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Spectroscopic modifications of Pippard relations: Tricritical phase transition in NH4Cl
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Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 513–518 www.elsevier.com/locate/jqsrt
Spectroscopic modifications of Pippard relations: Tricritical phase transition in NH4Cl H. Yurtseven, F. Eraslan Department of Physics, Middle East Technical University, 06531 Ankara, Turkey Received 12 October 2005; accepted 6 February 2006
Abstract We study here the Pippard relations modified spectroscopically for the NH4Cl crystal. By relating the specific heat CP to the frequency shift ð1=nÞðqn=qTÞP for the disorder-allowed Raman modes of n7 (93 cm1) and n5 (144 cm1) in NH4Cl close to the tricritical point (P ¼ 1.6 kbar, TC ¼ 257 K), we show that the first Pippard relation is valid for this crystalline system. r 2006 Elsevier Ltd. All rights reserved. Keywords: Pippard relations; Raman modes; NH4Cl
1. Introduction Ammonium chloride exhibits a l-phase transition at T ¼ 241 K at atmospheric pressure. This transition occurs from the disordered b phase to the ferro-ordered d phase, as the temperature is lowered. b phase has a CsCl structure with the O1h symmetry. In this disordered phase NH+ 4 ions are randomly distributed between up and down orientations [1]. At low temperatures there occurs a ferro-ordered d phase 1 where all the NH+ 4 ions orientate parallel to each other. This phase has a CsCl structure with the Td symmetry. The disordered b and the ferro-ordered d phases in NH4Cl have been obtained experimentally in a P–T phase diagram [2]. A phase diagram for NH4Cl [3] has been generalized for ammonium halides [4] and also it has been modified for ammonium and deutero-ammonium halides [5]. Apart from these experimental studies, theoretical phase diagrams have been obtained [6–9], as also reviewed in our previous study [10]. We have calculated a P–T phase diagram for ammonium halides (NH4Cl and NH4Br) [10] and a T–XBr phase diagram for a mixed crystal of NH4Cl1xBrx [11] by the Landau free energy expansion. The l-phase transition in NH4Cl (Tl ¼ 241 K, P ¼ 0), as we have reviewed in our earlier study [12], changes its character when the pressure increases to 1.6 kbar. As was first observed experimentally from the lengthchange measurements in NH4Cl at about 1.6 kbar [13], a discontinuous jump in the volume (first order phase transition at P ¼ 0) changed toward a continuous one (tricritical phase transition). As the pressure increased
Corresponding author.
E-mail address: [email protected] (H. Yurtseven). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.031
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further to about 2.8 kbar, volume exhibited eventually a continuous change with the temperature (second order phase transition) [13]. By means of correlations between the crystal volume and the disorder-induced Raman modes of n7 (93 cm1) and n5 (144 cm1) in NH4Cl, we have also predicted these discontinuous and continuous changes in the vibrational frequencies, as compared to our observed Raman frequencies for the first order, tricritical and the second order phase transitions in this crystalline system [12]. Correlations between the frequency shifts and the crystal volume have made it possible for us to predict the critical behavior of the thermodynamic functions such as the specific heat CP, thermal expansivity aP and the isothermal compressibility kT in the vicinity of the l phase transition in NH4Cl, by means of the Pippard relations. Using our observed Raman frequencies for the internal mode of n2 (1708 cm1) [14] and for the n5 (174 cm1) lattice mode [15] in NH4Cl, we have related the specific heat CP to the frequency shifts 1=nðqn=qTÞP in the vicinity of the l point (T ¼ 242.8 K, P ¼ 0). Also, using our observed frequencies for the disorderinduced Raman modes of n7 (93 cm1) and n5 (144 cm1), we have established the spectroscopic modification of the first Pippard relation (CP as a function of 1=nðqn=qTÞP ) in the vicinity of the first order (Tl ¼ 242.5 K, P ¼ 0) phase transition in NH4Cl [16]. This has been established using the n7 (93 cm1) mode for the second order (TC ¼ 268 K, P ¼ 2.8 kbar) in NH4Cl [16]. In this study we relate the specific heat CP to the frequency shifts 1=nðqn=qTÞP in the vicinity of the tricritical point in NH4Cl, where the two first order phase lines meet with a second order phase line, which leads to an interesting and new phase transition, as we have reviewed in our earlier study [17]. To examine the spectroscopic modification of the first Pippard relation in NH4Cl, we use our observed frequencies for the disorder-induced Raman modes of n7 (93 cm1) and n5 (144 cm1), and the specific heat CP data obtained at 1.5 kbar [18] for this crystal. Below, in Section 2 we present our calculations and results for the spectroscopic modifications of the Pippard relations close to the tricritical phase transition (P ¼ 1:6 kbar, T C ¼ 257 K) in NH4Cl. Section 3 gives our discussion on the calculations and results. Finally, in Section 4 we give our conclusions. 2. Calculations and results In the vicinity of the l point, the Pippard relations [19] can be expressed as C P ¼ VT ðdP=dTÞl aP þ TðdS=dTÞl
(1)
aP ¼ ðdP=dTÞl kT þ ð1=V ÞðdV =dTÞl .
