Pippard Relations Research Articles

Linear variations of the thermodynamic quantities in the liquid phase of carbon tetrachloride close to the melting point

We verify here a linear variation of the specific heat C p with the thermal expansivity α p and also a linear variation of α p with the isothermal compressibility κ T for the liquid phase of carbon tetrachloride close to the melting point. Linear relations (Pippard relations) are verified by analyzing the experimental data for the thermal expansivity α p at various pressures in the liquid phase of CCl 4 close to the melting point. Our linear plots of C p against α p and also α p against κ T give the values of the slope d P m/d T as ~ 8.5MPa/K and 9.5MPa/K, respectively, which can be compared with the experimental value obtained from a P– T phase diagram close to the melting point in carbon tetrachloride.

Pippard relations applied to the λ-phase transition in NH 4Br

We establish here a linear variation of the specific heat C P with the frequency shifts (1/ ν)(∂ ν/∂ T) P close to the λ-phase transition in NH 4Br ( T λ = 234 K). This linear relationship is based on our spectroscopically modified Pippard relations. For this verification of our relations, we use our observed Raman frequencies of the lattice mode of ν 7 (56 cm −1) and the internal mode of ν 2 (1684 cm −1) in the NH 4Br crystalline system. Our study given here indicates that the thermodynamic data can be obtained from the measured frequencies.

λ-PHASE TRANSITIONS IN ICE IH NEAR THE MELTING POINT

This study gives our predictions for the thermal expansivity α p , isothermal compressibility κ T , and specific heat Cp near the melting point for ice Ih. We calculate those thermodynamic quantities as functions of temperature and pressure near the melting point using the experimental data for the thermal expansivity as a function of pressure from the literature. The Pippard relations are validated by plotting Cp against α p and also α p against κ T linearly close to the melting point in ice Ih. We deduce the values of the slope ∂P m /∂T from those linear plots, which are very close to the experimental value obtained from the P-T phase diagram of ice Ih along the melting curve.

A linear variation of the thermal expansivity with frequency shifts for the translational mode in ammonia solid II near the melting point

We examine here our spectroscopic modification of the Pippard relation which is a linear variation of the thermal expansivity α p with the frequency shifts (1/ ν)(∂ ν/∂ p) T close to the melting point in ammonia solid II. For this we use our calculated frequencies for the Raman mode of ν (51 cm −1) in ammonia solid II for the pressures of 3.65, 5.02 and 6.57 kbars. We establish this linearity between α p and (1/ ν)(∂ ν/∂ p) T for the pressures studied in ammonia solid II close to the melting point. The observed behaviour of ammonia solid II is explained in terms of our spectroscopic relation given here.

Pippard Relations for the Solid Phase of Carbon Tetrachloride Close to the Melting Point

Pippard Relations for the Solid Phase of Carbon Tetrachloride Close to the Melting Point

Specific heat and frequency shifts for the translational modes in ammonia solid I close to phase transition

The specific heat C p is correlated to the frequency shifts (1/ ν)(∂ ν/∂ T) p in ammonia solid I close to the melting point. Our calculated Raman frequencies for the translational modes of ν T (100 cm −1) and ν T (130 cm −1), are used for this correlation for the pressures of 0, 1.93 and 3.07 kbar in this crystalline system. We obtain that the specific heat varied linearly with the frequency shifts for those pressures studied and values of the slope d P m/d T were extracted. The observed behaviour of the ammonia solid I near the melting point is explained on the basis of our spectroscopic modification of the first Pippard relation.

Temperature and pressure dependence of the mode Grüneisen parameters close to the melting point in hexagonal ice

We reexamine the Pippard relations in this study by relating the specific heat C P to the Raman frequency shifts 1 ν ∂ ν ∂ T P and the thermal expansivity α p to the 1 ν ∂ ν ∂ P T , when the mode Grüneisen parameter depends on the temperature and pressure close to the melting point in hexagonal ice. From linear relations between them, the values of the slope d P d T are deduced in this crystal. Our slope values calculated here do not change significantly compared to those obtained when the mode Grüneisen parameter is taken as a constant close to the melting point in hexagonal ice.

