Debye-Einstein Model Research Articles

Thermodynamic properties of stoichiometric lithium cobaltite LiCoO2

Lithium cobaltite LiCoO2 was prepared by solid state route and characterized using X-ray diffraction (XRD), scanning electron microscopy (SEM), magnetic properties measurement and simultaneous thermal analysis (STA). The heat capacity and enthalpy increments were measured by physical property measurement system (PPMS), differential scanning calorimetry (DSC) and drop calorimetry. The oxygen non-stoichiometry was determined using thermogravimetric measurement (TG). Above room temperature the temperature dependence of the molar heat capacity was derived by simultaneous least squares method from DSC and enthalpy data. The heat capacity was also analyzed in terms of a combined Debye–Einstein model. The molar entropy was evaluated from the low temperature heat capacity data.

A systematic first-principles study of the tungsten trioxide polymorphs

Tungsten trioxide (WO) is a compound that has a high degree of polymorphism, but a detailed phase diagram of this material is not available in the literature and a systematic ab-initio study including all its crystallographic phases does not exist. In this work, we have performed a first-principles density functional theory study of all WO crystalline phases within the generalized gradient and hybrid approximations of the exchange–correlation functional. We have calculated the equations of state, the isothermal bulk moduli, the -centered vibrational frequencies, and the phase diagram using the Debye–Einstein model. The calculated phase diagram shows only the crystallographic phase transition at high pressure and low temperature, being the -phase the stable polymorph at room temperature and ambient pressure, and the -phase stable at low temperatures and high pressures. In addition, the band structures were calculated observing a variation of the energy gap between 1.76 up to 2.79 eV with a similar topology of the top valence and bottom conduction bands across all the polymorphs.

Thermodynamic properties of tubular cobaltite Bi3.7Sr11.4Co8O29−δ

We synthetized a tubular cobaltite Bi3.7Sr11.4Co8O29−δ ceramics and characterized it using X-ray diffraction (XRD) and scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS). The simultaneous thermal analysis (STA) revealed a very unusual thermal behavior which was eventually ascribed to metastability of tubular phase at lower temperatures. The heat capacity and enthalpy increments were measured by physical property measurement system (PPMS), differential scanning calorimetry (DSC) and drop calorimetry. The oxygen non-stoichiometry was determined using thermogravimetric measurement (TG) and reduction in hydrogen atmosphere. Above room temperature the temperature dependence of the molar heat capacity was derived by simultaneous least squares method from DSC and enthalpy data. The heat capacity was also analyzed in terms of a combined Debye–Einstein model. The molar entropy Smo (298.15)=1349.45Jmol−1K−1 was evaluated from the low temperature heat capacity data.

Structure, oxygen non-stoichiometry and thermal properties of (Bi0.4Sr0.6)Sr2CoO5–δ

We synthesized and characterized (Bi0.4Sr0.6)Sr2CoO5−δ using X-ray diffraction (XRD) and scanning electron microscopy with energy dispersive spectroscopy (SEM–EDS). In the next step the heat capacity and enthalpy increments of (Bi0.4Sr0.6)Sr2CoO5−δ were measured by physical property measurement system (PPMS), differential scanning calorimetry (DSC) and drop calorimetry. Oxygen non-stoichiometry was determined using thermogravimetric measurement (TG) and reduction in hydrogen atmosphere. Above room temperature the temperature dependence of the molar heat capacity in the form Cpm=(201.0+0.09261×T−2184097×T−2)JK−1mol−1 was derived by the least squares method from the experimental data. The heat capacity was also analyzed in terms of a combined Debye–Einstein model. The molar entropy Sm° (298.15)=219.65Jmol−1K−1 was evaluated from the low temperature heat capacity data.

