Elastic Constants Of Diamond Research Articles

Ab initio calculations of second-, third-, and fourth-order partial and inner elastic constants of diamond.

By means of ab initio calculations, a unified framework is presented to investigate the effect of internal displacement on the linear and nonlinear elasticity of single diamond crystals. The calculated linear and nonlinear elastic constants, internal strain tensor and internal displacement in single diamond crystals are compatible with the available experimental data and other theoretical calculations. The complete set of second-, third- and fourth-order elastic constants and internal strain tensor not only offer a better insight into the nonlinear and anisotropic elasticity behaviors, but also shows us the basic internal mechanical response of diamond. This study provides a route to calculate the nonlinear internal and external elasticity response in a nonprimitive lattice.

Elastic and mechanical properties of cubic diamond and silicon using density functional theory and the random phase approximation

The elastic constants play a central role in the regulation of the thermo-mechanical and anisotropic response of materials. Nevertheless, there is still a lack of theoretical and experimental data on these constants which limits the possibility of developing new materials with targeted mechanical responses. The elastic constants of diamond and silicon as second and third-order are obtained using density functional theory (LDA, GGA, and HSE) and for the first time with the adiabatic-connection fluctuation–dissipation theorem in the random phase approximation. Our SOECs calculations show an excellent agreement with experimental data. In this framework, we study the mechanical properties such as Young’s modulus, Poisson’s ratio, bulk modulus, and shear modulus of diamond and silicon structures. Also, we visualize the 3D plot, as well as 2D for Young’s modulus, Poisson’s ratio, and others. For the TOECs, we show that can be taken our RPA results as a reference, since there is still a lack of reliable previous theoretical and experimental data of these materials.

Quantum effects on elastic constants of diamond by path-integral Monte Carlo simulations

Abstract Using the path-integral Monte Carlo method, we investigate the quantum effects on the elastic constants C 11 , C 12 , and C 44 of the diamond crystal in a wide temperature range at ambient pressure. The Tersoff potential is used to describe the interatomic interactions and the elastic constants are determined by a direct method derived from the stress-strain curve. We find that the elastic constants C 11 and C 44 behave as a quadratic function of temperature at low temperatures and C 12 is practically a constant being independent of temperature. The quantum effects are significant at low temperatures up to about 1000 K. The quantitative differences between the PIMC calculations and the experimental measurements of the elastic constants C 11 , C 12 , and C 44 at low temperatures are about 3 % , 17 % , and 9 % , respectively. Using the obtained elastic constants, we also estimate the bulk modulus B, anisotropic factor A, Cauchy pressure P Cauchy , and Poisson ratio ν of the diamond crystal.

Mindlin–Eringen anisotropic micromorphic elasticity and lattice dynamics representation

ABSTRACTTo account for certain essential features of material such as dispersive behaviour and optical branches in dispersion curves, a fundamental departure from classical elasticity to polar theories is required. Among the polar theories, micromorphic elasticity of appropriate grades and anisotropy is capable of capturing these physical phenomena completely. In the mathematical framework of micromorphic elasticity, in addition to the traditional elastic constants, some additional constants are introduced in the pertinent governing equations of motion. A precise evaluation of the numerical values of the aforementioned elastic constants in the realm of the experimentations poses serious difficulties. Thus this paper aims to provide a remedy as how to determine the micromorphic elastic constants theoretically in terms of the atomic force constants and lattice parameters of the crystalline solid with general anisotropy. In this treatment capture of the discrete nature of matter becomes an essential factor. To this end, the discrete lattice dynamics equations of a crystal are related to the pertinent anisotropic micromorphic equations of motion. This approach allows incorporating the symmetry groups of the crystals within lattice dynamics equations conveniently. For the illustration of the current theoretical developments, the micromorphic elastic constants of diamond and silicon crystals are computed in conjunction with ab initio density functional perturbation theory (DFPT). Moreover, the longitudinal and transverse optical and acoustic branches pertinent to [100] and [110] directions are presented. The accuracy of the results is verified by comparing the dispersion curves derived from the micromorphic theory, those of available experiments, and those directly obtained from DFPT calculations.