Spectroscopic modifications of Pippard relations: Tricritical phase transition in NH 4Cl

We study here the Pippard relations modified spectroscopically for the NH 4Cl crystal. By relating the specific heat C P to the frequency shift ( 1 / ν ) ( ∂ ν / ∂ T ) P for the disorder-allowed Raman modes of ν 7 (93 cm −1) and ν 5 (144 cm −1) in NH 4Cl close to the tricritical point ( P=1.6 kbar, T C=257 K), we show that the first Pippard relation is valid for this crystalline system.

Temperature and pressure dependence of the Raman frequency shifts near the melting point in ice I

This study examines the validity of the spectroscopic modification of the Pippard relations for the hexagonal ice (ice I) close to the melting point. A linear variation of the specific heat C P with the frequency shifts 1 ν ∂ ν ∂ T P is obtained for ice I. This linearity is also obtained between thermal expansivity α P and the frequency shifts 1 ν ∂ ν ∂ P T close to the melting point in this crystal.

Critical behaviour of Raman frequency shifts for translational modes near the melting point of ammonia solid I based on Pippard relations

A correlation between thermal expansivity α p and frequency shifts (1/ν) (∂ν/∂P) T was established for the ammonia solid I near its melting point. For this correlation, calculated frequencies for the translational modes of ν T (100 cm−1) and ν T (130 cm−1) were used to acquire the Raman frequency shifts at pressures of 0, 1.93 and 3.07 kbar in ammonia solid I. In this crystalline system, values of the slope dPm /dT were deduced for the fixed pressures from linear plots of α p vs. (1/ν) (∂ν/∂P) T . Slope values at zero pressure are in very good agreement with those obtained experimentally. It is also shown that the spectroscopic modification of the Pippard relations applies satisfactorily to ammonia solid I.

Specific heat related to the Raman frequency shifts for the rotatory mode in ammonia solid I close to the melting point

This study gives an evidence for the validity of our spectroscopic modification of the Pippard relation as applied to ammonia solid I close to the melting point. We use our calculated frequencies for the rotatory lattice (librational) mode in ammonia solid I for the pressures of 0, 1.93 and 3.07 kbars. We obtain that the specific heat C p varies linearly with the frequency shifts 1 / ν ( ∂ ν / ∂ T ) p for this Raman mode at those pressures studied in this crystalline system. Our values of the slope d P m / d T that we deduced from the linear plots, are not in satisfactory agreement with the experimental values in ammonia solid I near the melting point.

Spectroscopic modification of the Pippard relation applied for the translational mode in ammonia solid II near the melting point

This study gives our calculation for the frequency shifts 1 v ∂ v ∂ T P and the specific heat C P near the melting point in the ammonia solid II. We establish a linear relationship between C P and 1 v ∂ v ∂ T P using the Raman frequencies of the translational mode which we calculated in this system. This leads to the validity of the spectroscopic modification of the first Pippard relation in the ammonia solid II near the melting point. From this linear variation of C P with the 1 v ∂ v ∂ T P we deduce the slope values of d P m d T near the melting point for the fixed pressures of 3.65, 5.02 and 6.57 kbar in the ammonia solid II. They are compared with the experimental d P m d T values for this system.

Spectroscopic modifications of the Pippard relations for NaNO 2 in the sinusoidal antiferroelectric phase

We establish in this study a linear variation of the specific heat C p with the frequency shifts (1/ ν)(∂ ν/∂ T) in the sinusoidal antiferroelectric phase of NaNO 2, according to the first Pippard relation which we have modified spectroscopically. For this linear relationship we use the Brillouin frequencies of acoustic phonon modes for the q[010], q[001] and q[100] modes of NaNO 2 in the sinusoidal antiferroelectric phase. We then calculate d T/d P values due to those modes studied for NaNO 2, which agree with the literature values.

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

This study examines our spectroscopic modifications of the Pippard relations using the ultrasonic frequencies of the q[110] and q[11̄0] modes for the first order phase transition in NH4Cl. We also examine our relations using the ultrasonic frequencies of the q[110] mode for the tricritical phase transition in this crystal. From our plots of the specific heat Cp as a function of the frequency shifts (1/ν)(∂ν/∂T)P, we obtain that our spectroscopically modified Pippard relation considered here is satisfied for those phonon modes studied close to the first order and tricritical phase transitions in NH4Cl.

