Calculation of the specific heat close to the first-order and second-order phase transitions in ammonium halides

Journal of Molecular Structure 553 (2000) 267–279 www.elsevier.nl/locate/molstruc

Calculation of the specific heat close to the first-order and second-order phase transitions in ammonium halides H. Yurtseven*, A. Yanik Department of Physics, Istanbul Technical University, Maslak, Istanbul, Turkey Received 6 January 2000; revised 25 April 2000; accepted 25 April 2000

Abstract This study presents our calculation of the specific heat CVI under the prediction of an Ising model. For this calculation we used our observed Raman frequencies for the n 5LO (177 cm ⫺1) mode of NH4Br, the n 5TO (144 cm ⫺1) and n 7TA (93 cm ⫺1) modes of NH4Cl close to the first-order phase transitions …P ˆ 0†: We also calculate the specific heat CVI using our observed Raman frequencies for the n 7TA (93 cm ⫺1) mode of NH4Cl close to the second-order phase transition …P ˆ 2:8 kbar† in this crystalline system. Our results show that our CVI values agree with the observed CP data from the literature. The method of calculating the specific heat using the frequency data is discussed. 䉷 2000 Elsevier Science B.V. All rights reserved. Keywords: Raman; Phase transitions; Ammonium halides

1. Introduction Ammonium halides have been studied extensively in the literature since they exhibit the l-phase transitions. The l-phase transition takes place between the disordered b phase and the ferro-ordered d phase in NH4Cl at T 艑 242 K; whereas in NH4Br it is between the disordered b phase and the antiferro-ordered g phase at T 艑 234 K: Ignoring the NH⫹ 4 orientations, the disordered b phase (NH4Cl and NH4Br) has the CsCl type crystal structure with the O1h symmetry and in this phase the two possible orientations of the NH⫹ 4 ions are distributed randomly [1]. As the temperature decreases, there occurs the antiferro-ordered g phase (NH4Br) which has a tetragonal structure with the D4h * Corresponding author. Tel.: ⫹90-212-285-3209; fax: ⫹90-212286-6376.

symmetry. In this phase the NH⫹ 4 ions are orientated antiparallel in the a–b plane and they are parallel along the c-axis. As the temperature decreases further, the NH⫹ 4 ions orientated antiparallel in the a–b plane, all become parallel along the c axis, which form the ferro-ordered d phase with the CsCl structure that has the Td1 symmetry. The phase diagrams regarding the b, g and d phases in the ammonium halides have been given in the literature. The P–T phase diagrams of the ammonium halides have been obtained experimentally by Stevenson [2], which have been given in the modified form [3]. A master P–T phase diagram including the multicritical points in the ammonium and deuteroammonium halides has also been reported [4]. Apart from the P–T phase diagrams, the observed x–T phase diagrams for the NH4Cl1⫺xBrx system have been reported in the literature [5–8]. Some theoretical

0022-2860/00/$ - see front matter 䉷 2000 Elsevier Science B.V. All rights reserved. PII: S0022-286 0(00)00570-6

