Amonyum bileşiklerinde faz geçişi yakınında özgül ısı hesaplanması ve pippard bağıntılarının uygulanması
Amonyum bileşiklerinde faz geçişi yakınında özgül ısı hesaplanması ve pippard bağıntılarının uygulanması
Dosyalar
Tarih
2000
Yazarlar
Yanık, Aşkın
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Özet
Bu çalışmada, değişik basınçlardaki NELıBr, NH4CI, NH4 A1F4 ve NÜ4Br kristalleri için Ising modeline göre spin etkileşimlerinden kaynaklanan özgül ısı değerleri hesaplanmıştır. Ayrıca, NH4AIF4 ve NÜ4Br için Einstein modeline göre Cve ve NÜ4Br için Debye modeline göre Cvd değerleri de hesaplanmıştır. NE^Br' ün sıfir basınçta LOm(177 cm"1), Vs(134 cm"1) modlan, NH4CI' ün sıfir basınçta TAm(93 cm"1), TOm(144 cm"1) modlan ve P=2.8 kbar basınçta TAm(93 cm"1) modunun Raman frekanslarının analizi yapılarak, NtUBr ve NH4CI için özgül ısı değerleri hesaplanmıştır. NELjBr ve NH4CI' ün bu Raman modlannın frekansları kullanılarak, bu kristallerde X faz geçişi civarında dv I ÖT frekans kaymasının kritik davranışını tanımlayan kritik üs a değeri saptandı. Bir katı cismi oluşturan atomların herbirine eşlik olunan N sayıda özdeş harmonik salınıcılann, aynı v frekansında titreşim yapmaları durumunda Einstein modeline göre sabit hacimde özgül ısı Cvl örgü titreşimleri bölgesinde; Cve = 3 Nnk (Oe/T)2 / (eQvfT-lf (1) şeklinde tanımlanır. Burada 9e Jive/Tc Einstein sıcaklığı, ve Einstein frekansı, N örgü sayısı ve n ise birim hücredeki molekül sayısıdır. Planck değişmezi h=6.626 x 10"34 J.sn ve Boltzmann değişmezi k=1.3807 x 10"23 JK"1' dir. Debye modeline göre atomlar tek bir frekansla titreşmezler, bir frekans aralığında tanımlıdırlar. Bu frekans aralığında maksimum frekans değeri Vd Debye frekansıdır. 6D=hvD/k ise Debye sıcaklığı olarak bilinir. Debye modeline göre sabit hacimde özgül ısı Cvd aşağıdaki gibi verilir: j eDiT Cm=9Nnk( - )3 JjfV l{ex -\)-dx (2) &d o Burada x=0e/T' dir. Modifiye Pippard bağıntılarının doğruluğu kabul edilmek üzere kritik üs a, özgül ısı, ısısal genleşme ve eşsıcaklıklı sıkıştınlabilirlik gibi bazı termodinamik büyüklüklerin kritik davranışlarını tanımlar. Bu faz geçişi boyunca, Grüneisen parametresinin sabit kaldığını varsayar. Ising modeline göre özgül ısı Cvı' nın kritik davranışı aşağıdaki gibi verilir; Cvı = - J2A. (T/Tc2) (l-ö) (2-a) | e I* (3) Burada Tc kritik sıcaklık, e = (T-Tc)/Tc indirgenmiş sıcaklık ve a ise özgül ısı için kritik üstür. Etkileşim parametresi J, en yakın komşu atomlar arası etkileşimi tanımlar. Bu çalışmada, MLtBr' ün sıfir basınçta LOM(177 cm"1), v5(134 cm"1) modlan, NH4CI' ün sıfir basınçta TAM(93 cm"1), TOm(144 cm'1) modlan ve P=2.8 kbar basınçta TAm(93 cm"1) modunun Raman frekanslarının kritik davranışları, aşağıdaki kuvvet yasasına göre analiz edildi. ln(v/vc) = Bİe|1" (4) Burada vc kritik sıcaklıktaki kritik frekans ve B genliktir. Eş.(4) ile Eş.(3)' ü ilişkilendirirken, dv I ÖT frekans kaymasının, Tc geçiş sıcaklığı yakınında Cvı ile aynı davranışı gösterdiği varsayılır. Bölüm 3.1' de, sıfir basınçta NHıBr' ün LOm(177 cm"1) Raman modu için Raman frekanslarının kritik davranışı, kritik sıcaklığın altında ve üzerinde analiz edildi. LOm(177 cm"1) Raman modunun farklı sıcaklıklara karşı gelen frekans değerleri literatürden alındı. Eş. (4) kullanılarak kritik üs "a" değeri hesaplandı. ln[ln(v/vx)]' nın ln|e|' a çizilen grafiğinde eğim " \-a "' yi verir. Kritik frekans vx=170.06 cm"1 alınarak kritik üs 0.08 olarak bulundu. Kritik sıcaklık 1\ =234.3 K olmak üzere Cp = 86.534 J K"1 mol"1 deneysel özgül ısı değeri Cvı kabul edilerek, T1\ için J2A değeri xı J2 A = -8357.02 J mol"1 olarak hesaplandı. a=0.06 olarak alınarak Cvı değerleri, T<1\ ve T>1\ için J2A değerleri değişmez tutularak hesaplandı. Bölüm 3.3' de sıfir basınçta NH4CF ün TOm(144 cm"1) Raman modu için Raman frekanslarının kritik davranışı, kritik sıcaklığın altında ve üzerinde analiz edildi. Bu mod için Raman frekansları literatürden alındı. Eş.