ANALYSIS OF THE FREQUENCY SHIFTS CLOSE TO THE SECOND ORDER PHASE TRANSITION IN NH4Cl

In this study we analyzed observed Raman data for the frequencies of the disorder-induced modes of ν7 (93 cm−1) and ν5 (144 cm−1) in the second order phase region of NH4Cl (P = 2.8 kbar). By means of a power-law formula describing the critical behaviour of the frequency shifts for those phonon modes studied, we extract the critical exponent value of a = 0.25, which can also describe the critical behaviour of the specific heat C p close to the second order phase transition in NH4Cl. Within this content, our spectroscopically modified Pippard relations are introduced here.

Application of the spectroscopic modifications of Pippard relations to NaNO2 in the ferroelectric phase

This study examines a linear variation of the specific heat CP with the frequency shifts 1/ν(∂ν/∂T) for the Brillouin frequencies of the L-mode [010], [001] and [100] in the ferroelectric phase of NaNO2 according to our spectroscopically modified Pippard relation. We obtain this linear relationship for those modes studied and calculate dTC/dP in the ferroelectric phase of NaNO2. Our calculated values of dTC/dP for the [001] and [100] modes are in good agreement with the values given in the literature.

Spectroscopic modifications of Pippard relations: first-order and second-order phase transitions in NH4Cl

In this study we introduce the spectroscopic modifications of Pippard relations and apply them to the disorder-induced Raman modes of NH4Cl in the first-order (P=0) and second-order (2.8kbar) phase regions in this crystalline system. We obtain linear variations of the specific heat CP with our observed frequency shifts [(1/ν)(∂ν/∂T)P] of those Raman modes studied for the first-order and second-order phase transitions in NH4Cl. This will be discussed in detail.

APPLICATION OF THE PIPPARD RELATIONS TO AMMONIA NEAR THE MELTING POINT

This study gives our application of the Pippard relations to the ammonia near its melting point. The Pippard relations between the specific heat C P , the thermal expansivity α P and the isothermal compressibility κ T , have been established for the ammonia near its melting point. We have obtained that the Pippard relations are valid for the ammonia. We give here an evidence that on the basis of the experimental data the critical transition in this system near its melting point can be considered as the λ-type phase transition.

Spectroscopic modifications of Pippard relations for ammonium halides

Spectroscopic modifications of the Pippard relations have been established for ammonium halides near their λ-phase transitions. A linear relationship has been obtained between the specific heat and the frequency shifts using our observed Raman frequencies of the ν 5TO(134 cm −1) and ν 5LO(177 cm −1) lattice modes of NH 4Br near its first-order phase transition ( P = 0). Similar linear relationships have also been obtained using our observed Raman frequencies of the ν 5TO(174 cm −1) lattice mode and the ν 2(1708 cm −1) internal mode of NH 4Cl near the tricritical ( P ⋍ 1.6 kbar ) and the second-order ( P ⋍ 2.8 kbar ) phase transitions in this crystal system. Our values of the slope ( dP dT ) λ deduced from the spectroscopic modifications of the Pippard relations are comparable with the literature value for the NH 4Br crystal near its first-order phase transition. We also give in this study our ( dP dT ) λ values which are the first reported in the literature for the tricritical and the second-order phase transitions in NH 4Cl.

Spectroscopic modifications of Pippard relations and their application to NH4Cl

Abstract The Pippard relations have been reformulated in terms of the spectroscopic parameters and applied to the NH 4 Cl system in the vicinity of its λ-transition. We have effectively tested some of these relations in an earlier study where we have used, for the frequency shift, data for the ν 5 lattice mode of NH 4 Cl. Here, using our observed frequency shifts for the internal mode, ν 2 of NH + 4 , we have obtained a good fit to the Pippard relations for temperatures below T λ for the NH 4 Cl system. We have then been able to extract the frequency shift data for the ν 2 internal mode as a function of temperature, using the specific heat and also thermal expansion data from the literature. The method of analysis for obtaining the thermodynamic data from the spectroscopically measured quantities or vice versa is given here.