the l phase transitions [19]. He applied his relations to NH4Cl and by following him, the Pippard relations have been applied to various physical systems, as we have reviewed in our previous work [20]. Spectroscopically, the Pippard relations can be modi®ed by obtaining a linear variation of the speci®c heat Cp with the frequency shifts …1=n†…2n=2T†P : Also, a linear variation of the thermal expansivity aP can be obtained with the frequency shifts …1=n†…2n=2P†T close to the phase transitions. We have presented our spectroscopic modi®cations of the Pippard relations and applied them (linear variation of Cp with …1=n†…2n=2T†) to NH4Cl at P ˆ 0 (®rst order phase region) [20]. We have also applied our spectroscopically modi®ed Pippard relation (linear variation of Cp with …1=n†…2n=2T†) to NH4Br [21], and to NH4Cl at P ù 1:6 kbar (tricritical phase region) and at P ù 2:8 kbar (second order phase region) [21,22]. Regarding the other physical systems, we have established our spectroscopic modi®cations of the Pippard relations for ammonia [23] and for NaNO2 [24] as well. In this study we obtain a linear relationship between the speci®c heat Cp and the frequency shifts  modes close …1=n†…2n=2T†P for the q[110] and q‰110Š to the ®rst order phase transition …P ˆ 0† in NH4Cl. We also obtain this linear relationship between Cp and …1=n†…2n=2T†P using the ultrasonic frequencies of the q[110] mode close to the tricritical phase transition

…P ˆ 1:5 kbar† in NH4Cl. From those linear relationships, we calculate the slope dP=dT close to the phase transitions considered for the NH4Cl system. In Section 2 we give an outline of the theory. In Section 3 we give our calculations and results. Discussion and conclusions are given in Sections 4 and 5, respectively. 2. Theory The thermodynamic functions such as the speci®c heat Cp, the thermal expansivity aP and the isothermal compressibility kT can be related to one another by means of the Pippard relations close to the l-phase transitions [19]. Those relations are given by     dP dS V aP 1 T …1† Cp ˆ T dT l dT l     dP 1 dV aP ˆ kT 1 …2† dT l V dT l where …dP=dT†l is the slope at the l point, V and S are the volume and entropy, respectively. The thermal expansivity aP and the isothermal compressibility kT can be related to the frequency shifts by de®ning the mode GruÈneisen parameter as

gˆ2

V dn n dV

…3†

Speci®cally, by de®ning the isobaric mode GruÈneisen parameter as   1 1 2n gP ; 2 …4† aP n 2T P the thermal expansivity, aP ; …1=V†…2V=2T†P ; can be related to the temperature dependence of the frequency at constant pressure. Also, by de®ning the isothermal mode GruÈneisen parameter as   1 1 2n gT ; …5† kT n 2P T the isothermal compressibility, kT ; 2…1=V†  …2V=2P†T ; can be related to the pressure dependence of the frequency at constant temperature. By substituting those de®nitions of the mode GruÈneisen parameters gP (Eq. (4)) and gT (Eq. (5)) into the Pippard relations, Eqs. (1) and (2), respectively, we get the spectroscopic modi®cations of the Pippard relations

H. Yurtseven et al. / Journal of Molecular Structure 598 (2001) 109±116

given by Cp ˆ 2

TV gP

and 1 aP ˆ gT





dP dT

dP dT







l

1 2n n 2T



 P

1T

dS dT



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

l

…6†

…7†

as we have already presented in our previous studies [20±24]. The above relations then relate the speci®c heat Cp to the frequency shifts …1=n†  …2n=2T†P and also the thermal expansivity aP to the frequency shifts …1=n†…2n=2P†T close to the l transition point. In order to establish our spectroscopic modi®cations of the Pippard relations (Eqs. (6) and (7)), we require the expressions for the temperature dependence of the frequency (at constant pressure) and also for the pressure dependence of the frequency (at constant temperature). By means of Eq. (4), the temperature dependence of the frequency can be obtained using the length-change data LP(T )/L1 for a cubic crystal as

nP …T† ˆ DP 1 A…P† 1 n1 exp‰23gP ln…LP …T†=L1 †Š …8† where n1 and L1 are the values of the frequency and the length, respectively, at room temperature …T ˆ 296 K; P ˆ 0†: In this relation the order±disorder contribution DP is zero for the disordered phase …T . Tc † and it is nonzero for the ordered phase including the transition temperature …T # Tc †: This contribution can be taken to depend upon the temperature as