C2/m-carbon: structural, mechanical, and electronic properties

The structural, mechanical, and electronic properties of C2/m-carbon were studied utilizing the first-principles calculations. The calculated lattice parameters and elastic constants of diamond are in excellent agreement with the available experimental data, indicating our calculations for C2/m-carbon are valid and believable. The calculated elastic constants indicate that C2/m-carbon is mechanically stable according to the elastic stability criteria under pressure. Furthermore, the elastic anisotropy has been visualized in detail by plotting the directional dependence of Poisson’s ratio, Young’s modulus, and shear modulus, whereas the calculated values of Poisson’s ratio and B/G present their brittle manner. B/G increases under increasing pressure with B/G = 1.75 at about 260.74 GPa and v increases observed with increasing pressure with v = 0.26 at about 261.35 GPa for C2/m-carbon, respectively. Our calculations predict that C2/m-carbon is an indirect semiconductor with wide band gap of 4.197 eV.

First-Principle Calculations for Elastic and Thermodynamic Properties of Diamond

The elastic constants and thermodynamic properties of diamond are investigated by using the CRYSTAL03 program. The lattice parameters, the bulk modulus, the heat capacity, the Grüneisen parameter, and the Debye temperature are obtained. The results are in good agreement with the available experimental and theoretical data. Moreover, the relationship between V/V0 and pressure, the elastic constants under high pressure are successfully obtained. Especially, the elastic constants of diamond under high pressure are firstly obtained theoretically. At the same time, the variations of the thermal expansion α with pressure P and temperature Tare obtained systematically in the ranges of 0–870 GPa and 0–1600 K.

Elastic constants of diamond from molecular dynamics simulations

The elastic constants of diamond between 100 and 1100 K have been calculated for the firsttime using molecular dynamics and the second-generation, reactive empirical bond-orderpotential (REBO). This version of the REBO potential was used because it was redesignedto be able to model the elastic properties of diamond and graphite at 0 K whilemaintaining its original capabilities. The independent elastic constants of diamond,C11,C12,and C44, and the bulk modulus were all calculated as a function of temperature, and the resultsfrom the three different methods are in excellent agreement. By extrapolating the elasticconstant data to 0 K, it is clear that the values obtained here agree with the previouslycalculated 0 K elastic constants. Because the second-generation REBO potentialwas fit to obtain better solid-state force constants for diamond and graphite,the agreement with the 0 K elastic constants is not surprising. In addition, thefunctional form of the second-generation REBO potential is able to qualitativelymodel the functional dependence of the elastic constants and bulk modulus ofdiamond at non-zero temperatures. In contrast, reactive potentials based on otherfunctional forms do not reproduce the correct temperature dependence of the elasticconstants. The second-generation REBO potential also correctly predicts thatdiamond has a negative Cauchy pressure in the temperature range examined.

Elastic Constants of Diamond Like Carbon Films by Surface Brillouin Scattering

AbstractThe elastic constants of thin Diamond-Like Carbon (DLC) films supply important information, but their measurement is difficult. Standard nanoindentation does not directly measure the elastic constants and has strong limitations particularly in the case of hard thin films on softer substrates, such as tetrahedral amorphous carbon on Si. Surface acoustic waves provide a better mean to investigate elastic properties. Surface Brillouin scattering (SBS) intrinsically probes acoustic waves of the wavelength which is appropriate to test the properties of films in the tens to hundreds of nanometers thickness range. SBS can be used to derive all the isotropic elastic constants of hard-on-soft and soft-on-hard amorphous carbon films of different kinds, with thickness down to less than 10 nm. The results help to resolve the previous uncertainties in mechanical data. The Young's modulus of tetrahedral amorphous carbon (ta-C) turns out to be lower than that of diamond, while the moduli of hydrogenated ta-C (ta-C:H) are considerably lower than those of ta-C because of the weakening effect of C-H bonding.