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models predicting the phase diagram in the ammonium halides have also been given in the literature [9–12]. Various physical properties of the ammonium halides have been studied both experimentally and theoretically. We have reviewed some of those studies regarding the l-phase transitions in NH4Cl [13] and in NH4Br [14]. The specific heat can be calculated using the predictions of some models. In our earlier study [15] we have developed the Ising model superimposed on the Einstein and/or the Debye model. Using our observed Raman frequencies for the n 5TO (174 cm ⫺1) mode that corresponds to the Einstein frequency of nE ˆ 179 cm⫺1 [16], we have calculated the specific heat CVI due to the nearest neighbour spin interactions in NH4Cl [17]. In our recent study [18] we have used our observed frequencies for the disorder-induced Raman modes of n 7TA (93 cm ⫺1) and n 5TO (144 cm ⫺1) of NH4Cl to calculate the specific heat CVI for the first-order, tricritical and second-order phase transitions in this crystal system under the Ising model studied. We have applied our model to the NH4Br to calculate the specific heat close to the lphase transition in this crystal [19]. For this calculation we have used our observed frequencies of the n 5LO (177 cm ⫺1) Raman mode of NH4Br [20]. In this study we reanalysed our frequency data for the n 5LO (177 cm ⫺1) Raman mode of NH4Br and for the n 5TO (144 cm ⫺1) and n 7TA (93 cm ⫺1) Raman modes of NH4Cl to calculate the specific heat CVI in these crystalline systems. In our previous study [18] in order to calculate the specific heat, we have first calculated the Raman frequencies of the n 7TA (93 cm ⫺1) and n 5TO (144 cm ⫺1) modes of NH4Cl for the first-order, tricritical and second-order phase transitions, using the length-change measurements [21]. Here, in this study our analysis was based on our observed frequency data, which were used to deduce the critical exponent for calculating the specific heat. This exponent value was then used to predict the specific heat CVI in the ammonium halides. Thus, this method of analysis given here directly uses the measured frequencies to calculate the specific heat, instead of using the length-change data for predicting the frequencies [18]. This work thus provides a direct way of calculating the specific heat by means of the critical exponent deduced from the frequency

measurements. Our calculated specific heat CVI for the NH4Cl and NH4Br are then compared with the experimentally measured CP data for NH4Br [22] and NH4Cl [23]. In Section 2 we introduce an Ising model from which we calculate the specific heat. In this section our calculations and results are given. In Section 3 we discuss our results. Conclusions are given in Section 4. 2. Calculations and results We present here the specific heat expression derived from the free energy of an Ising model described by the Hamiltonian X …1† HI ˆ ⫺J…V† s i s j ij

In this expression s i and s j are the spin variables for the nearest-neighbour interactions, and J(V) is the interaction parameter that depends upon the volume. The free energy of an Ising system can be obtained from the partition function F I …J…V†; T† ˆ ⫺kT lnZ

…2†

where the partition function is defined as X ⫺H =kT e I Zˆ

…3†

i; j

By defining the logarithm of the partition function F ˆ ln Z; we then obtain the free energy as F I …J…V†; T† ˆ ⫺kT F…J…V†=kT†

…4†

By taking the second derivative of the free energy FI, the specific heat at constant volume will be obtained as C VI ˆ k…J=kT†2 F 00

…5†

00

where F corresponds to the second derivative of F…J…V†=kT†: This is the analytical expression for the specific heat CVI for an Ising model. We can analyse the critical behaviour of the specific heat CVI by employing the power-law formula for the free energy F I ˆ A 00 ⫹ A 0 兩e兩

2⫺a

…6†

where a is the critical exponent. In Eq. (6) A 00 ˆ JA0 and A 0 ˆ JA; where A0 and A are dimensionless

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constants. The reduced temperature is e ˆ …T ⫺ TC †=TC where TC is the critical temperature. By Eq. (4), we have

Fˆ⫺

J 2⫺a ‰A ⫹ A兩e兩 Š kT 0

…7†

By taking the first derivative of F with respect to J=kT; we get

F 0 ˆ 2F=2…J=kT† ˆ ⫺‰A 0 ⫹ A兩e兩2⫺a Š ⫹

AT 1⫺a …2 ⫺ a†兩e兩 TC

…8†

The second derivative of F with respect to J=kT gives kT 2 …2 ⫺ a† JTC   TA …1 ⫺ a†兩e兩⫺a  …A ⫺ 1†兩e兩1⫺a ⫹ TC

F 00 ˆ ⫺

…9†

We can now use Eq. (5) to predict the critical behaviour of the specific heat CVI which gives us CVI ˆ ⫺