DP ˆ a…DT†2 1 b…DT† 1 c

…9†

where DT ˆ T 2 Tc ; and a, b and c are constants, as also given in our recent studies [16±18]. In Eq. (8) the pressure-dependent term can be in the form of A…P† ˆ a0 1 a1 P 1 a2 P2

…10†

where a0, a1 and a2 are constants. Thus, the temperature dependence of the length-change LP(T )/L1 will determine the frequency as a function temperature at constant pressures, according to Eq. (8). On the other hand, the temperature dependence of the smoothed length-change data below and above Tc can be expressed as LP …T†=L1 ˆ ap DT 1 bp

…11†

111

where ap and bp are constants. Therefore, using Eq. (11) in Eq. (8), the frequency shifts …1=n†  …2n=2T†P can be obtained and our spectroscopically modi®ed Pippard relation (Eq. (6)) can be established. Similarly, the pressure dependence of the frequency at constant temperatures can be obtained. By means of Eq. (5), the pressure dependence of the frequency can be obtained for a cubic crystal as

nT …P† ˆ DT 1 A…T† 1 n1 exp‰23gT ln…LT …P†=L1 †Š …12† where the temperature-dependent term A…T† can be taken in the form A…T† ˆ a 00 1 a 01 …T 2 Tc † 1 a 02 …T 2 Tc †2 a 00 ;

a 01

…13†

a 02

and are constants. In Eq. (12) the Here, order±disorder contribution to the frequency is nonzero …DT ± 0† for P $ Pc and it is zero …DT ˆ 0† for P , Pc : D T can be taken to depend upon the pressure as

DT ˆ a 0 …DP†2 1 b 0 …DP† 1 c 0

…14†

where DP ˆ P 2 Pc with the critical pressure Pc, a 0 ; b 0 and c 0 are constants. Therefore, we can determine the pressure dependence of the frequency by means of that dependence of the length change LT …P†=L1 according to Eq. (12). Similar to Eq. (11), the pressure dependence of the smoothed length-change data can be expressed as LT …P†=L1 ˆ aT DP 1 bT

…15†

So, using the pressure dependence of the lengthchange (Eq. (15)), the frequency can be obtained as a function of pressure at constant temperatures according to Eq. (12). By obtaining the frequency shifts …1=n†…2n=2P†T from Eq. (12), our spectroscopically modi®ed Pippard relation (Eq. (7)) can be established. 3. Calculations and results We established here the ®rst Pippard relation which we modi®ed spectroscopically (Eq. (6)), using the ultrasonic frequency data [25] and the speci®c heat Cp data [26,27] close to the ®rst order …P ˆ 0† and the tricritical …P ˆ 1:5 kbar† phase transitions

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H. Yurtseven et al. / Journal of Molecular Structure 598 (2001) 109±116

Fig. 1. The speci®c heat Cp as a function of the frequency shifts …1=n†…2n=2T†P for the q[110] mode of NH4Cl in the ®rst order phase region …P ˆ 0; Tl ˆ 242:9 K† according to Eq. (6). The Cp data are taken from Ref. [26].

in NH4Cl. In our analysis we ®rst calculated the ultrasonic frequencies for the q[110] mode for the pressures of P ˆ 0 and 1.5 kbar [16] and also for the  mode for P ˆ 0 [17] using the observed ultraq‰110Š sonic data [25], according to Eq. (8). By establishing a linear variation of Cp with the frequency shifts …1=n†…2n=2T†P for each mode, we were able to calculate the values of the slope dP=dT for NH4Cl crystal close to the ®rst order …P ˆ 0† and tricritical …P ˆ 1:5 kbar† phase transitions in this system. 3.1. Analysis of the q[110] mode at P ˆ 0 We analysed here the q[110] mode of NH4Cl for P ˆ 0 to obtain a linear relationship between the speci®c heat Cp and its frequency shifts …1=n†…2n=2T†P according to Eq. (6). For this analysis we used our calculated ultrasonic frequencies for the q[110] mode [16] by means of Eq. (8) where we used the length-change data [1] for NH4Cl. The lengthchange data were smoothed below and above Tc, according to Eq. (11) where we had the values of the coef®cients ap ˆ 21:628 £ 1025 K21 and bp ˆ 0:99556 for DT , 0; ap ˆ 4:975 £ 1025 K21 and bp ˆ 0:99730 for DT . 0: We then calculated the ultrasonic frequencies for the q[110] mode using the length-change data [1] by means of Eq. (8) where we