(4)' de kritik frekans vr=142.99 cm"1 olarak alınarak kritik üs o=0.13 bulundu. Kritik sıcaklık 1\= 242.5 K olmak üzere, TTX için J2A değerleri değişmez tutularak Cvı değerleri hesaplandı. Bölüm 3.4' de sıfir basınçta NH4CI' ün TAm(93 cm"1) Raman modu için Raman frekanslarının kritik davranışının analizi yapıldı. Söz konusu frekans değerleri literatürden alındı. Eş (4) kullanılarak kritik üs "a" değeri Vx=93.60 cm"1 kritik frekansı için hesaplanarak 0.15 olarak bulundu. Kritik sıcaklık Tr= 242.5 K olmak üzere deneysel özgül ısı değeri CP = 131.42 J K"1 mol"1, T1\ için Cvı kabul edilip J2A = -6,941.51 J mol"1 bulundu. Kritik üs a=0.15 olarak alınarak Cvı değerleri, J2A değerleri değişmez tutularak kritik sıcaklığın altında ve üzerinde hesaplandı. Bölüm 3.5' de 2.8 kbar basınçta NH4CI' ün TAM(93 cm"1) Raman modu frekanslarının kritik davranışı, kritik sıcaklığın altında ve üzerinde analiz edildi. Söz konusu frekans değerleri literatürden elde edildi. lnpn(v/vx)]' nın İn j e [ ' a olan grafiğinde eğim "1-a"' yi verdiği için, Eş (4) kullanılarak kritik üs "a" hesaplandı. Kritik frekans Va=95.47 cm"1 alınarak a=0.09 olarak hesaplandı. P=2.8 kbar basınçta kritik sıcaklık 1\= 268 K olarak deneysel özgül ısı değeri CP = 104.79 J K"1 mol"1 T
In this thesis we have calculated the specific heat Cvi due to the spin interactions in an Ising model for the NH4B1*, NH4CI, NH4AIF4 and ND^Br crystals under various pressures. Furthermore specific heat Cve for NH4AIF4 and ND4Br under the prediction of Einstein model and CTO using the prediction of Debye model for ND4Br were also calculated. Here we calculated the specific heat Cvi for NILBr and NH4CI by analysing the observed Raman frequencies of the LOm(177 cm'1), v5(134 cm"1) modes for NKtBr at zero pressure and the Raman frequencies of the TAM(93 cm"1), TOm(144 cm"1) modes for NH4CI at zero pressure and the Raman frequencies of the TAm(93 cm"1) mode at P=2.8 kbar in NH4CI. Using the frequencies of these Raman modes of NILiBr and NH4CI we deduced the critical exponent a which describes the critical behaviour of the frequency shifts dv I dl near the X phase transitions in these crystals. According to the prediction of an Einstein model, all of the atoms in a crystal system vibrate with the same frequency v. In the lattice vibration zone with constant volume the specific heat can be expressed as: CvE=3NrJc(9E/T)2/(eeE/T-l)2 (1) Here 0e =hvE/k is the Einstein temperature, Ve is the Einstein frequency, N is the lattice number and n is the molecule number in unit cell. Planck constant is h=6.626 x 10"34 J.sn and Boltzmann constant is k=1.3807 x 10"23 JK"1. Additionally, as for Debye model, atoms do not vibrate with a single frequency. Their frequencies are defined in a frequency interval. The maximum frequency value of this interval is equal to the Debye frequency vD. GD=hvD/k is known as the Debye temperature. According to the Debye model, the specific heat Cvd at constant volume can be expressed as: T 0D'T Cm=9Nnk(-f fx*ex/(e*-l)-dx W &d 0 where x = 9e/T. xv On the basis of the validity of the modified Pippard relations, the critical exponent a can describe the critical behaviour of thermodynamic quantities such as the specific heat, thermal expansivity and isothermal compressibility. This assumes that a single mode Grüneisen parameter remains constant right through the phase transitions. The critical behavior of the specific heat Cvi can be expressed as Cvi = - J2A. (T/Tc2) (1 -a) (2-a) | e | " (3) using the prediction of an Ising model Here Tc is the critical temperature, e = (T-Tc)/Tc is the reduced temperature and a is the critical exponent for the specific heat. The interaction parameter J describes the nearest neighbour spin interactions in an Ising model Here we employ Eq.