The elastic moduli and their pressure and temperature derivatives for calcium oxide

The room temperature adiabatic elastic moduli of CaO were obtained using a pulse superposition technique to measure the ultrasonic velocities. The elastic moduli were found to be (in GPa) at 298 K.

     

4.

Third-order elastic constants and pressure derivatives of the second-order elastic constants of β-tin     

V.M. MathewC.S. Menon  K.P. Jayachandran 《Journal of Physics and Chemistry of Solids》2002,63(10):1835-1838

The second- and third-order elastic constants and pressure derivatives of second-order elastic constants of tetragonal β-tin have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The 12 non-vanishing third-order elastic constants and the six pressure derivatives of the second-order elastic constants in tetragonal β-tin are obtained in the present work and are compared with the available experimental values. The second-order elastic constant C₃₃ obtained in the present study is in reasonable agreement with the experimental values. The third-order elastic constants are generally one order of magnitude greater than the second-order elastic constants as expected of a crystalline solid. The third-order elastic constant C₃₃₃ is higher in magnitude than all other values. This shows a greater anharmonicity of β-tin along the c-axis direction of the crystal.     

5.

LiI elastic constants and temperature derivatives at 295 K     

K.O. McLeanCharles S. Smith 《Journal of Physics and Chemistry of Solids》1972,33(2):275-278

     

6.

BaO single crystal elastic constants and their temperature dependence     

V.H. Vetter  R.A. Bartels 《Journal of Physics and Chemistry of Solids》1973,34(8):1448-1449

     

7.

Temperature and pressure derivatives of the elastic constants of cubic rubidium cyanide     

S. Haussühl 《Solid State Communications》1979,32(2):181-183

The temperature and pressure derivatives of all elastic constants of rubidium cyanide have been measured from 133 to 380 K respectively from 0 to 1500 × 10⁵ Nm^-2 by ultrasonic methods. The thermoelastic behaviour resembles that of KCN and NaCN. A strong softening of the shear resistance c₄₄ approaching the transition temperature from higher temperatures is observed. The pressure derivatives are also quite similar to those of KCN and NaCN, but fully different from those of normal alkali halides of rocksalt-type. This behaviour confirms a rule already observed in other isotypic crystal groups: the quasi-invariant pressure derivatives are shifted in a characteristic way for a certain structure type, if rare-gas-like ions are replaced by asymmetric ions. The nonlinear elastic behaviour is qualitatively interpreted by interactions of volume-conserving type as existing in fluids.     

8.

The pressure and temperature derivatives of the elastic moduli of lithium hydride     

D. Gerlich  C.S. Smith 《Journal of Physics and Chemistry of Solids》1974,35(12):1587-1592

The pressure and temperature derivatives of the elastic moduli of single crystal LiH have been determined at room temperature, by the ultrasonic pulse echo method. From the pressure data, the mode Grüneisen parameters as well as the low and high temperature limiting values of the Grüneisen constant were computed; the latter were only slightly different. An equation of state has been constructed for LiH, and found to be in agreement with the experimental data. Based on a non-central force model, the values of the elastic moduli and their pressure derivatives have been calculated theoretically, and the latter values compared with the experimental data.     

9.

Temperature derivatives at constant volume of the elastic constants of alkali halide crystals     

J. Shanker  R.K. Varshney 《Solid State Communications》1983,46(3):263-264

A new method has been developed for calculating the temperature derivatives at constant volume of the elastic constants of the alkali halides using the concept of phonon pressure. Calculated values of temperature derivatives of elastic constants are compared with the corresponding values recently derived from ultrasonic experimental data.     

10.