had n1 ˆ 205:96 kHz and gP ˆ 0:71 for the q[110] mode [16]. We have reported our calculated ultrasonic frequencies for the q[110] mode in our recent work [16], where the values of the parameters a, b and c (Eq. (9)) for the order±disorder contribution DP (Eq. (8)), were given (see Table 3 in Ref. [16]). We took a0 ˆ 0 for A…P† given in Eq. (10) for P ˆ 0…Tc ˆ 242:9 K†: From our calculated frequency values, we extracted the frequency shifts …1=n†…2n=2T†P for the q[110] mode and we plotted the speci®c heat Cp against …1=n†…2n=2T†P according to Eq. (6), as given in Fig. 1. From this linear plot, by taking V ˆ Vl ˆ 34:72 cm3 mol21 and T ˆ Tl ˆ 242:9 K we obtained the slope value of …dP=dT†l ˆ 81:2 bar K21 for NH4Cl. The intercept value we obtained was Tl …dS=dT†l ˆ 18:97 cal: 3.2. Analysis of the q[110] mode at P ˆ 1:5 kbar The ultrasonic frequencies were calculated for the q[110] mode as a function of temperature at 1.5 kbar by using Eq. (8). In this equation the length-change data [1] were used, which were smoothed above and below Tc according to Eq. (11). This gave us the values of the parameters ap ˆ 21:995 £ 1025 K21 and bp ˆ 0:99555 for DT , 0; ap ˆ 4:978 £ 1025 K21 and bp ˆ 0:99730 for DT . 0 in the tricritical phase region …P ˆ 1:5 kbar†: Using our value of

H. Yurtseven et al. / Journal of Molecular Structure 598 (2001) 109±116

113

Fig. 2. The speci®c heat Cp as a function of the frequency shifts …1=n†…2n=2T†P for the q[110] mode of NH4Cl in the tricritical phase region …P ˆ 1:5 kbar; Tc ˆ 254:7 K† according to Eq. (6). The Cp data are taken from Ref. [26].

A…P† ˆ 0:651 kHz and the temperature dependence of DP with the coef®cients a, b and c for P ˆ 1:5 kbar (see Table 3 in Ref. [16]), the ultrasonic frequencies for the q[110] mode were calculated for P ˆ 1:5 kbar; as we have given previously [16]. We then obtained the frequency shifts …1=n†…2n=2T†P from our calculated ultrasonic frequencies for the q[110] mode at P ˆ 1:5 kbar: Fig. 2 gives

our plot of Cp against …1=n†…2n=2T†P for this mode in the tricritical phase region of NH4Cl, according to Eq. (6). From this plot, we obtained the slope value of …dP=dT†c ˆ 104 bar K21 by means of the values of T ˆ Tl ˆ 254:7 K; V ˆ Vl ˆ 34:403 cm3 mol21 and gp ˆ 0:71 as before, which we used in Eq. (6). This plot gave us the intercept value of Tc …dS=dT†c ˆ 73:48 J:

 mode of NH4Cl in the ®rst order phase region Fig. 3. The speci®c heat Cp as a function of the frequency shifts …1=n†…2n=2T†P for the q‰110Š …P ˆ 0; Tl ˆ 242:9 K† according to Eq. (6). The Cp data are taken from Ref. [26].