(3) to calculate the specific heat Cvi as a function of temperature for the NELtBr at zero pressure and the NH4CI system at zero and P= 2.8 kbar. We analysed the critical behaviour of the Raman frequencies of the LOm(177 cm1), v5(134 cm"1) modes of NELiBr at zero pressure and the Raman frequencies of the TAm(93 cm"1), TOm(144 cm"1) modes at zero pressure and the Raman frequencies of the TAM(93 cm"1) mode for NH4CI at P=2.8 kbar, according to the power law formula: ln(v/vc) = B|e|1"a (4) In Eq.(4) vc represents the critical frequency at T = Tc and B is the amplitude. In comparison Eq.(4) with Eq.(3), we assume that the frequency shifts dv I &I have the same critical behaviour as Cvi near the transition temperature Tc. In Section 3.1 we analysed the critical behaviour of the Raman frequencies for the LOM(177 cm"1) Raman mode of NELtBr at P=0 for T, and T>Tx. The frequencies against various temperatures for the LOm(177 cm"1) Raman mode of NELtBr at P=0 were obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/vx)) vs ln| 8 1 plot gave us the value of" \-a ". Taking the critical frequency as Vx=170.06 cm"1, the critical exponent was found to be 0.08. Taking the critical temperature Tx =234.3 K, the experimental specific heat value Cp = 86.534 J K"1 mol"1 was accepted to be Cvi for TTx by taking the J2A values as constants. The critical behaviour of frequencies for the v5 (134 cm"1) Raman mode of NHtBr at P=0 for TTx was analysed in Section 3.2. The frequencies against various temperatures for the Vs (134 cm"1) Raman mode of NHtBr at P=0 were obtained from the literature. The slope of ln(ln(vM.)) vs ln|s| plot gave us the value of "1-a". Taking the critical frequency as v*. = 138.52 cm"1, the critical exponent was XVI found to be 0.06. Taking the critical temperature T*. =234.3 K, the experimental specific heat value CP = 1 16.98 J K"1 mol"1 was accepted to be Cvi for TTx. by taking the J2A values as constants. In Section 3.3 we analysed the critical behaviour of frequencies for the TOm(144 cm"1) Raman mode of NH4CI at P=0 for TTx. The frequencies against various temperatures for the TOm(144 cm'1) Raman mode of NH4CI at P=0 were obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/vx)) vs ln| 8 1 plot gave us the value of" \-a ". Taking the critical frequency as v?=142.99 cm"1, the critical exponent was found to be 0.13. Taking the critical temperature T^= 242.5 K, the experimental specific heat value CP = 131.4202 J K"1 mol"1 was accepted to be Cvi for T. and T>Tx by taking the J2A values as constants. In Section 3.4 we analysed the critical behaviour of frequencies for the TAm(93 cm"1) Raman mode of NH4CI at P=0 for TTx. The frequencies against various temperatures for TAM(93 cm"1) Raman mode of NH4CI at P=0 were obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/vx)) vs İn | e | plot gave us the value of" \-a ". Taking the critical frequency as Vr=93.60 cm"1, the critical exponent was found to be 0.15. Taking the critical temperature T^= 242.5 K, the experimental specific heat value Cp = 131.42 J K"1 mol"1 was accepted to be Cvi for T, and T>Tx, by taking the J2 A values as constants. In Section 3.5 we analysed the critical behaviour of frequencies for the TAm(93 cm"1) Raman mode of NH4CI at P=2.8 kbar for TT\. The frequencies against various temperatures for TAm(93 cm"1) Raman mode of NH4CI at P=2.8 kbar was obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/v^)) vs ln| s | plot gave us the value of "1-a". Taking the critical frequency as Vx=95.47 cm"1 the critical exponent was found to be 0.09. Taking the critical temperature Tx= 268 K, experimental specific heat value Cp = 104.79 J K'1 mol"1 was accepted to be Cvi for TT>. by taking the J A values as constants. As a result we obtained that the calculated Cyi values agree with the Cp data measured experimentally. This indicates that the observed behaviour of NH»Br and NH4CI can be described adequately by calculating the specific heat from the spectroscopic data for NHjBr and NH4CI using the method employed here. Using the Eq.(l) we have calculated the specific heat Cve in Section 3.6.1 for NH4AIF4. Here the Einstein frequency is taken to be Ve=280 cm"1 which is equvalent to the Einstein temperature of 0e - 400 K. The specific heat Cvi due to spin interactions was calculated in Section 3.6.2 for NH4AIF4. Taking the critical temperature Tc=157 K, the experimental specific heat value CP=86.94 J K"1 mol"1 at T=148.3 K was accepted to be Cvi for T, and 0=0.01 for T>Tx. we calculated the Cvi values which are the Ising contribution to Cv near the Appoint for T<[\ and T>Tx. by taking the J2A values as constants. As a result, we obtained that the calculated Cvi values agree with the Cp data measured experimentally. We have calculated the specific heat Cve in Section 3.7.1 for ND4Br using the Eq.(l). Here the Einstein temperatures are taken to be 9e=225 K and 9e=345 K and the values of N.n are 2 x 6.02 x 1023. Additionally, the specific heat Cvd values were calculated according to Eq.(2) by taking the Debye temperature 9d as 141 K. Taking into consideration that according to the Ising model two different phase transitions were in question between 17 and 300 K for NTXjBr, the critical behaviour of specific heat was analysed between 140 K and 195 K and between 200 K and 300 K separately in Section 3.7.2. For the region between 140 K and 195 K, the critical temperature was taken to be Tc=166.7 K. The experimental specific heat value Cp=78.58 J K"1 mol"1 measured at T=165 K was accepted to be Cvi for TTx we calculated the Cvi values which are the Ising contribution to Cv near the X-point for TTa. by taking the J2A values as constants. The critical temperature was taken to be Tc=214.86 K for the region between 200 K and 300 K. By accepting the experimental specific heat value of Cp=92.132 J K"1 mol"1 as Cvi for T
In this thesis we have calculated the specific heat Cvi due to the spin interactions in an Ising model for the NH4B1*, NH4CI, NH4AIF4 and ND^Br crystals under various pressures. Furthermore specific heat Cve for NH4AIF4 and ND4Br under the prediction of Einstein model and CTO using the prediction of Debye model for ND4Br were also calculated. Here we calculated the specific heat Cvi for NILBr and NH4CI by analysing the observed Raman frequencies of the LOm(177 cm'1), v5(134 cm"1) modes for NKtBr at zero pressure and the Raman frequencies of the TAM(93 cm"1), TOm(144 cm"1) modes for NH4CI at zero pressure and the Raman frequencies of the TAm(93 cm"1) mode at P=2.8 kbar in NH4CI. Using the frequencies of these Raman modes of NILiBr and NH4CI we deduced the critical exponent a which describes the critical behaviour of the frequency shifts dv I dl near the X phase transitions in these crystals. According to the prediction of an Einstein model, all of the atoms in a crystal system vibrate with the same frequency v. In the lattice vibration zone with constant volume the specific heat can be expressed as: CvE=3NrJc(9E/T)2/(eeE/T-l)2 (1) Here 0e =hvE/k is the Einstein temperature, Ve is the Einstein frequency, N is the lattice