The hydrostatic pressure derivatives of the elastic constants of indium and an indium cadmium alloy     

S.C. Flower  G.A. Saunders  Y.K. YoḠurtçu 《Journal of Physics and Chemistry of Solids》1985,46(1):97-102

The hydrostatic pressure derivatives of the elastic stiffness constants of indium and indium-3.4 at.% cadmium alloy single crystals have been obtained from pulse echo overlap measurements of the dependence of ultrasonic wave velocities upon pressure. The softest zone centre acoustic phonon mode in indium is a shear mode propagating k along the [101] direction rather than that (k[110], e[11?0]) which drives the ferroelastic phase transition in the indium-cadmium alloys. The derivative δ((C₁₁ – C₁₂)/2)/δP is positive, accounting for the stability of the fct structure of indium under high pressure. Using the quasiharmonic, anisotropic continuum model the acoustic mode Grüneisen parameters have been calculated and are discussed in terms of mode softening. The high temperature limiting value$\text{\$}\text{?}$γ_H (= 2.56) of the mean acoustic mode Grüneisen parameter is found to be close to the thermodynamic Grüneisen parameter γ^th (=2.5).     

11.

The static and elastic properties of mixed diatomic crystals     

R.K. Singh  Shankar P. Sanyal  J.K. Nayak 《Journal of Physics and Chemistry of Solids》1982,43(1):21-24

An interionic potential model has been proposed to study the static and elastic properties of mixed diatomic crystals. The interaction system of this potential consists of the long-range Coulomb and three-body interaction and the short-range overlap repulsion. This potential has been used to calculate the cohesive energy, phase-transition pressure and volume, third-order elastic constants and pressure derivatives of the effective second-order elastic constants for NaCl-NaBr mixed crystals. These results agree reasonably well with the available experimental results on the host crystals and allow us to draw some meaningful conclusions for the mixed alkali halide crystals.     

12.

Molecular forces and elastic constants of polyethylene single crystals     

Joginder N. Anand 《Journal of Macromolecular Science: Physics》2013,52(3):445-458

Polyethylene has an orthorhombic lattice for which nine elastic constants exist; they are obtained in terms of the intra- and intermolecular forces. Constants involved in the 6-12 Lennard-Jones potential approximating the London dispersion type of van der Waals' forces are obtained by computing the crystal potential energy and comparing it with the cohesive energy. First and second nearest-neighbor interactions are considered to establish relationship between the elastic constants and the interaction constants. The latter are obtained in terms of the C—C bond, stretching, bending, and repulsive force constants and the L-J potential constants. A limited type of central force assumption is applied. Values of Young and shear moduli are obtained along the three axes. The value along the chain compares with the experimentally determined and calculated values for oriented polyethylene. Young's modulus along the lateral direction is of the order of Young's modulus of bulk polyethylene, showing that intermo-lecular forces are the ones that determine the Young modulus of bulk polyethylene.     

13.

The temperature dependence of the elastic constants of MnO     

S.B. Palmer  A. Waintal 《Solid State Communications》1980,34(8):663-666

We have measured the temperature dependence of the elastic constants of a monodomain single crystal of MnO, obtained by applying a unixial stress, from 4.2 to 300 K. Measurements were made by standard ultrasonic techniques and the acoustic waves were generated and detected by CdS transducers. Problems associated with the coupling between transducer and sample encountered by previous workers were therefore overcome. For propagation down the [111] direction, only the slow shear mode [$\text{1}\text{2}$(c₁₁?c₁₂)] was observable below the Néel temperature (T_N. The fast shear and longitudinal modes were heavily attenuated. Complex behaviour in the region of T_N and a further anomaly at 42 K were observed.     

14.

The effect of temperature on elastic constants and bulk modulus     

J. K. Baria  A. R. Jani 《Indian Journal of Physics》2010,84(11):1509-1516

A model pseudopotential depending on an effective core radius is proposed to study the elastic constants C ₁₁, C ₁₂, C ₄₄ and bulk modulus B of Cu, Ag, Au, Ni, Pt and Rh at various temperatures (300–1500 K). The contribution of d-like electron is taken into account by introducing repulsive short-range Born-Mayer term. An excellent agreement between theoretical investigation and experimental findings has confirmed our formalism.     

15.

Temperature variations in elastic constants and Debye temperature of single crystals of a KBr-KI solid solution     

A. A. Botaki  I. N. Gyrbu  A. V. Sharko 《Russian Physics Journal》1972,15(6):917-919

     

16.