(2)
and
Eq. (1) expresses a linear variation of the specific heat CP with the thermal expansivity aP, whereas Eq. (2) is a linear variation of aP with the isothermal compressibility kT in the vicinity of the l point. Pippard applied his first relation (Eq. (1)) to the NH4Cl system [19]. From Eqs. (1) and (2) the slope (dP/dT)l can be calculated, and also the intercept values of T(dS/dT)l (Eq. (1)) and (1/V) (dV/dT)l (Eq. (2)) can be calculated in the vicinity of the l-transition point. The above relations can be modified spectroscopically by defining the mode Gru¨neisen parameters. From the definitions of the isobaric mode Gru¨neisen parameter gP ¼ ð1=aP Þ 1=nðqn=qTÞP
(3)
and the isothermal mode Gru¨neisen parameter gT ¼ 1=kT 1=nðqn=qPÞT
(4)
the Pippard relations (Eqs. (1) and (2)) can be modified spectroscopically in the vicinity of the l point. By substituting Eq. (3) into Eq. (1) and Eq. (4) into Eq. (2), we get C P ¼ TV =gP ðdP=dTÞl 1=nðqn=qTÞP þ TðdS=dTÞl
(5)
ARTICLE IN PRESS H. Yurtseven, F. Eraslan / Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 513–518
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and aP ¼ 1=gT ðdP=dTÞl 1=nðqn=qPÞT þ 1=V ðdV =dTÞl .
(6)
Eqs. (5) and (6) are the spectroscopic modifications of the Pippard relations in the vicinity of the l point, as we have also presented in our previous study [16]. In this study we analyzed our observed frequency shifts 1=nðqn=qTÞP for the n7 (93 cm1) and n5 (144 cm1) Raman modes of NH4Cl close to its tricritical phase transition (P ¼ 1.6 kbar, TC ¼ 257.17 K). By relating those Raman frequency shifts to the specific heat CP, we established the first Pippard relation in its spectroscopic form (Eq. (5)) close to the tricritical phase transition in NH4Cl. This linear relationship between CP and 1=nðqn=qTÞP is shown in Fig. 1, using the Raman frequency shifts for the n7 (93 cm1) mode in NH4Cl above TC (TC ¼ 257.17 K, P ¼ 1.6 kbar) according to Eq. (5). Figs. 2 and 3 show our plots for CP as a function of 1=nðqn=qTÞP for the n5 (144 cm1) Raman mode, below and above TC, respectively (TC ¼ 257.17 K, P ¼ 1.6 kbar), in NH4Cl, according to Eq. (5). From this linear variation of the specific heat CP with the frequency shifts 1=nðqn=qTÞP (Eq. (5)), we extracted the values of the slope (dP/dT)C and the intercept T(dS/dT)C close to the tricritical phase transition in NH4Cl. Table 1 gives the values obtained for the slope (dP/dT)C and the intercept T(dS/dT)C, using the Raman frequency shifts 1=nðqn=qTÞP for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) close to the tricritical temperature TC in NH4Cl.
74
72 T>TC (dP/dT)C=151 bar/K 70
Cp(J/mol.K)
68
66
64
62
60
58 0
1
2
3 4 5 -1/(∂/∂T)P*10-3(K-1)
6
7
Fig. 1. Specific heat CP as a function of 1=nðqn=qTÞP for the n7 (93 cm1) Raman mode in NH4Cl close to the tricritical phase transition (T4T C ) (P ¼ 1.6 kbar, TC ¼ 257.17 K).