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H. Yurtseven et al. / Journal of Molecular Structure 598 (2001) 109±116

Table 1 The slope values of dP=dT and the intercept values of T…dS=dT† which we obtained using the ultrasonic frequency shifts for the q[110] and  modes according to Eq. (6) close to the ®rst order …P ˆ 0; Tl ˆ 242:9 K† and the tricritical …P ˆ 1:5 kbar; Tc ˆ 254:7 K† phase q‰110Š transitions in NH4Cl Mode

…dP=dT†l …bar K21 † P ˆ 0

…dP=dT†c …bar K21 † P ˆ 1:5 kbar

Tl …dS=dT†l …cal† P ˆ 0

Tc …dS=dT†c …cal† P ˆ 1:5 kbar

q[110]  q‰110Š

81.2 83.55

104 ±

18.97 16.38

17.5 ±

 mode at P ˆ 0 3.3. Analysis of the q‰110Š We also established our spectroscopically modi®ed Pippard relation (Eq. (6)) using the speci®c heat Cp data [26] and our calculated frequency shifts for the  mode of NH4Cl at P ˆ 0: The ultrasonic q‰110Š frequencies of this mode were calculated using Eq. (8) where we used the length-change data [1]. The length-change data were smoothed according to Eq. (11) where we had ap ˆ 15:024 £ 1025 K21 and bp ˆ 0:99535 for DT , 0; ap ˆ 4:98 £ 1025 K21 and bp ˆ 0:99730 for DT . 0: The ultrasonic frequencies  mode of NH4Cl at were then calculated for the q‰110Š P ˆ 0 using Eq. (8) where we had n1 ˆ 385:80 kHz and gp ˆ 0:96 for this mode, A…P† ˆ 0 and the temperature dependence of DP has been given in our recent work (see Table III in Ref. [17]). We have  reported our calculated frequencies for the q‰110Š mode in our recent work [17]. Using our calculated frequencies, we obtained the frequency shifts …1=n†…2n=2T†P as a function of temperature for the  q‰110Š mode. A plot of Cp is given against …1=n†…2n=2T†P according to Eq. (6) in Fig. 3. This linear plot gave us the slope value of …dP=dT†l ˆ 83:55 bar K21 for NH4Cl using the values of V ˆ Vl ˆ 34:72 cm3 mol21 and Tl ˆ 242:9 K in Eq. (6). We obtained the intercept value as Tl …dS=dT†l ˆ 16:38 cal:

4. Discussion We examined here the validity of our spectroscopic modi®cation of the Pippard relation (Eq. (6)) for NH4Cl by means of a linear plot of the speci®c heat Cp as a function of the frequency shifts …1=n†…2n=2T†P for the q[110] mode at the pressures of 0 kbar (Fig. 1)  mode at and 1.5 kbar (Fig. 2) and also for the q‰110Š 0 kbar (Fig. 3) in this crystal. As shown in those

®gures, a linear variation of the speci®c heat Cp with the frequency shifts …1=n†…2n=2T†P is reasonably good within the temperature regions close to the phase transitions. In our analysis the temperature regions we studied, were 23:1 K , DT , 2:3 K for P ˆ 0 kbar …Tl ˆ 242:9 K† and 21:65 K , DT , 0:2 K for P ˆ 1:5 kbar …Tc ˆ 254:7 K†: Beyond these temperature regions above and below the transition temperatures, a linear relationship between Cp and …1=n†…2n=2T†P was no longer valid and there occur deviations from this linearity. We were then restricted to the temperature regions where relatively limited number of data were used in our analysis. Our slope values for …dP=dT†l (Table 1) can be compared with those given in our previous studies. First of all, our values of 81.2 and 83.55 bar K 21, which we obtained from the ultrasonic frequencies  modes, respectively, at P ˆ for the q[110] and q‰110Š 0 (®rst order phase region), are close to each other. They can be compared with our values of 105.5 bar K 21 [30] and 94.9 bar K 21 [20], which were deduced from our observed Raman frequencies for the n5 TO (174 cm 21) and n2 (1708 cm 21) modes of NH4Cl, respectively. So, our …dP=dT†l values we obtained in this study are smaller than those given above. All of our …dP=dT†l values given here can also be compared with those values of 62.1 and 90.1 bar K 21, which we calculated using our observed Raman frequencies for the n5 TO (134 cm 21) and n5 LO (177 cm 21) modes of NH4Br [21]. When compared with the literature values of 108 bar K 21 [28] and 116.9 bar K 21 [29] for NH4Cl in the ®rst order phase region …P ˆ 0†; our present results are again smaller. In the tricritical region …P ˆ 1:5 kbar† our slope value of 104 bar K 21 which we obtained using the ultrasonic frequencies of the q[110] mode of NH4Cl, is close to our previous values of 109.4 and 114.4 bar K 21 due to the n5 TO (174 cm 21) and n2