JAT …1 ⫺ a†…2 ⫺ a†兩e兩⫺a TC2

…10†

where a is now the critical exponent for the specific heat. In Eq. (10) we neglect the weakly divergent 兩e兩1⫺a term in comparison with the 兩e兩⫺a term near TC, that appears in Eq. (9) which is used in Eq. (4). Thus, the divergence behaviour of the specific heat CVI can be described by Eq. (10) given above. For this calculation of the specific heat CVI, we use the critical behaviour of the frequency shifts. Close to the phase transitions, frequency can be described by a power-law formula 1⫺a

ln…n=nc † ˆ B兩e兩

…11†

Here n c is the critical frequency and B is the amplitude. a describes the critical exponent and e is the reduced temperature, as before. From Eq. (11) the frequency shifts can be expressed as   1 2n B…1 ⫺ a† ⫺a ˆ 兩e兩 …12† TC n 2T P We assume here that the frequency shifts have similar critical behaviour as the specific heat since both quantities are described by the same critical exponent a, according to Eqs. (10) and (12). This is

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particularly valid according to the spectroscopically modified Pippard relations which we have developed in our earlier study [24]. Thus, by means of the powerlaw relations (Eqs. (10) and (12)), CVI and ‰…1=n†…2n=2T†P Š can be plotted against the reduced temperature e to estimate the critical exponent a. Alternatively, by using the frequency data, the exponent a can be estimated through Eq. (11), which in turn leads one to calculate the specific heat CVI according to Eq. (10) or vice versa. As the frequencies are measurable to high accuracy, the specific heat data can then be obtained accurately within the temperature interval close to the phase transitions. In this study we first extracted the values of the critical exponent a according to Eq. (11) using our observed Raman frequencies for the lattice modes of n 5LO (177 cm ⫺1) of NH4Br and, n 5TO (144 cm ⫺1) and n 7TA (93 cm ⫺1) modes of NH4Cl close to the first-order phase transitions …P ˆ 0† in these crystals. We also extracted the value of the critical exponent a, using our observed Raman frequencies for the n 7TA (93 cm ⫺1) mode of NH4Cl close to the second-order phase transition …P ˆ 2:8 kbar† in this crystal. We then used the values of the critical exponent a to calculate the specific heat CVI, according to Eq. (10) for NH4Br …P ˆ 0† and NH4Cl …P ˆ 0 and P ˆ 2:8 kbar† crystals. For this calculation of the specific heat CVI through the critical exponent a, we first determined the critical frequency n C given in Eq. (11) by plotting our observed frequencies as a function of temperature. Our observed frequencies were dependent upon the temperature non-linearly close to phase transitions. As our experimental data were scattered in the transition region, we smoothed our data within the temperature interval over which our observed frequencies varied linearly with the temperature This linear plot of our observed frequencies versus temperature then gave us the critical frequency n C at the transition temperature TC. This determination of the critical frequency was appropriate in our analysis because of the variation of its location on the transition line, as one approaches the transition point from the sides (above and below TC). 2.1. Analysis of the n 5LO (177 cm ⫺1) Raman mode of NH4 Br(P ˆ 0) Our observed Raman frequencies for the n 5LO

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Fig. 1. Our observed frequencies (smoothed) for the n 5LO (177 cm ⫺1) Raman mode of NH4Br as a function of the reduced temperature e ˆ …T ⫺ Tl †=Tl in the log–log scale close to the l-phase transition in this crystal …Tl ˆ 234:3 K; P ˆ 0†:

(177 cm ⫺1) mode of NH4Br, were used to obtain the critical exponent a for the frequency shifts according to Eq. (11). We then used this exponent value for the specific heat, by means of Eq. (10) …Tl ˆ 234:3 K; P ˆ 0†: For this analysis we obtained the critical frequency as nl ˆ 170:06 cm⫺1 by means of a linear plot of n versus T. We plot in the log–log scale ln(n /n l ) against e , as given in Fig. 1. This plot gave us the critical exponent value of a ˆ 0:08; within the temperature interval of 227:2 K ⬍ T ⬍ 240 K: By knowing the critical exponent value, we were able to predict the specific heat CVI as a function of

temperature by determining the JA value above and below Tl , according to the power-law formula (Eq. (10)). In our previous study [19] we have calculated the specific heat CVI using the value of a ˆ 0:06 which we have extracted from the analysis of the Raman frequencies for the n 5LO (177 cm ⫺1) mode of NH4Br [20]. Here, in this study for the analysis of the n 5LO (177 cm ⫺1) mode, we took the temperature interval of 227 K ⬍ T ⬍ 240 K to extract the critical exponent a and we then calculated the specific heat CVI. We used the observed value of CP ˆ 86:5 J K⫺1 mol⫺1 [22] at T ˆ 210 K and our value