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300
Cp(.J/mol.K)
250
200
150
100
50
0 0
5
10 15 20 -1/(∂/∂T)P*10-4(K-1)
25
30
Fig. 2. Specific heat CP as a function of 1=nðqn=qTÞP for the n5 (144 cm1) Raman mode in NH4Cl close to the tricritical phase transition (ToT C ) (P ¼ 1.6 kbar, TC ¼ 257.17 K).
3. Discussion We established here the first Pippard relation in its spectroscopic form (Eq. (5)) using the Raman frequency shifts for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) in NH4Cl close to the tricritical phase transition (P ¼ 1.6 kbar, TC ¼ 257.17 K). Because of the experimental data which were scattered below TC for the n7 (93 cm1) Raman mode of NH4Cl, we were not able to analyze the frequency shifts of this mode in this region. As shown in Figs. 1–3, a linear variation of the specific heat CP with the frequency shifts 1=nðqn=qTÞP was obtained for the n7 (93 cm1) and n5 (144 cm1) Raman modes close to the tricritical temperature (TC ¼ 257.17 K), according to Eq. (5). The specific heat CP has been related linearly to our observed frequency shifts 1=nðqn=qTÞP for the n7 (93 cm1) and n5 (144 cm1) modes close to the first order (P ¼ 0, Tl ¼ 242.5 K), and close to the second order (P ¼ 2.8 kbar, TC ¼ 268 K) for the n7 (93 cm1) mode in NH4Cl in our earlier study [16]. By means of this linear relationship between CP and 1=nðqn=qTÞP (Eq. (5)), we were able to extract the values of the slope (dP/dT)C and the intercept T(dS/dT)C close to the tricritical phase transition in NH4Cl, as given in Table 1. Our slope values of (dP/dT)C for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) can be compared with our earlier values of dP/dT for the tricritical phase transition in NH4Cl [20]. Previously, we have obtained the values of 109.4 and 114.4 bar/K for the n5 (174 cm1) and n2 (1708 cm1) Raman modes, respectively, close to the tricritical phase transition (P ¼ 1:6 kbar, T C ¼ 257 K) in NH4Cl [20]. Our dP/dT values given in Table 1 can also be compared with the dP/dT values which we have deduced from the n5 (174 cm1) and n2 (1708 cm1) modes in the vicinity of first order l-phase transition and second order phase transition in NH4Cl.
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517
76
74 T>TC 72
(dP/dT)C=129.5 bar/K
Cp (J/(mol.K)
70
68
66
64
62
60
58 255
260 265 -1/(∂/∂T)P*10-5(K-1)
270
Fig. 3. Specific heat CP as a function of 1=nðqn=qTÞP for the n5 (144 cm1) Raman mode in NH4Cl close to the tricritical phase transition (T4T C ) (P ¼ 1.6 kbar, TC ¼ 257.17 K).
Table 1 Values of the slope (dP/dT)C and the intercept T(dS/dT)C which we obtained from the spectroscopic modification of the Pippard relation (Eq. (5)) for the Raman modes indicated close to the tricritical phase transition in NH4Cl (P ¼ 1.6 kbar, TC ¼ 257.17 K) NH4Cl
Raman modes
(dP/dT)C (bar/K)
T(dS/dT)C (J/mol K)
T4T C ToT C T4T C
n7 (93 cm1) n5 (144 cm1) n5 (144 cm1)
151 93.9 129.5
59.7 17.1 307.8
Previously, we have obtained using our observed Raman frequencies for the internal mode of n2 (1708 cm1) in NH4Cl, the value of (dP/dT)l ¼ 94.9 bar/K in the vicinity of the l point at atmospheric pressure (Tl ¼ 242.8 K) [14]. Also using our observed Raman frequencies for the n5 (174 cm1) mode in NH4Cl, we have extracted the slope value of ðdP/dT)l ¼ 105.5 bar/K (P ¼ 0, Tl ¼ 242.8 K) by means of the spectroscopic modifications of the Pippard relations [15]. For the second order (TC ¼ 268 K, P ¼ 2.8 kbar) phase transition in NH4Cl, the values of the slope dP/dT ¼ 58.3 and 145.5 bar/K have been deduced using our observed frequencies of the n5 (174 cm1) and n2 (1708 cm1) modes, respectively, in NH4Cl, as given in our earlier study [20]. When we deduced the values of the slope (dP/dT)C according to Eq. (5), we used the values of the mode Gru¨neisen parameter for the n7 (93 cm1) and n5 (144 cm1) modes in NH4Cl. Those values which we determined, as given in our earlier study [12], were gP ¼ 3.1 for the n7 (93 cm1) and gP ¼ 2.13 for the n5 (144 cm1) modes in NH4Cl. These gP values with the crystal volume V ¼ 34.72 cm3/mol [21] for NH4Cl were used in Eq. (5) to extract the values of the slope (dP/dT)C and the intercept T(dS/dT)C, as given in Table 1.