Young's modulus and Poisson's ratio of CVD diamond

The elastic modulus E of diamond is often set equal to 1 s 11 = 1050 GPa , which assumes that it does not vary much with orientation, and many authors use v = 0.2 as an appropriate average value of Poisson's ratio, which is incorrect. In fact, since the elastic constants of diamond are known with great accuracy, it is a straightforward matter to derive exact numbers for E and v that take into consideration the stress direction and the intrinsic anisotropy as well as the crystalline configuration. For diamonds synthesized by chemical vapor deposition (CVD) we find that in a first approximation the Hershey-Kröner-Eshelby averaging procedure yields acceptable numbers, E ̄ = 1143 GPa and v ̄ = 0.0691 , which are quite compatible with available experimental evidence. Our measurements of the biaxial modulus E′ = E (1 − v) make use of the bulge test method to characterize the elastic behavior of both microwavepower- and hot-filament-assisted CVD diamond films. High quality deposits yield ′ E ≈ 1180 and 1220 GPa for randomly orientated and (110)-textured deposits respectively; these results confirm that state-of-the-art deposits exhibit elastic properties that are in accord with the measured stiffnesses of natural single-crystal diamond. The residual hydrogen content strongly impacts the elastic behavior and appears to be responsible for the degradation of the modulus observed in this and previous work.

Anisotropy of Young's modulus and Poisson's ratio in diamond

The elastic modulus E of diamond is often set equal to 1 s 11 = 1050 GPa, which assumes that it does not vary much with orientation, and many authors use v = 0.2 as an appropriate average value of Poisson's ratio, which is incorrect. In this paper, we show that, since the elastic constants of diamond are known with great accuracy, it is a straightforward matter to derive exact numbers for E and v that take into consideration the stress direction, the intrinsic anisotropy, as well as the crystalline configuration. For textured CVD diamond deposits, we find that, in a first approximation, the Hershey-Kröner-Eshelby averaging procedure yields acceptable numbers, E ̄ = 1143 GPa and v ̄ = 0.0691 , which are quite compatible with available experimental evidence; this confirms that high-quality deposits have elastic constants similar to those of single-crystal natural diamond.

Elastic constants of diamond up to 1600 K

Abstract The recent achievement of hydrostatic pressures of the order of 1 Mbar within the Diamond Anvil Cell suggests that it may be possible to approach in the laboratory both the pressure and temperature conditions prevailing in the earth's interior. Thus it would be desirable to know if diamond retains its superhard characteristics at high temperatures. The elastic stiffness constants of a crystalline material are related to its stability and hardness. If the material is transparent, as is the case for diamond, Brillouin scattering is a convenient method for the determination of its elastic constants at high temperatures. The upper temperature limit (1600 K) of our measurements was set by the softening and surface deterioration of the quartz crucibles in which the diamond samples were held under vacuum, and not by the diamond samples themselves. Additionally, the measurement of a particular phonon branch in two different scattering geometries allowed us to measure the index of refraction of diamond up...

Dispersion Curves for Waves in a Cubic Micropolar Medium with Reference to Estimations of the Material Constants for Diamond

The authors start with an analytical study on the propagation of plane harmonic waves in an infinitely extended, cubi-anisotropic, elastic, and micropolar medium. It is possible to have six fundamental waves. Of these, a longitudinal uncoupled displacement wave and two coupled transverse waves of displacement and microrotation which become uncoupled with become uncoupled displacement waves at the limit of long wavelength are compared with the acoustical waves in the classical theory. The micropolar elastic constants of diamond, as well as its local moment of inertia, are estimated under the requirement that the theoretical dispersion curves for these waves fit the experimental data. Numerical computations are carried out and the results are made into a table.

Brillouin scattering in diamond

The elastic constants of diamond are determined from the shifts of the Brillouin components. Attention is drawn to the discrepancy between the intensities of the Brillouin components observed experimentally and those expected on the basis of the elasto-optic constants of diamond given in the literature.

The variation of the dielectric constant of diamond with pressure

Abstract The variation of dielectric constant with pressure of two diamonds has been measured. If an atomic polarizability α is calculated from the Clausius–Mossotti equation, it is found that V/α(∂α/∂V) T has the value 1·2, which conforms closely to its value for silicon and germanium. Some of the results of other workers on the photoelastic properties of diamond are also in conformity with the present results. There is some uncertainty, however, as to the elastic constants of diamond and it seems possible that both these and V/α(∂α/∂V) T may depend significantly on the impurity content of the stones.