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271

Table 1 Our calculated values of the critical exponent a using our observed Raman frequencies of the lattice modes indicated for the first-order phase transitions in NH4Br and NH4Cl …P ˆ 0†; and for the second-order phase transition in NH4Cl …P ˆ 2:8 kbar†: Critical frequencies n c, critical temperatures TC and the temperature intervals are given here. Our values of JA and the observed CP data for NH4Br and NH4Cl are also given here Crystal

Raman Mode ⫺1

P (kbar)

T (K)

CP (J mol ⫺1 K ⫺1)

⫺ JA (J mol ⫺1)

TC (K)

a

n c (cm ⫺1)

Temperature interval

210.0 234.4 241.4 251.9 241.4 251.9 254.9 277.0

86.50 115.00 131.40 77.70 131.40 77.68 104.79 81.25

10550.19 7898.05 9739.77 7307.28 9138.60 6941.51 12936.90 9125.98

234.3

0.08

170.06

227.2 K ⬍ T ⬍ 240 K

242.5

0.13

142.99

237.6 K ⬍ T ⬍ 242.5 K

242.5

0.15

93.60

237.6 K ⬍ T ⬍ 242.5 K

268

0.09

95.47

257.1 K ⬍ T ⬍ 275.9 K

NH4Br

n 5 (177 cm )

0

NH4Cl

n 5 (144 cm ⫺1)

0

NH4Cl

n 7 (93 cm ⫺1)

0

NH4Cl

n 7 (93 cm ⫺1)

2.8

of a ˆ 0:08 with Tl ˆ 234:3 K in Eq. (10), which gave us JA ˆ ⫺10 550:19 J mol ⫺1. By holding this JA value constant, we calculated the specific heat CVI as a function of temperature below Tl . Similarly, by using the observed value of CP ˆ 115:6 J K⫺1 mol⫺1 [22] taken at T ˆ 234:4 K and our exponent value of a ˆ 0:08; we determined the value of JA ˆ ⫺7898:05 J mol⫺1 according to Eq. (10). By taking this JA value as a constant, we then calculated the CVI values as a function of temperature above Tl . For calculating the CVI values, the observed CP data, and our values of JA and a, which we calculated using the Raman frequencies of the n 5LO (177 cm ⫺1) mode (Eq. (11)), are given in Table 1. We plot our calculated CVI values as a function of temperature in Fig. 2. The observed CP values [22] are also plotted in this figure. ⫺1

2.2. Analysis of the n 5TO (144 cm ) Raman mode of NH4Cl (P ˆ 0) We analysed the n 5TO (144 cm ⫺1) Raman mode of NH4Cl by extracting the critical exponent for the specific heat using the frequencies near the l-phase transition …Tl ˆ 242:5 K†: We first obtained the critical frequency from a plot of our observed frequencies for this mode as a function of temperature, which gave us nl ˆ 142:99 cm⫺1 : We then plotted ln[ln(n / n l )] against lne according to Eq. (11). This plot is given in Fig. 3. From this plot, we got the critical exponent value of a ˆ 0:13: This exponent value was used to evaluate the specific heat CVI by means of Eq. (10) where the JA values were obtained above