Structure, non-stoichiometry and thermodynamic properties of Bi1.85Sr2Co1.85O7.7− ceramics

Abstract The structure and oxygen non-stoichiometry of misfit layered cobaltite Bi1.85Sr2Co1.85O7.7−δ was determined by Rietveld analysis and by thermogravimetric measurements. The heat capacity and enthalpy increments of pseudo-ternary oxide Bi1.85Sr2Co1.85O7.7−δ was measured by the relaxation time method (PPMS) from 2 K to 256 K, by differential scanning calorimetry (DSC) from 258 K to 355 K and by the drop calorimetry from 573 K to 1153 K. Above room temperature the temperature dependence of the molar heat capacity in the form Cpm = (305.8 + 0.07325 · T − 4702536 · T−2) J K−1 mol−1 was derived by the least-squares method from the experimental data. The heat capacity was analyzed in terms of a combined Debye–Einstein model. The molar entropy S m ∘ ( 298.15 ) = 317.7 J mol−1 K−1 was evaluated from the low temperature heat capacity measurements.

Oxygen non-stoichiometry and thermodynamic properties of Bi2Sr2CoO6+ ceramics

In the literature the structure Bi 2 Sr 2 CoO 6+ δ has been described with a variable oxygen nonstoichiometry δ . We identified δ parameter by thermogravimetric measurements and determined thermodynamic properties of Bi 2 Sr 2 CoO 6+ δ . The heat capacity and enthalpy increments of Bi 2 Sr 2 CoO 6+ δ were measured by the relaxation time method (PPMS) from 2 to 256 K, by differential scanning calorimetry (DSC) from 258 to 355 K and by drop calorimetry from 573 to 1123 K. Above-room temperature, the dependence of molar heat capacity under isodynamical (constant p O 2 ) and conventional isoplethal conditions (constant composition) was derived from the experimental data by the least-squares method. The low temperature heat capacity was analyzed in terms of a combined Debye–Einstein model. The molar entropy S m ∘ ( 298.15 ) = 318.9 J K − 1 mol − 1 was evaluated from the low-temperature heat-capacity measurements.

Heat capacity, enthalpy and entropy of Sr14Co11O33 and Sr6Co5O15

Heat capacity and enthalpy increments of two ternary strontium cobalt mixed oxides Sr14Co11O33 and Sr6Co5O15 were measured by the relaxation time method (2–250K), DSC (258–355K) and drop calorimetry (573–1173K). Temperature dependencies of the molar heat capacity in the form Cpm=1359.46+0.355826·T–19.5816·106·T−2JK−1mol−1 for Sr14Co11O33 and Cpm=515.81+ 0.313326·T–5.07872·106·T−2JK−1mol−1 for Sr6Co5O15 respectively, were derived by the least-squares method from the experimental data. The heat capacity was analyzed in terms of a combined Debye–Einstein model. The molar entropies at 298.15K, Sm(298.15K)=1339.76JK−1mol−1 for Sr14Co11O33 and Sm(298.15K)=578.77JK−1mol−1 for Sr6Co5O15 were evaluated from the low temperature heat capacity measurements.

Theoretical study of the thermodynamic properties of lithium, sodium, and potassium nitrates

The thermal properties of lithium, sodium, and potassium nitrates have been studied in a gradient approximation of the density functional theory using the method of linear combination of atomic orbitals of the CRYSTAL09 program package. The long-wave frequencies and corresponding mode Gruneisen parameters are calculated. The quasi-harmonic Debye-Einstein model is used to calculate the parameters of the equation of states and also the dependences of the thermodynamic potentials, the entropy, the heat capacity, the thermal expansion coefficient, and the Gruneisen parameter on pressure and temperature. The role of external and intramolecular vibrations in the interpretation of thermodynamic properties is determined. The obtained results agree well with the available experimental data.

First-Principles Mode Gruneisen Parameters and Negative Thermal Expansion inα−ZrW2O8

Mode Grüneisen parameters were estimated for α-ZrW(2)O(8) zone-center modes by means of density functional theory calculations and the temperature dependence of the coefficient of thermal expansion was obtained according to the Debye-Einstein model of the quasiharmonic approximation. The lowest energy optic modes were identified at 45 and 46 cm(-1), and were shown to be the main modes responsible for negative thermal expansion at low temperature. Experimental evidence of the lowest energy, triply degenerated infrared active optic mode, was also found in the far infrared spectrum of α-ZrW(2)O(8). Upon increasing temperature, other optic modes with E<25 meV (particularly at 96, 100, 133, 161, and 164 cm(-1)) also contribute significantly to the coefficient of thermal expansion near room temperature. An analysis was made of selected zone-center modes in light of previously proposed models for explaining negative thermal expansion in open framework materials.