H. Yurtseven et al. / Journal of Molecular Structure 598 (2001) 109±116 21

(1708 cm ) Raman modes, respectively, as given in our earlier study [21]. Considering the values of intercept T…dS=dT†; as given in Table 1, our values of nearly 19 and  modes, respec16.4 cal due to the q[110] and q‰110Š tively, are close to our earlier value of 13.8 cal due to the n2 (1708 cm 21) Raman mode [20] and also the literature value of 15.67 cal at zero pressure [29]. Our value of 17.5 cal at P ˆ 1:5 kbar is also close to the values at P ˆ 0: In our analysis we also studied the q[100] mode by calculating its ultrasonic frequencies as a function of temperature using Eq. (8). For this calculation the same method was employed as for the q[110] mode  mode [17] using the observed [16] and for the q‰110Š ultrasonic frequency data for P ˆ 0 and 1.5 kbar [25], which we have reported in our recent work [18]. By extracting the frequency shifts for the q[100] mode, we attempted to establish our spectroscopically modi®ed Pippard relation (Eq. (6)). Although Cp varied almost linearly with the …1=n†…2n=2T†P within the temperature regions considered for both pressures of P ˆ 0 and 1.5 kbar, we were not able to get any meaningful values for the slope dP=dT: This is mainly due to the fact that the q[100] mode exhibits unusual critical behaviour when compared with that of the  modes of NH4Cl. In fact, the ultraq[110] and q‰110Š sonic frequency of the q[100] mode decreases abruptly as the temperature decreases towards the transition temperature, it then increases below the transition temperature, as obtained from the observed frequency data for the phase transitions of the ®rst order, tricritical and the second order in NH4Cl [25]. Because of this unusual frequency behaviour of the q[100] mode, it has negative mode GruÈneisen parameter …gp ˆ 20:7† through which we get negative small value for the slope dP=dT at the pressures of P ˆ 0 and 1.5 kbar. This indicates that the q[100] mode is not appropriate to examine our spectroscopic modi®cation of the Pippard relation (Eq. (6)). We also note here that we were unable to test Eq. (6) using the  mode of NH4Cl in ultrasonic frequencies of the q‰110Š the tricritical phase region …P ˆ 1:5 kbar† since no observed frequency data for this mode were available in this transition region [25]. Finally, by obtaining the ultrasonic frequency shifts  and …1=n†…2n=2P†T for the modes of q‰110Š; q‰110Š q[100] in NH4Cl, our spectroscopically modi®ed

115

Pippard relation (Eq. (7)) can also be examined using the thermal expansivity ap or the length-change LT …P†=L1 data. For this analysis the frequencies can be calculated as a function of pressure (at constant temperatures) by means of Eq. (12) where the length-change data LT …P†=L1 should be used according to Eq. (15). 5. Conclusions We examined here the Pippard relation which we modi®ed spectroscopically by obtaining a linear variation of the speci®c heat Cp with the frequency shifts …1=n†…2n=2T†P close to the ®rst order …P ˆ 0† and the tricritical …P ˆ 1:5 kbar† phase transition in NH4Cl. For this analysis, we used the ultrasonic frequency shifts which we calculated from the observed frequency data for the q[110] mode (P ˆ 0  mode …P ˆ 0† in and 1.5 kbar) and for the q‰110Š NH4Cl. Our plots show that Cp varies linearly with the …1=n†…2n=2T†P close to ®rst order and the tricritical phase transition in NH4Cl. It is indicated here our spectroscopic modi®cations of the Pippard relations are satisfactory for some appropriate phonon modes in systems such as ammonium halides, ammonia and NaNO2. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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