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Regarding our spectroscopically modified Pippard relations (Eqs. (5) and (6)), the critical behavior of the thermodynamic quantities such as the specific heat CP, thermal expansivity aP and the isothermal compressibility kT can be predicted using the frequency shifts in the vicinity of the phase transitions in NH4Cl. By assuming a constant value of the isobaric mode Gru¨neisen parameter gP, the thermal expansivity aP can be predicted using the frequency shifts 1=nðqn=qTÞP , according to Eq. (3). Also, using the frequency shifts 1=nðqn=qPÞT with the isothermal mode Gru¨neisen parameter gT that remains constant through the phase transitions the isothermal compressibility kT can be predicted according to Eq. (4). Thus, the second Pippard relation in the spectroscopically modified form (Eq. (6)) can be established close to the tricritical phase transition in NH4Cl. By plotting aP predicted from Eq. (3) as a function of the frequency shifts 1=nðqn=qPÞT which can be obtained from the Raman measurements, the slope (dP/dT)C can then be calculated close to the tricritical phase transition in NH4Cl. This can be done by using the Raman frequencies measured as functions of pressure at the tricritical temperature (T C ¼ 257 K) for the n7 (93 cm1) and n5 (144 cm1) modes in NH4Cl, as we obtained 1=nðqn=qTÞP of those modes at the tricritical pressure (P ¼ 1:6 kbar) in this study. Thus, by means of the spectroscopically modified Pippard relations (Eqs. (5) and (6)), we can calculate the values of CP, aP and kT using the frequency shifts 1=nðqn=qTÞP and 1=nðqn=qPÞT close to the tricritical phase transition in NH4Cl. 4. Conclusions We studied here correlation between the specific heat CP and the frequency shifts 1=nðqn=qTÞP close to the tricritical phase transition in NH4Cl (P ¼ 1:6 kbar, T C ¼ 257:17 K). A linear variation of CP with the 1=nðqn=qTÞP was obtained for the Raman modes of n7 (93 cm1) and n5 (144 cm1) in NH4Cl for the phase transition studied here. The values of the slope (dP/dT)C deduced from our linear plots for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) agree with the experimental value and they can be compared with our earlier values for the first order and second order phase transitions in NH4Cl. Our method of analysis given here provides us to predict the thermodynamic quantities such as the specific heat CP, thermal expansivity aP and the isothermal compressibility kT close to the tricritical phase transition in NH4Cl. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Levy HA, Peterson SW. Phys Rev 1952;86:766. Stevenson R. J Chem Phys 1961;34:346. Garland CW, Renard R. J Chem Phys 1966;44:1120. Leung RC, Zahradnik C, Garland CW. Phys Rev B 1979;19:2612. Press W, Eckert J, Cox DE, Rotter C, Kamitakahara W. Phys Rev B 1976;14:1983. Yamada Y, Mori M, Noda Y. J Phys Soc Japan 1972;32:1565. Hu¨ller A. Z Physik 1974;270:343. Vaks VG, Schneider VE. Phys Stat Sol (a) 1976;35:61. Vlasova AA, Tornau EE, Schneider VE. Fiz Tverd Tela (Leningrad) 1978;20:858. Yurtseven H, Salihog˘lu S, Tu¨blek A. Phase Trans 1995;54:151. Tarı O¨, Yurtseven H, Salihog˘lu S. Phase Trans 2000;72:351. Yurtseven H, Sherman WF. Phase Trans 1995;54:1. Weiner BB, Garland CW. J Chem Phys 1972;56:155. Yurtseven H, Sherman WF. J Mol Struct 1994;323:243. Yurtseven H, Sherman WF. J Mol Struct 1984;115:173. Yurtseven H, Yanık A. J Mol Struct 2001;560:161. Salihog˘lu S, Yurtseven H. Phase Trans 1997;60:173. Garland CW, Baloga JD. Phys Rev B 1977;16:331. Pippard AB. The elements of classical thermodynamics. New York: University of Cambridge; 1957. Yurtseven H, Sherman WF. J Mol Struct 1997;435:143. Sakamoto Y. J Sci Hiroshima Univ Ser A 1954;18:95.
Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 513–518 www.elsevier.com/locate/jqsrt
Spectroscopic modifications of Pippard relations: Tricritical phase transition in NH4Cl H. Yurtseven, F. Eraslan Department of Physics, Middle East Technical University, 06531 Ankara, Turkey Received 12 October 2005; accepted 6 February 2006
Abstract We study here the Pippard relations modified spectroscopically for the NH4Cl crystal. By relating the specific heat CP to the frequency shift ð1=nÞðqn=qTÞP for the disorder-allowed Raman modes of n7 (93 cm1) and n5 (144 cm1) in NH4Cl close to the tricritical point (P ¼ 1.6 kbar, TC ¼ 257 K), we show that the first Pippard relation is valid for this crystalline system. r 2006 Elsevier Ltd. All rights reserved. Keywords: Pippard relations; Raman modes; NH4Cl
1. Introduction Ammonium chloride exhibits a l-phase transition at T ¼ 241 K at atmospheric pressure. This transition occurs from the disordered b phase to the ferro-ordered d phase, as the temperature is lowered. b phase has a CsCl structure with the O1h symmetry. In this disordered phase NH+ 4 ions are randomly distributed between up and down orientations [1]. At low temperatures there occurs a ferro-ordered d phase 1 where all the NH+ 4 ions orientate parallel to each other. This phase has a CsCl structure with the Td symmetry. The disordered b and the ferro-ordered d phases in NH4Cl have been obtained experimentally in a P–T phase diagram [2]. A phase diagram for NH4Cl [3] has been generalized for ammonium halides [4] and also it has been modified for ammonium and deutero-ammonium halides [5]. Apart from these experimental studies, theoretical phase diagrams have been obtained [6–9], as also reviewed in our previous study [10]. We have calculated a P–T phase diagram for ammonium halides (NH4Cl and NH4Br) [10] and a T–XBr phase diagram for a mixed crystal of NH4Cl1xBrx [11] by the Landau free energy expansion. The l-phase transition in NH4Cl (Tl ¼ 241 K, P ¼ 0), as we have reviewed in our earlier study [12], changes its character when the pressure increases to 1.6 kbar. As was first observed experimentally from the lengthchange measurements in NH4Cl at about 1.6 kbar [13], a discontinuous jump in the volume (first order phase transition at P ¼ 0) changed toward a continuous one (tricritical phase transition). As the pressure increased
Corresponding author.