and below Tl . Using the observed value of CP ˆ 131:4 J K⫺1 mol⫺1 [23] at T ˆ 241:4 K below Tl , we obtained JA ˆ ⫺9739:77 J mol ⫺1. By using this value, we evaluated the CVI values below Tl through Eq. (10). Similarly, above Tl , we used the observed CP value [23] of 77.7 J K ⫺1 mol ⫺1 at T ˆ 251:9 K; which gave us the value of JA ˆ ⫺7307:28 J mol⫺1 ; according to Eq. (10). With this constant value of JA and the exponent value of a ˆ 0:13; we calculated the specific heat CVI above Tl by Eq. (10). Our JA values which we calculated using the frequencies of the n 5TO (144 cm ⫺1) Raman mode of NH4Cl …P ˆ 0†; are given with the observed CP values [23] in Table 1. We plot our calculated CVI values as a function of temperature in Fig. 4. The observed CP data [23] are given in this plot for comparison. 2.3. Analysis of the n 7TA (93 cm ⫺1) Raman mode of NH4Cl (P ˆ 0) We used our observed Raman frequencies for the n 7TA (93 cm ⫺1) mode of NH4Cl to extract the critical exponent a for the specific heat …P ˆ 0†: By plotting our observed frequencies for this mode as a function of temperature, we got the value of the critical frequency as nl ˆ 93:60 cm⫺1 at Tl ˆ 242:5 K: We then plotted ln‰ln…n=nl †Š as a function of lne using the Raman frequencies of the n 7TA (93 cm ⫺1) mode for P ˆ 0; as given in Fig. 5, according to Eq. (11). This plot gave us the critical exponent value of a ˆ 0:15 for the frequency shifts (1/n )(2n /2T) and also for the specific heat. Using this exponent value, we were able to predict the specific heat CVI values above

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Fig. 2. Our calculated CVI values as a function of temperature for NH4Br due to the n 5LO (177 cm ⫺1) Raman mode …Tl ˆ 234:3 K; P ˆ 0†: The observed CP data plotted here, are taken from the work of Lushington and Garland [22].

and below Tl . For this prediction we used the observed value of CP ˆ 131:4 J K⫺1 mol⫺1 [23] at T ˆ 241:4 K below Tl , from which we got JA ˆ ⫺9138:60 J mol ⫺1. By taking this JA value and the exponent a value, we then predicted the CVI values below Tl using Eq. (10). Similar analysis was performed for the above Tl where we used the measured value of CP ˆ 77:68 J K⫺1 mol⫺1 [23] at T ˆ 251:9 K to obtain JA

value that was ⫺6941.51 J mol ⫺1. JA values which we calculated for the n 7TA (93 cm ⫺1) Raman mode of NH4Cl …P ˆ 0†; are tabulated with the observed CP values [23] in Table 1. With this constant JA value, we used a ˆ 0:15 in Eq. (10) to obtain the CVI values. Our calculated CVI values are plotted as a function of temperature in Fig. 6. The observed CP data [23] are also plotted in this figure …P ˆ 0†:

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273

Fig. 3. Our observed frequencies (smoothed) for the n 5TO (144 cm ⫺1) Raman mode of NH4Cl as a function of the reduced temperature e ˆ …T ⫺ Tl †=Tl in the log–log scale close to the l-phase transition in this crystal …Tl ˆ 242:5 K; P ˆ 0†:

2.4. Analysis of the n 7TA (93 cm ⫺1) Raman mode of NH4Cl (P ˆ 2.8 kbar) Our analysis for the n 7TA (93 cm ⫺1) Raman mode of NH4Cl was also performed for the second-order phase region …P ˆ 2:8 kbar† in this crystalline system. Our observed frequencies for this mode were plotted as a function of temperature, where we got the value of nC ˆ 95:47 cm⫺1 for the critical frequency at the critical temperature TC ˆ 268 K: We then plotted ln‰ln…n=nc †Š against lne , which gave us the exponent

value of a ˆ 0:09; according to Eq. (11). This plot is given in Fig. 7. By knowing the exponent value, we calculated the specific heat CVI. Using the measured value of CP ˆ 104:79 J K ⫺1 mol⫺1 [23] at T ˆ 254:9 K for T ⬍ Tc for the pressure of P ˆ 2:8 kbar; we obtained the JA value as –12936.9 J mol ⫺1, according to Eq. (10). By using this JA value and also the exponent value in Eq. (10) we calculated the CVI values below TC. Similarly, we predicted the CVI values above TC. By means of the observed value of CP ˆ

274

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Fig. 4. Our calculated CVI values as a function of temperature for NH4Cl due to the n 5TO (144 cm ⫺1) Raman mode …Tl ˆ 242:5 K; P ˆ 0†: The observed CP data plotted here, are taken from the work of Garland and Baloga [23].