number and n is the molecule number in unit cell. Planck constant is h=6.626 x 10"34 J.sn and Boltzmann constant is k=1.3807 x 10"23 JK"1. Additionally, as for Debye model, atoms do not vibrate with a single frequency. Their frequencies are defined in a frequency interval. The maximum frequency value of this interval is equal to the Debye frequency vD. GD=hvD/k is known as the Debye temperature. According to the Debye model, the specific heat Cvd at constant volume can be expressed as: T 0D'T Cm=9Nnk(-f fx*ex/(e*-l)-dx W &d 0 where x = 9e/T. xv On the basis of the validity of the modified Pippard relations, the critical exponent a can describe the critical behaviour of thermodynamic quantities such as the specific heat, thermal expansivity and isothermal compressibility. This assumes that a single mode Grüneisen parameter remains constant right through the phase transitions. The critical behavior of the specific heat Cvi can be expressed as Cvi = - J2A. (T/Tc2) (1 -a) (2-a) | e | " (3) using the prediction of an Ising model Here Tc is the critical temperature, e = (T-Tc)/Tc is the reduced temperature and a is the critical exponent for the specific heat. The interaction parameter J describes the nearest neighbour spin interactions in an Ising model Here we employ Eq.(3) to calculate the specific heat Cvi as a function of temperature for the NELtBr at zero pressure and the NH4CI system at zero and P= 2.8 kbar. We analysed the critical behaviour of the Raman frequencies of the LOm(177 cm1), v5(134 cm"1) modes of NELiBr at zero pressure and the Raman frequencies of the TAm(93 cm"1), TOm(144 cm"1) modes at zero pressure and the Raman frequencies of the TAM(93 cm"1) mode for NH4CI at P=2.8 kbar, according to the power law formula: ln(v/vc) = B|e|1"a (4) In Eq.(4) vc represents the critical frequency at T = Tc and B is the amplitude. In comparison Eq.(4) with Eq.(3), we assume that the frequency shifts dv I &I have the same critical behaviour as Cvi near the transition temperature Tc. In Section 3.1 we analysed the critical behaviour of the Raman frequencies for the LOM(177 cm"1) Raman mode of NELtBr at P=0 for T, and T>Tx. The frequencies against various temperatures for the LOm(177 cm"1) Raman mode of NELtBr at P=0 were obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/vx)) vs ln| 8 1 plot gave us the value of" \-a ". Taking the critical frequency as Vx=170.06 cm"1, the critical exponent was found to be 0.08. Taking the critical temperature Tx =234.3 K, the experimental specific heat value Cp = 86.534 J K"1 mol"1 was accepted to be Cvi for TTx by taking the J2A values as constants. The critical behaviour of frequencies for the v5 (134 cm"1) Raman mode of NHtBr at P=0 for TTx was analysed in Section 3.2. The frequencies against various temperatures for the Vs (134 cm"1) Raman mode of NHtBr at P=0 were obtained from the literature. The slope of ln(ln(vM.)) vs ln|s| plot gave us the value of "1-a". Taking the critical frequency as v*. = 138.52 cm"1, the critical exponent was XVI found to be 0.06. Taking the critical temperature T*. =234.3 K, the experimental specific heat value CP = 1 16.98 J K"1 mol"1 was accepted to be Cvi for TTx. by taking the J2A values as constants. In Section 3.3 we analysed the critical behaviour of frequencies for the TOm(144 cm"1) Raman mode of NH4CI at P=0 for TTx. The frequencies against various temperatures for the TOm(144 cm'1) Raman mode of NH4CI at P=0 were obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/vx)) vs ln| 8 1 plot gave us the value of" \-a ". Taking the critical frequency as v?