Adiabatic and isothermal elastic constants for highly anharmonic crystals     

Dr. T. Paszkiewicz 《Zeitschrift für Physik B Condensed Matter》1974,18(1):1-15

The mass operator of the phonon Green function for a highly anharmonic crystal in the ladder approximation is derived. In order to calculate the mass operator in the collision-dominated regime it is necessary to solve the Boltzmann-like equations. These solutions allow us to find the thermal conductivity coefficient and to prove that the thermodynamic relation between adiabatic and isothermal elastic constants is satisfied.     

17.

Low temperature elastic constants and Debye temperature of NbO2     

J.G. Bennett  R.J. Sladek 《Solid State Communications》1978,25(12):1035-1037

The transit times of ultrasonic waves have been measured in single crystal NbO₂ from 295 K down to 1.5 K for quasilongitudinal and shear waves propagating in the [100] direction and down to 160 K for eight other waves. Values are obtained for the C₄₄ elastic constant and for an elastic constant combination which is approximately equal to C₁₁ for temperatures down to 1.5 K and for C₁₁, C₁₂, C₁₃, C₁₆, C₃₃, and C₆₆ down to 160 K. These results are used to deduce 0 K values for the elastic constants and an elastic Debye temperature of 596 ± 7 K at 1.5 K. The acoustic mode heat capacity calculated from the latter is significantly smaller than the heat capacity measured by Wenger and Keesom at low temperatures. Following Wenger and Keesom, the difference is attributed to phasons (excitations involving the phase modulation of charge density waves). An average velocity is deduced for the phasons.     

18.

Many body forces and elastic constants of rare gas crystals     

Alfred Hüller 《Zeitschrift für Physik A Hadrons and Nuclei》1971,245(4):324-332

The effect of many body Van der Waals forces on the elastic constants of solid rare gases is calculated. These results are combined with contributions from the two body potential and zero point energy. The violation of the Cauchy relation is studied.     

19.

具有立方晶系结构的多晶体材料的弹性常数——Y弹性常数   总被引：2，自引：0，他引：2       下载免费PDF全文

林政  刘旻 《物理学报》2009,58(6):4096-4102

提出了一个新的物理参量“Ｙ弹性常数”,并阐述了其物理含义.并将其应用于具有立方晶系结构的多晶体材料,推导了立方晶系结构的多晶体材料的Ｙ弹性常数,通过算例与具有立方晶系结构的多晶体材料的X射线弹性常数进行了比较.运用这个Ｙ弹性常数进一步推导出的多晶体材料整体的机械弹性常数的表达式与Krner的研究结果完全符合. 关键词： Y弹性常数 立方晶系 多晶体材料     

20.

Analysis of the pressure derivatives of the second order elastic constants in alkaline earth oxides     

S.C. Goyal  S.P. Tripathi 《Journal of Physics and Chemistry of Solids》1977,38(4):351-353

The three body force shell model (TSM) has been applied to evaluate the magnitude of the pressure derivatives of second order elastic constants of alkaline earth oxides and a good agreement with the experimental values has been obtained. It is also pointed out that the calculated values of dC₄₄?dp show a smooth variation with ionicities. The theory has been modified by taking into account next nearest neighbour interaction.     

c11	c12	c44
226.2 ±0.9	62.4 ±0.9	80.6 ±0.3 The pressure derivatives of the elastic moduli, which were measured at and below room temperature, decrease with decreasing temperature. The temperature derivatives of the elastic moduli were also obtained for several temperatures below room temperature.

Third-order elastic constants and pressure derivatives of the second-order elastic constants of β-tin

The second- and third-order elastic constants and pressure derivatives of second-order elastic constants of tetragonal β-tin have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The 12 non-vanishing third-order elastic constants and the six pressure derivatives of the second-order elastic constants in tetragonal β-tin are obtained in the present work and are compared with the available experimental values. The second-order elastic constant C₃₃ obtained in the present study is in reasonable agreement with the experimental values. The third-order elastic constants are generally one order of magnitude greater than the second-order elastic constants as expected of a crystalline solid. The third-order elastic constant C₃₃₃ is higher in magnitude than all other values. This shows a greater anharmonicity of β-tin along the c-axis direction of the crystal.     