81:25 J K ⫺1 mol⫺1 [23] at T ˆ 277 K for T ⬎ TC for the pressure of P ˆ 2:8 kbar; we got the value of JA ˆ ⫺9125:98 J mol⫺1 according to Eq. (10). Our JA values calculated using the Raman frequencies of the n 7TA (93 cm ⫺1) mode of NH4Cl …P ˆ 2:8 kbar†; are tabulated with the observed CP values [23] in Table 1. By inserting this value of JA and also the exponent value into Eq. (10), we were then able to calculate the specific heat CVI due to the n 7TA (93 cm ⫺1) Raman mode of NH4Cl for the secondorder phase transition …P ˆ 2:8 kbar†: Our calculated

CVI values are plotted as a function of temperature for P ˆ 2:8 kbar in Fig. 8. The experimental CP data [23] are also plotted in this figure.

3. Discussion We calculated here the specific heat CVI as a function of temperature using our observed Raman frequency data. As seen from Fig. 2, our calculated CVI values agree well with the observed CP data below

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275

Fig. 5. Our observed frequencies (smoothed) for the n 7TA (93 cm ⫺1) Raman mode of NH4Cl as a function of the reduced temperature e ˆ …T ⫺ Tl †=Tl in the log–log scale close to the l-phase transition in this crystal …Tl ˆ 242:5 K; P ˆ 0†:

Tl . Above Tl this agreement is still good up to 245 K. Above this temperature there is a discrepancy between our calculated CVI and the observed CP values. This critical behaviour of CVI is almost the same as that obtained in our previous calculation for CVI due to the n 5LO (177 cm ⫺1) Raman mode of NH4Br, where we have had a ˆ 0:06 [19]. Here, we reanalysed the n 5LO (177 cm ⫺1) mode of NH4Br by smoothing our observed Raman frequencies for this mode, as we explained in Section 2 since our frequency data were scattered. So that those values of the Raman frequencies were used to determine the critical exponent a for the specific heat CVI.

For NH4Cl we used our observed Raman frequencies for the n 5TO (144 cm ⫺1) and n 7TA (93 cm ⫺1) modes to obtain the values of the critical exponent a. As we did for the n 5LO (177 cm ⫺1) Raman mode of NH4Br, we also smoothed our observed frequencies for the n 5TO (144 cm ⫺1) mode …P ˆ 0† and for the n 7TA (93 cm ⫺1) mode …P ˆ 0 and P ˆ 2:8 kbar† because of the fact that our frequency data were scattered for these modes as well. We then used those smoothed values of our frequencies and we determined the values of the critical exponent a for the specific heat CVI. The values of a ˆ 0:13 and 0.15 for the n 5TO (144 cm ⫺1) and for the n 7TA

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Fig. 6. Our calculated CVI values as a function of temperature for NH4Cl due to the n 7TA (93 cm ⫺1) Raman mode …Tl ˆ 242:5 K; P ˆ 0†: The observed CP data plotted here, are taken from the work of Garland and Baloga [23].