=142.99 cm"1, the critical exponent was found to be 0.13. Taking the critical temperature T^= 242.5 K, the experimental specific heat value CP = 131.4202 J K"1 mol"1 was accepted to be Cvi for T. and T>Tx by taking the J2A values as constants. In Section 3.4 we analysed the critical behaviour of frequencies for the TAm(93 cm"1) Raman mode of NH4CI at P=0 for TTx. The frequencies against various temperatures for TAM(93 cm"1) Raman mode of NH4CI at P=0 were obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/vx)) vs İn | e | plot gave us the value of" \-a ". Taking the critical frequency as Vr=93.60 cm"1, the critical exponent was found to be 0.15. Taking the critical temperature T^= 242.5 K, the experimental specific heat value Cp = 131.42 J K"1 mol"1 was accepted to be Cvi for T, and T>Tx, by taking the J2 A values as constants. In Section 3.5 we analysed the critical behaviour of frequencies for the TAm(93 cm"1) Raman mode of NH4CI at P=2.8 kbar for TT\. The frequencies against various temperatures for TAm(93 cm"1) Raman mode of NH4CI at P=2.8 kbar was obtained from the literature. Using Eq(4) the critical exponent "a" was calculated since the slope of ln(ln(v/v^)) vs ln| s | plot gave us the value of "1-a". Taking the critical frequency as Vx=95.47 cm"1 the critical exponent was found to be 0.09. Taking the critical temperature Tx= 268 K, experimental specific heat value Cp = 104.79 J K'1 mol"1 was accepted to be Cvi for TT>. by taking the J A values as constants. As a result we obtained that the calculated Cyi values agree with the Cp data measured experimentally. This indicates that the observed behaviour of NH»Br and NH4CI can be described adequately by calculating the specific heat from the spectroscopic data for NHjBr and NH4CI using the method employed here. Using the Eq.(l) we have calculated the specific heat Cve in Section 3.6.1 for NH4AIF4. Here the Einstein frequency is taken to be Ve=280 cm"1 which is equvalent to the Einstein temperature of 0e - 400 K. The specific heat Cvi due to spin interactions was calculated in Section 3.6.2 for NH4AIF4. Taking the critical temperature Tc=157 K, the experimental specific heat value CP=86.94 J K"1 mol"1 at T=148.3 K was accepted to be Cvi for T, and 0=0.01 for T>Tx. we calculated the Cvi values which are the Ising contribution to Cv near the Appoint for T<[\ and T>Tx. by taking the J2A values as constants. As a result, we obtained that the calculated Cvi values agree with the Cp data measured experimentally. We have calculated the specific heat Cve in Section 3.7.1 for ND4Br using the Eq.(l). Here the Einstein temperatures are taken to be 9e=225 K and 9e=345 K and the values of N.n are 2 x 6.02 x 1023. Additionally, the specific heat Cvd values were calculated according to Eq.(2) by taking the Debye temperature 9d as 141 K. Taking into consideration that according to the Ising model two different phase transitions were in question between 17 and 300 K for NTXjBr, the critical behaviour of specific heat was analysed between 140 K and 195 K and between 200 K and 300 K separately in Section 3.7.2. For the region between 140 K and 195 K, the critical temperature was taken to be Tc=166.7 K. The experimental specific heat value Cp=78.58 J K"1 mol"1 measured at T=165 K was accepted to be Cvi for TTx we calculated the Cvi values which are the Ising contribution to Cv near the X-point for TTa. by taking the J2A values as constants. The critical temperature was taken to be Tc=214.86 K for the region between 200 K and 300 K. By accepting the experimental specific heat value of Cp=92.132 J K"1 mol"1 as Cvi for T
Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2000
Anahtar kelimeler
Amonyum bileşikleri,
Ising modeli,
Isı değeri,
Kristaller,
Ammonium compounds,
Ising model,
Heat value,
Crystals