E-mail address: [email protected] (H. Yurtseven). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.031
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further to about 2.8 kbar, volume exhibited eventually a continuous change with the temperature (second order phase transition) [13]. By means of correlations between the crystal volume and the disorder-induced Raman modes of n7 (93 cm1) and n5 (144 cm1) in NH4Cl, we have also predicted these discontinuous and continuous changes in the vibrational frequencies, as compared to our observed Raman frequencies for the first order, tricritical and the second order phase transitions in this crystalline system [12]. Correlations between the frequency shifts and the crystal volume have made it possible for us to predict the critical behavior of the thermodynamic functions such as the specific heat CP, thermal expansivity aP and the isothermal compressibility kT in the vicinity of the l phase transition in NH4Cl, by means of the Pippard relations. Using our observed Raman frequencies for the internal mode of n2 (1708 cm1) [14] and for the n5 (174 cm1) lattice mode [15] in NH4Cl, we have related the specific heat CP to the frequency shifts 1=nðqn=qTÞP in the vicinity of the l point (T ¼ 242.8 K, P ¼ 0). Also, using our observed frequencies for the disorderinduced Raman modes of n7 (93 cm1) and n5 (144 cm1), we have established the spectroscopic modification of the first Pippard relation (CP as a function of 1=nðqn=qTÞP ) in the vicinity of the first order (Tl ¼ 242.5 K, P ¼ 0) phase transition in NH4Cl [16]. This has been established using the n7 (93 cm1) mode for the second order (TC ¼ 268 K, P ¼ 2.8 kbar) in NH4Cl [16]. In this study we relate the specific heat CP to the frequency shifts 1=nðqn=qTÞP in the vicinity of the tricritical point in NH4Cl, where the two first order phase lines meet with a second order phase line, which leads to an interesting and new phase transition, as we have reviewed in our earlier study [17]. To examine the spectroscopic modification of the first Pippard relation in NH4Cl, we use our observed frequencies for the disorder-induced Raman modes of n7 (93 cm1) and n5 (144 cm1), and the specific heat CP data obtained at 1.5 kbar [18] for this crystal. Below, in Section 2 we present our calculations and results for the spectroscopic modifications of the Pippard relations close to the tricritical phase transition (P ¼ 1:6 kbar, T C ¼ 257 K) in NH4Cl. Section 3 gives our discussion on the calculations and results. Finally, in Section 4 we give our conclusions. 2. Calculations and results In the vicinity of the l point, the Pippard relations [19] can be expressed as C P ¼ VT ðdP=dTÞl aP þ TðdS=dTÞl
(1)
aP ¼ ðdP=dTÞl kT þ ð1=V ÞðdV =dTÞl .
(2)
and
Eq. (1) expresses a linear variation of the specific heat CP with the thermal expansivity aP, whereas Eq. (2) is a linear variation of aP with the isothermal compressibility kT in the vicinity of the l point. Pippard applied his first relation (Eq. (1)) to the NH4Cl system [19]. From Eqs. (1) and (2) the slope (dP/dT)l can be calculated, and also the intercept values of T(dS/dT)l (Eq. (1)) and (1/V) (dV/dT)l (Eq. (2)) can be calculated in the vicinity of the l-transition point. The above relations can be modified spectroscopically by defining the mode Gru¨neisen parameters. From the definitions of the isobaric mode Gru¨neisen parameter gP ¼ ð1=aP Þ 1=nðqn=qTÞP
(3)
and the isothermal mode Gru¨neisen parameter gT ¼ 1=kT 1=nðqn=qPÞT
(4)
the Pippard relations (Eqs. (1) and (2)) can be modified spectroscopically in the vicinity of the l point. By substituting Eq. (3) into Eq. (1) and Eq. (4) into Eq. (2), we get C P ¼ TV =gP ðdP=dTÞl 1=nðqn=qTÞP þ TðdS=dTÞl
(5)
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and aP ¼ 1=gT ðdP=dTÞl 1=nðqn=qPÞT þ 1=V ðdV =dTÞl .
(6)
Eqs. (5) and (6) are the spectroscopic modifications of the Pippard relations in the vicinity of the l point, as we have also presented in our previous study [16]. In this study we analyzed our observed frequency shifts 1=nðqn=qTÞP for the n7 (93 cm1) and n5 (144 cm1) Raman modes of NH4Cl close to its tricritical phase transition (P ¼ 1.6 kbar, TC ¼ 257.17 K). By relating those Raman frequency shifts to the specific heat CP, we established the first Pippard relation in its spectroscopic form (Eq. (5)) close to the tricritical phase transition in NH4Cl. This linear relationship between CP and 1=nðqn=qTÞP is shown in Fig. 1, using the Raman frequency shifts for the n7 (93 cm1) mode in NH4Cl above TC (TC ¼ 257.17 K, P ¼ 1.6 kbar) according to Eq. (5). Figs. 2 and 3 show our plots for CP as a function of 1=nðqn=qTÞP for the n5 (144 cm1) Raman mode, below and above TC, respectively (TC ¼ 257.17 K, P ¼ 1.6 kbar), in NH4Cl, according to Eq. (5). From this linear variation of the specific heat CP with the frequency shifts 1=nðqn=qTÞP (Eq. (5)), we extracted the values of the slope (dP/dT)C and the intercept T(dS/dT)C close to the tricritical phase transition in NH4Cl. Table 1 gives the values obtained for the slope (dP/dT)C and the intercept T(dS/dT)C, using the Raman frequency shifts 1=nðqn=qTÞP for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) close to the tricritical temperature TC in NH4Cl.