Temperature and pressure derivatives of the elastic constants of cubic rubidium cyanide

The temperature and pressure derivatives of all elastic constants of rubidium cyanide have been measured from 133 to 380 K respectively from 0 to 1500 × 10⁵ Nm^-2 by ultrasonic methods. The thermoelastic behaviour resembles that of KCN and NaCN. A strong softening of the shear resistance c₄₄ approaching the transition temperature from higher temperatures is observed. The pressure derivatives are also quite similar to those of KCN and NaCN, but fully different from those of normal alkali halides of rocksalt-type. This behaviour confirms a rule already observed in other isotypic crystal groups: the quasi-invariant pressure derivatives are shifted in a characteristic way for a certain structure type, if rare-gas-like ions are replaced by asymmetric ions. The nonlinear elastic behaviour is qualitatively interpreted by interactions of volume-conserving type as existing in fluids.     

The pressure and temperature derivatives of the elastic moduli of lithium hydride

The pressure and temperature derivatives of the elastic moduli of single crystal LiH have been determined at room temperature, by the ultrasonic pulse echo method. From the pressure data, the mode Grüneisen parameters as well as the low and high temperature limiting values of the Grüneisen constant were computed; the latter were only slightly different. An equation of state has been constructed for LiH, and found to be in agreement with the experimental data. Based on a non-central force model, the values of the elastic moduli and their pressure derivatives have been calculated theoretically, and the latter values compared with the experimental data.     

Temperature derivatives at constant volume of the elastic constants of alkali halide crystals

A new method has been developed for calculating the temperature derivatives at constant volume of the elastic constants of the alkali halides using the concept of phonon pressure. Calculated values of temperature derivatives of elastic constants are compared with the corresponding values recently derived from ultrasonic experimental data.     

The hydrostatic pressure derivatives of the elastic constants of indium and an indium cadmium alloy

The hydrostatic pressure derivatives of the elastic stiffness constants of indium and indium-3.4 at.% cadmium alloy single crystals have been obtained from pulse echo overlap measurements of the dependence of ultrasonic wave velocities upon pressure. The softest zone centre acoustic phonon mode in indium is a shear mode propagating k along the [101] direction rather than that (k[110], e[11?0]) which drives the ferroelastic phase transition in the indium-cadmium alloys. The derivative δ((C₁₁ – C₁₂)/2)/δP is positive, accounting for the stability of the fct structure of indium under high pressure. Using the quasiharmonic, anisotropic continuum model the acoustic mode Grüneisen parameters have been calculated and are discussed in terms of mode softening. The high temperature limiting value$\text{\$}\text{?}$γ_H (= 2.56) of the mean acoustic mode Grüneisen parameter is found to be close to the thermodynamic Grüneisen parameter γ^th (=2.5).     

The static and elastic properties of mixed diatomic crystals

An interionic potential model has been proposed to study the static and elastic properties of mixed diatomic crystals. The interaction system of this potential consists of the long-range Coulomb and three-body interaction and the short-range overlap repulsion. This potential has been used to calculate the cohesive energy, phase-transition pressure and volume, third-order elastic constants and pressure derivatives of the effective second-order elastic constants for NaCl-NaBr mixed crystals. These results agree reasonably well with the available experimental results on the host crystals and allow us to draw some meaningful conclusions for the mixed alkali halide crystals.     

Molecular forces and elastic constants of polyethylene single crystals

Polyethylene has an orthorhombic lattice for which nine elastic constants exist; they are obtained in terms of the intra- and intermolecular forces. Constants involved in the 6-12 Lennard-Jones potential approximating the London dispersion type of van der Waals' forces are obtained by computing the crystal potential energy and comparing it with the cohesive energy. First and second nearest-neighbor interactions are considered to establish relationship between the elastic constants and the interaction constants. The latter are obtained in terms of the C—C bond, stretching, bending, and repulsive force constants and the L-J potential constants. A limited type of central force assumption is applied. Values of Young and shear moduli are obtained along the three axes. The value along the chain compares with the experimentally determined and calculated values for oriented polyethylene. Young's modulus along the lateral direction is of the order of Young's modulus of bulk polyethylene, showing that intermo-lecular forces are the ones that determine the Young modulus of bulk polyethylene.     