Comparative Experimental and Density Functional Theory (DFT) Study of the Physical Properties of MgB2 and AlB2

In the present study, we report an intercomparison of various physical and electronic properties of MgB2 and AlB2. In particular, the results of phase formation, resistivity ρ(T), thermoelectric power S(T), magnetization M(T), heat capacity (C P ), and electronic band structure are reported. The original stretched hexagonal lattice with a=3.083 Å, and c=3.524 Å of MgB2 shrinks in c-direction for AlB2 with a=3.006 Å, and c=3.254 Å. The resistivity ρ(T), thermoelectric power S(T) and magnetization M(T) measurements exhibited superconductivity at 39 K for MgB2. Superconductivity is not observed for AlB2. Interestingly, the sign of S(T) is +ve for MgB2 the same is −ve for AlB2. This is consistent with our band structure plots. We fitted the experimental specific heat of MgB2 to Debye–Einstein model and estimated the value of Debye temperature (Θ D) and Sommerfeld constant (γ) for electronic specific heat. Further, from γ, the electronic density of states (DOS) at Fermi level N(E F) is calculated. From the ratio of experimental N(E F) and the one being calculated from DFT, we obtained value of λ to be 1.84, thus placing MgB2 in the strong coupling BCS category. The electronic specific heat of MgB2 is also fitted below T c using α-model and found that it is a two gap superconductor. The calculated values of two gaps are in good agreement with earlier reports. Our results clearly demonstrate that the superconductivity of MgB2 is due to very large phonon contribution from its stretched lattice. The same two effects are obviously missing in AlB2, and hence it is not superconducting. DFT calculations demonstrated that for MgB2, the majority of states come from σ and π 2p states of boron on the other hand σ band at Fermi level for AlB2 is absent. This leads to a weak electron phonon coupling and also to hole deficiency as π bands are known to be of electron type, and hence obviously the AlB2 is not superconducting. The DFT calculations are consistent with the measured physical properties of the studied borides, i.e., MgB2 and AlB2.

Thermal expansivity ofGe1−ySnyalloys

The temperature dependence of the lattice parameter of ${\text{Ge}}_{1\ensuremath{-}y}{\text{Sn}}_{y}$ alloys deposited on Si substrates has been determined from an analysis of their x-ray reciprocal-space maps. It is found that over the range $0&lt;y&lt;0.03$ the alloy thermal expansivity increases by up to 20% as a function of $y$. This implies a strong deviation from a linear interpolation between the end compounds since the thermal expansivities of pure Ge and $\ensuremath{\alpha}\text{-Sn}$ are nearly the same. Alternative interpolation formulas based on a Debye model and a mixed Debye-Einstein model of the phonon structure are tested and it is found that they also fail to explain the observed increase in thermal expansivity.

Heat capacity and phonon spectra of A IIIN

The low temperature heat capacities of three AIIInitrides, AIII=Al, Ga and In, were measured by relaxation method in the temperature range 2–300 K and the corresponding entropies at the reference temperature 298.15 K were evaluated from the experimental data. The lattice heat capacity at constant volume was also assessed theoretically within harmonic crystal approximation by direct method using a combination of VASP software package to obtain the Hellmann-Feynman forces and the Phonon program to calculate the phonon spectra. The experimental data were analyzed by means of a Debye-Einstein model taking use of the calculated heat capacity and involving additionally an anharmonic contribution.

Orientational Ordering and Low-Temperature Libration in the Rotor−Stator Cocrystals of Fullerenes and Cubane