74
72 T>TC (dP/dT)C=151 bar/K 70
Cp(J/mol.K)
68
66
64
62
60
58 0
1
2
3 4 5 -1/(∂/∂T)P*10-3(K-1)
6
7
Fig. 1. Specific heat CP as a function of 1=nðqn=qTÞP for the n7 (93 cm1) Raman mode in NH4Cl close to the tricritical phase transition (T4T C ) (P ¼ 1.6 kbar, TC ¼ 257.17 K).
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300
Cp(.J/mol.K)
250
200
150
100
50
0 0
5
10 15 20 -1/(∂/∂T)P*10-4(K-1)
25
30
Fig. 2. Specific heat CP as a function of 1=nðqn=qTÞP for the n5 (144 cm1) Raman mode in NH4Cl close to the tricritical phase transition (ToT C ) (P ¼ 1.6 kbar, TC ¼ 257.17 K).
3. Discussion We established here the first Pippard relation in its spectroscopic form (Eq. (5)) using the Raman frequency shifts for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) in NH4Cl close to the tricritical phase transition (P ¼ 1.6 kbar, TC ¼ 257.17 K). Because of the experimental data which were scattered below TC for the n7 (93 cm1) Raman mode of NH4Cl, we were not able to analyze the frequency shifts of this mode in this region. As shown in Figs. 1–3, a linear variation of the specific heat CP with the frequency shifts 1=nðqn=qTÞP was obtained for the n7 (93 cm1) and n5 (144 cm1) Raman modes close to the tricritical temperature (TC ¼ 257.17 K), according to Eq. (5). The specific heat CP has been related linearly to our observed frequency shifts 1=nðqn=qTÞP for the n7 (93 cm1) and n5 (144 cm1) modes close to the first order (P ¼ 0, Tl ¼ 242.5 K), and close to the second order (P ¼ 2.8 kbar, TC ¼ 268 K) for the n7 (93 cm1) mode in NH4Cl in our earlier study [16]. By means of this linear relationship between CP and 1=nðqn=qTÞP (Eq. (5)), we were able to extract the values of the slope (dP/dT)C and the intercept T(dS/dT)C close to the tricritical phase transition in NH4Cl, as given in Table 1. Our slope values of (dP/dT)C for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) can be compared with our earlier values of dP/dT for the tricritical phase transition in NH4Cl [20]. Previously, we have obtained the values of 109.4 and 114.4 bar/K for the n5 (174 cm1) and n2 (1708 cm1) Raman modes, respectively, close to the tricritical phase transition (P ¼ 1:6 kbar, T C ¼ 257 K) in NH4Cl [20]. Our dP/dT values given in Table 1 can also be compared with the dP/dT values which we have deduced from the n5 (174 cm1) and n2 (1708 cm1) modes in the vicinity of first order l-phase transition and second order phase transition in NH4Cl.
ARTICLE IN PRESS H. Yurtseven, F. Eraslan / Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 513–518
517
76
74 T>TC 72
(dP/dT)C=129.5 bar/K
Cp (J/(mol.K)
70
68
66
64
62
60
58 255
260 265 -1/(∂/∂T)P*10-5(K-1)
270
Fig. 3. Specific heat CP as a function of 1=nðqn=qTÞP for the n5 (144 cm1) Raman mode in NH4Cl close to the tricritical phase transition (T4T C ) (P ¼ 1.6 kbar, TC ¼ 257.17 K).