The temperature dependence of the elastic constants of MnO

We have measured the temperature dependence of the elastic constants of a monodomain single crystal of MnO, obtained by applying a unixial stress, from 4.2 to 300 K. Measurements were made by standard ultrasonic techniques and the acoustic waves were generated and detected by CdS transducers. Problems associated with the coupling between transducer and sample encountered by previous workers were therefore overcome. For propagation down the [111] direction, only the slow shear mode [$\text{1}\text{2}$(c₁₁?c₁₂)] was observable below the Néel temperature (T_N. The fast shear and longitudinal modes were heavily attenuated. Complex behaviour in the region of T_N and a further anomaly at 42 K were observed.     

The effect of temperature on elastic constants and bulk modulus

A model pseudopotential depending on an effective core radius is proposed to study the elastic constants C ₁₁, C ₁₂, C ₄₄ and bulk modulus B of Cu, Ag, Au, Ni, Pt and Rh at various temperatures (300–1500 K). The contribution of d-like electron is taken into account by introducing repulsive short-range Born-Mayer term. An excellent agreement between theoretical investigation and experimental findings has confirmed our formalism.     

Adiabatic and isothermal elastic constants for highly anharmonic crystals

The mass operator of the phonon Green function for a highly anharmonic crystal in the ladder approximation is derived. In order to calculate the mass operator in the collision-dominated regime it is necessary to solve the Boltzmann-like equations. These solutions allow us to find the thermal conductivity coefficient and to prove that the thermodynamic relation between adiabatic and isothermal elastic constants is satisfied.     

Low temperature elastic constants and Debye temperature of NbO2

The transit times of ultrasonic waves have been measured in single crystal NbO₂ from 295 K down to 1.5 K for quasilongitudinal and shear waves propagating in the [100] direction and down to 160 K for eight other waves. Values are obtained for the C₄₄ elastic constant and for an elastic constant combination which is approximately equal to C₁₁ for temperatures down to 1.5 K and for C₁₁, C₁₂, C₁₃, C₁₆, C₃₃, and C₆₆ down to 160 K. These results are used to deduce 0 K values for the elastic constants and an elastic Debye temperature of 596 ± 7 K at 1.5 K. The acoustic mode heat capacity calculated from the latter is significantly smaller than the heat capacity measured by Wenger and Keesom at low temperatures. Following Wenger and Keesom, the difference is attributed to phasons (excitations involving the phase modulation of charge density waves). An average velocity is deduced for the phasons.     

Many body forces and elastic constants of rare gas crystals

The effect of many body Van der Waals forces on the elastic constants of solid rare gases is calculated. These results are combined with contributions from the two body potential and zero point energy. The violation of the Cauchy relation is studied.     

具有立方晶系结构的多晶体材料的弹性常数——Y弹性常数

提出了一个新的物理参量“Ｙ弹性常数”,并阐述了其物理含义.并将其应用于具有立方晶系结构的多晶体材料,推导了立方晶系结构的多晶体材料的Ｙ弹性常数,通过算例与具有立方晶系结构的多晶体材料的X射线弹性常数进行了比较.运用这个Ｙ弹性常数进一步推导出的多晶体材料整体的机械弹性常数的表达式与Krner的研究结果完全符合. 关键词： Y弹性常数 立方晶系 多晶体材料     

Analysis of the pressure derivatives of the second order elastic constants in alkaline earth oxides

The three body force shell model (TSM) has been applied to evaluate the magnitude of the pressure derivatives of second order elastic constants of alkaline earth oxides and a good agreement with the experimental values has been obtained. It is also pointed out that the calculated values of dC₄₄?dp show a smooth variation with ionicities. The theory has been modified by taking into account next nearest neighbour interaction.