Cocrystals of cubane and fullerenes, C60-cubane and C70-cubane, show distinct rotational ordering transitions. We studied the corresponding structural changes with temperature-dependent X-ray diffraction and the thermodynamics of the phase transitions with adiabatic microcalorimetry and differential scanning calorimetry. C60-cubane has one phase transition around 130 K from a high-temperature fcc phase with freely rotating C60 to a low-temperature orthorombic phase in which the fullerene rotation is frozen. The corresponding enthalpy change is approximately 1170 J/mol, and the entropy change is 9.6 J/(mol K). C70-cubane has two phase transitions. Around 380 K, the high-temperature fcc phase with freely rotating C70 transforms into a bct phase in which the C70 rotates uniaxially around an axis that precesses around the c direction with a full opening angle of 40 degree. Around 170 K, the uniaxial rotation also freezes out, with an accompanying structural transition to monoclinic and enthalpy and entropy changes of 620 J/mol and 8.7 J/(mol K), respectively. The low-temperature specific heat was analyzed in terms of the Debye-Einstein model to estimate the librational energies of the fullerenes and Debye temperatures. We found very similar values for the two cocrystals, approximately Elib = 2.2 meV and TDebye = 23 K. For reference, we also measured the specific heats of pure C60 and C70 and found Elib = 2.96 meV and TDebye = 32 K for C60 and Elib = 1.9 meV and TDebye = 20 K for C70.

Pyroelectric properties of real LiNbO3 single crystals grown from a congruent melt

The temperature dependences of the pyroelectric coefficient of lithium niobate single crystals grown from a congruent melt have been investigated in the range of 4.2–300 K. No anomalies were found at low temperatures, and the experimental dependence is described well by the Debye-Einstein model, with TD = 357 K and two pyroactive frequencies of 692 and 869 cm−1. Specific features of lithium niobate have been analyzed. Two sublattices, formed by two pairs of mesotetrahedra with (according to the symmetry conditions) dipole and octupole moments, were selected in the structure. Their contributions to the total polarization differ by an order of magnitude. Vacuum annealing of the samples leads to the occurrence of anomalies only at temperatures over 280 K; these anomalities are interpreted as a manifestation of superionic conductivity.

Calculation of the thermodynamic properties of A III nitrides

The enthalpies of formation of three A III nitrides, AlN, GaN, and InN with wurtzite-type structure, were assessed using an ab initio density functional theory approach. A full-potential LAPW method within the generalized gradient approximation implemented in the WIEN2k package was employed to obtain the total cohesive energies of A IIIN, the corresponding metallic elements and the N 2 molecule. Furthermore, the entropies and specific heats at T = 298.15 K were obtained from an analysis of the low temperature heat capacity data using a Debye–Einstein model. The total energies and the integrated low-temperature heat capacities were eventually combined to evaluate the resulting heats of formation of A IIIN at T = 298.15 K .

Magnetic phase diagram ofEu4Ga8Ge16by magnetic susceptibility, heat capacity, and Mössbauer measurements

The magnetic ordering in the ${\mathrm{Eu}}_{4}{\mathrm{Ga}}_{8}{\mathrm{Ge}}_{16}$ clathrate compound was studied using magnetic susceptibility, M\"ossbauer spectroscopy, and heat-capacity measurements. Antiferromagnetic ordering is observed at 8 K. M\"ossbauer spectroscopy shows the Eu atoms to exist as ${\mathrm{Eu}}^{2+}$ with strong room-temperature absorption at -11.8 mm/s vs ${\mathrm{EuF}}_{3}.$ The temperature dependence of the Lamb-M\"ossbauer factor can be fitted to an Einstein model with an effective Einstein temperature of 80 K. In combination with heat-capacity measurements, this result suggests that the Eu atoms behave like isolated oscillators that are effectively decoupled from the lattice. The magnetic susceptibility data are interpreted using the Curie-Weiss law for ordering with ${\ensuremath{\mu}}_{\ensuremath{\parallel},\mathrm{eff}}=8.00\ifmmode\pm\else\textpm\fi{}0.03{\ensuremath{\mu}}_{B}/{\mathrm{Eu}}^{2+}$ ion and $\ensuremath{\theta}=4.43\ifmmode\pm\else\textpm\fi{}0.08\mathrm{K}.$ The positive Weiss constant is rationalized by a nearest-neighbor ferromagnetic exchange interaction within an overall antiferromagnetically ordered system. The spin-flop phase diagram is established and used to extract the next-nearest-neighbor interaction energy. The location of the tri-critical point in H,T space is determined. The magnetic heat capacity ${C}_{m}$ is extracted by modeling the lattice contribution using a weighted Debye-Einstein model with fitted Debye temperature ${\ensuremath{\Theta}}_{D}=314\ifmmode\pm\else\textpm\fi{}4\mathrm{K}$ and using a fixed Einstein temperature ${\ensuremath{\Theta}}_{E}=80\mathrm{K}.$ Integration of ${C}_{m}/T$ gave 15.1 J/K mol ${\mathrm{Eu}}^{2+}$ in fair accordance with expectations based on configurational considerations.