Table 1 Values of the slope (dP/dT)C and the intercept T(dS/dT)C which we obtained from the spectroscopic modification of the Pippard relation (Eq. (5)) for the Raman modes indicated close to the tricritical phase transition in NH4Cl (P ¼ 1.6 kbar, TC ¼ 257.17 K) NH4Cl
Raman modes
(dP/dT)C (bar/K)
T(dS/dT)C (J/mol K)
T4T C ToT C T4T C
n7 (93 cm1) n5 (144 cm1) n5 (144 cm1)
151 93.9 129.5
59.7 17.1 307.8
Previously, we have obtained using our observed Raman frequencies for the internal mode of n2 (1708 cm1) in NH4Cl, the value of (dP/dT)l ¼ 94.9 bar/K in the vicinity of the l point at atmospheric pressure (Tl ¼ 242.8 K) [14]. Also using our observed Raman frequencies for the n5 (174 cm1) mode in NH4Cl, we have extracted the slope value of ðdP/dT)l ¼ 105.5 bar/K (P ¼ 0, Tl ¼ 242.8 K) by means of the spectroscopic modifications of the Pippard relations [15]. For the second order (TC ¼ 268 K, P ¼ 2.8 kbar) phase transition in NH4Cl, the values of the slope dP/dT ¼ 58.3 and 145.5 bar/K have been deduced using our observed frequencies of the n5 (174 cm1) and n2 (1708 cm1) modes, respectively, in NH4Cl, as given in our earlier study [20]. When we deduced the values of the slope (dP/dT)C according to Eq. (5), we used the values of the mode Gru¨neisen parameter for the n7 (93 cm1) and n5 (144 cm1) modes in NH4Cl. Those values which we determined, as given in our earlier study [12], were gP ¼ 3.1 for the n7 (93 cm1) and gP ¼ 2.13 for the n5 (144 cm1) modes in NH4Cl. These gP values with the crystal volume V ¼ 34.72 cm3/mol [21] for NH4Cl were used in Eq. (5) to extract the values of the slope (dP/dT)C and the intercept T(dS/dT)C, as given in Table 1.
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Regarding our spectroscopically modified Pippard relations (Eqs. (5) and (6)), the critical behavior of the thermodynamic quantities such as the specific heat CP, thermal expansivity aP and the isothermal compressibility kT can be predicted using the frequency shifts in the vicinity of the phase transitions in NH4Cl. By assuming a constant value of the isobaric mode Gru¨neisen parameter gP, the thermal expansivity aP can be predicted using the frequency shifts 1=nðqn=qTÞP , according to Eq. (3). Also, using the frequency shifts 1=nðqn=qPÞT with the isothermal mode Gru¨neisen parameter gT that remains constant through the phase transitions the isothermal compressibility kT can be predicted according to Eq. (4). Thus, the second Pippard relation in the spectroscopically modified form (Eq. (6)) can be established close to the tricritical phase transition in NH4Cl. By plotting aP predicted from Eq. (3) as a function of the frequency shifts 1=nðqn=qPÞT which can be obtained from the Raman measurements, the slope (dP/dT)C can then be calculated close to the tricritical phase transition in NH4Cl. This can be done by using the Raman frequencies measured as functions of pressure at the tricritical temperature (T C ¼ 257 K) for the n7 (93 cm1) and n5 (144 cm1) modes in NH4Cl, as we obtained 1=nðqn=qTÞP of those modes at the tricritical pressure (P ¼ 1:6 kbar) in this study. Thus, by means of the spectroscopically modified Pippard relations (Eqs. (5) and (6)), we can calculate the values of CP, aP and kT using the frequency shifts 1=nðqn=qTÞP and 1=nðqn=qPÞT close to the tricritical phase transition in NH4Cl. 4. Conclusions We studied here correlation between the specific heat CP and the frequency shifts 1=nðqn=qTÞP close to the tricritical phase transition in NH4Cl (P ¼ 1:6 kbar, T C ¼ 257:17 K). A linear variation of CP with the 1=nðqn=qTÞP was obtained for the Raman modes of n7 (93 cm1) and n5 (144 cm1) in NH4Cl for the phase transition studied here. The values of the slope (dP/dT)C deduced from our linear plots for the n7 (93 cm1) mode (T4T C ) and for the n5 (144 cm1) mode (ToT C and T4T C ) agree with the experimental value and they can be compared with our earlier values for the first order and second order phase transitions in NH4Cl. Our method of analysis given here provides us to predict the thermodynamic quantities such as the specific heat CP, thermal expansivity aP and the isothermal compressibility kT close to the tricritical phase transition in NH4Cl. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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