(93 cm ⫺1) modes …P ˆ 0†; respectively, are close to each other, which can describe the first-order phase transition in NH4Cl. This is expected because both modes are the disorder-induced Raman modes and they show similar critical behaviour. For the pressure of P ˆ 2:8 kbar; our exponent value is a ˆ 0:09 which can describe the second-order phase transition in NH4Cl. These exponent values seem to be reasonable to describe the critical behaviour of the specific heat CVI regarding the observed CP, as given in Figs. 4, 6 and 8. From

these figures, we see that our calculated CVI values are in very good agreement with the observed CP data below Tl (Figs. 4 and 6) close to the first-order phase transition …P ˆ 0† in NH4Cl, whereas close to the second-order phase transition in NH4Cl …P ˆ 2:8 kbar† agreement is also good between our calculated CVI values and the observed CP above TC (Fig. 8). For the first-order phase transition in NH4Cl …P ˆ 0†; as shown in Figs. 4 and 6, our calculated CVI values do not follow the trend of the observed CP above ⬃243 K up to ⬃248 K. For the second-order phase transition

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277

Fig. 7. Our observed frequencies (smoothed) for the n 7TA (93 cm ⫺1) Raman mode of NH4Cl as a function of the reduced temperature e ˆ …T ⫺ Tc †=Tc in the log–log scale close to the second-order phase transition in this crystal …Tc ˆ 268 K; P ˆ 2:8 kbar†:

in NH4Cl …P ˆ 2:8 kbar† there is a discrepancy between our calculated CVI values and the observed CP data below TC, where they match at around T ˆ 255 K; as shown in Fig. 8. We can make a comment about this discrepancy that occurred above Tl (Figs. 4 and 6) and below TC (Fig. 8). As we showed in these figures, we compared our calculated CVI values with the observed CP data from the literature. This comparison between CVI and CP can be debated as Pippard [25] pointed out that CP is significantly different from CV and that in ammonium chloride an analysis of the data leads no evidence of any significant change in CV as Tl is

approached. This has been shown experimentally in NH4Cl that CP changes enormously compared to CV which is weakly divergent, because of the measurements of Sakamoto [26]. However, under the Ising model superimposed on the Einstein and/or the Debye model, which we have developed the specific heat CVI shows the divergence behaviour whereas CPI remains finite but it exhibits an anomalous behaviour near Tl [15]. Thus the comparison of our calculated CVI with the observed CP data is appropriate in the sense that our calculated CVI and the experimentally observed CP both change abruptly near Tl . On the other hand, as the specific heat CPI calculated from

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Fig. 8. Our calculated CVI values as a function of temperature for NH4Cl due to the n 7TA (93 cm ⫺1) Raman mode …Tc ˆ 268 K; P ˆ 2:8 kbar†: The observed CP data plotted here, are taken from the work of Garland and Baloga [23].

CVI according to our model [15] shows anomalous behaviour near Tl as stated above, the CPI can be compared with the experimental CV which can be determined through the thermodynamic relation CP ⫺ CV ˆ VT a2P =kT by means of the measurements of the specific heat CP, the thermal expansivity a P and the isothermal

compressibility k T, as has been obtained for NH4Cl by Sakamoto [26]. Another comment for this discrepancy would be due to the method that we used here for calculating the specific heat CVI. We assumed that the main contribution to the specific heat is due to the vibrational mode considered, whose frequencies were used for calculating CVI. In this sense, we calculate the specific heat of the crystalline system as a whole, by using the frequencies of the vibrational mode studied.

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4. Conclusions We calculated here the specific heat CVI using our observed Raman frequencies for the NH4Br …P ˆ 0† and NH4Cl …P ˆ 0 and 2.8 kbar) crystals under the prediction of the Ising model. Our calculated values of CVI are generally in good agreement with the observed CP data from the literature. This indicates that the specific heat for the ammonium halides can be calculated for the first-order and second-order phase transitions using the frequency data by means of our method employed here. Our method can be applied to some other systems to calculate the specific heat and their observed behaviour can then be explained by means of this calculation. Acknowledgements We are grateful to Dr S. Salihoglu for his criticisms on the model studied here. We are also indebted to the referee for his critical reading and valuable comments on the manuscript. References [1] H.A. Levy, S.W. Peterson, Phys. Rev. 86 (1952) 766. [2] R. Stevenson, J. Chem. Phys. 34 (1961) 346. [3] W. Press, J. Eckert, D.E. Cox, C. Rotter, W. Kamitakahara, Phys. Rev. B 14 (1976) 1983.

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