Strong electron-phonon coupling in superconductingMgB2: A specific heat study

We report on measurements of the specific heat of the recently discovered superconductor MgB$_2$ in the temperature range between 3 and 220 K. Based on a modified Debye-Einstein model, we have achieved a rather accurate account of the lattice contribution to the specific heat, which allows us to separate the electronic contribution from the total measured specific heat. From our result for the electronic specific heat, we estimate the electron-phonon coupling constant $\lambda$ to be of the order of 2, significantly enhanced compared to common weak-coupling values $\leq 0.4$. Our data also indicate that the electronic specific heat in the superconducting state of MgB$_2$ can be accounted for by a conventional, s-wave type BCS-model.

Equations of state for rare gas solids under strong compression

Experimental data on equations of state for rare gas solids are analysed within an “extended” Mie-Gruneisen approach, which uses a Modified Debye-Einstein model, MoDE2 for the quasiharmonic phonon contribution to the thermal pressure with additional corrections for intrinsic anharmonicities. The static lattice pressure is represented thereby with an Adapted Polynomial expansion of 3rd order, AP3, which uses as adjustable parameters the atomic number Z, the atomic volume Vsl0, the bulk modulus Ks10, and its first and second pressure derivatives K′s10, and K″s10 at zero pressure. Within this model, not only the data for selected p(V,T) isotherms, but also for the specific heat capacities Cpo andCvo, the volume V0(T), and the bulk modulus K0(T), at ambient pressure, are analysed in detail. Good agreement of the model with the experimental data is found for Ar, Kr, and Xe. The data for Ne, however, point to significant anharmonicity and vacancy contributions at ambient pressure, which go beyond the limits of the present model.

Intra- and inter-molecular modes reflected in femtosecond time-resolved optical heterodyne-detected optical Kerr effect signal of molecules in liquids

We present a theoretical work on the femtosecond time-resolved optical heterodyne-detected optical Kerr effect (OHD-OKE) measurement of molecules in liquids to clarify the role of intra- and inter-molecular low-frequency modes. An expression for the signal of the OHD-OKE measurement is derived on the basis of the double space representation. The theory is applied to the system of neat chloroform at room temperature. A Raman active vibrational mode with a frequency of 262cm −1 is considered, together with rotational degrees of freedom or Raman active inter-molecular vibrational modes. All the vibrational modes are assumed to be harmonic. For simplicity, a free and diffusive rotor model is adopted. A quantum mechanical treatment is also used to take into account rovibrational dephasings. The reorientational relaxation is treated in the Debye-Einstein model. Effects of the rovibrational dephasing appear in the OHD-OKE signal in a subpicosecond time regime, but intra-molecular quantum beats disappear in the early time region compared with the experimental signal. The effects of the reorientational relaxation appear in a picosecond time regime. Inclusion of inter-molecular vibrational modes qualitatively explains the OHD-OKE measurement, although calculations by the present model are only approximate.

Comment on the linear resistivity of high-temperature superconducting compounds

Abstract We report calculations of the phonon-induced electrical resistivity ρ(T) of YBa2Cu3O7 using Allen's expression for inverse transport relaxation time τph −1. ρ(T) is found to be a quasi-linear function of T(ρ ∞ T 0.98) for T ≥ 90 K, i.e. for T ≥ T c. The calculation does not take into account Umklapp scattering, which is likely to be important in the high-temperature superconductors (HTSCs) because of the low-lying optical phonon branches found in these compounds. Therefore, linear T behaviour of ρ(T) might well extend substantially below the transition temperatures of most HTSCs. Results for ρ (T) for a Debye-Einstein model of HTSCs (which gives realistic results for cu (T) and θDebye(T)) show linear behaviour of ρ(T) starting at even lower tempertures (T ≥ 30 K). We conclude from these results that linear T behaviour of the resistivity of HTSCs is fully compatible with the Bloch-Grineisen theory of resistivity in ordinary metals, and is, in itself, not an indication that a non-phonon mechanism m...