Heat Capacity of Diamond at High Temperatures

The enthalpy of gem diamonds has been measured from 273° to 1073°K using a ``drop'' method and a Bunsen ice calorimeter. The derived heat-capacity values, which are believed to be accurate to ±0.5%, are used to calculate the thermal functions of diamond above 298.15°K. Values of Cv0 (harmonic) derived from the present investigation and recent measurements of the thermal expansion and the elastic constants of diamond are compared with theoretical treatments. Reasonably good agreement in the temperature range of this study can be obtained with a single Debye function (ΘD=1860) obtained from the measured elastic constants. Extrapolation of the data gives Θ∞=1880±10.

The Four Elastic Constants of Diamond

THE thermal scattering of light in diamond has been studied in great detail by R. S. Krishnan1 and by V. Chandrasekharan2. These authors pointed out that the frequency shifts observed for the Brillouin components are not in good agreement with the values calculated using the elastic constants data of Bhagavantam and Bhimasenachar3. The values of the elastic constants have since been determined by Prince and Wooster4 and recently much more accurately by McSkimin and Bond5. McSkimin and Bond's values do not agree with those of Bhagavantam and Bhimasenachar and of Prince and Wooster, especially as regards C 12 and C 44. Using the values of McSkimin and Bond, the frequency shifts of the Brillouin components for the various orientations studied by Chandrasekharan with the bigger speetrograph were calculated, but the agreement with the experimental values was still unsatisfactory. We have therefore evaluated the elastic constants of diamond directly from the observed frequency shifts6.

The theory of the vibrations and the Raman spectrum of the diamond lattice

The crystal structure of diamond was first determined by Bragg in 1913 from X-ray photographs; the carbon atoms are arranged at the apices and median points of interlinked tetrahedra. Born (1914) derived expressions for the three elastic constants of diamond in terms of two force constants related to the valency bonds between neighbouring atoms. But, at that time, the only experimental data available were the compressibility and the Debye characteristic temperature 0, and precise determination of the valence force constants was not possible. Meanwhile, investigation of the optical properties of diamond had produced evidence for the existence of two distinct types, one with an absorption band at 8 [i in the infra-red, the other transparent at this point. Robertson, Fox & Martin (1934) took up this problem and found that absorption in the infra-red is associated with absorption in the ultra-violet; diamonds transparent at 8y transmit much farther into the ultra-violet. Both types of diamond have Bragg’s tetrahedral structure, the same refractive index, specific gravity, dielectric constant and electron diffraction. Their infra-red spectra are identical up to 7y, and the frequency shift of the principal Raman line is the same. The derivation of the elastic constants was again considered by Nagendra Nath (1934). He extended the theory to include central forces between second neighbours in the lattice. He also suggested that the frequency shift of the principal Raman line corresponds to the relative vibration of the two carbon atoms in the unit cell, along the line joining their nuclei. Raman and his collaborators have recently (1941) put forward a new theory of lattice dynamics according to which the vibrational spectrum of a crystal consists of a few discrete lines. This is in direct contradiction to the quasi-continuous vibrational spectrum predicted by classical or quantum mechanics. On this new theory there are eight fundamental frequencies of vibration for diamond; the values of these frequencies are deduced from the observed specific heat, ultra-violet absorption and Raman spectrum, which, it is claimed, cannot be explained by ‘orthodox’ lattice dynamics. Raman (1944) has suggested that there are, not two, but four types of diamond, two with tetrahedral symmetry and two with octahedral symmetry depending on the electronic configurations, but X-ray analysis gives no indication of this and the attempts of his school to explain the observed infra-red spectra on the basis of their new lattice theory have been, up to now, unsuccessful.

The scattering of light in diamond and its Raman spectrum

A detailed study of the scattering of light in diamond and its Raman spectrum has been made using the λ 2536·5 mercury resonance radiation as exciter. The seattered spectrum exhibits two pairs of Doppler-shifted components, one pair due to the longitudinal sound waves, and the other due to the two sets of transverse sound waves which have very nearly the same velocity. The velocities of the longitudinal and transverse sound waves estimated from the observed frequency shifts of the displaced components are in agreement with those calculated from the elastic constants of diamond. The directional dependence of sound velocity in diamond has been quantitatively verified. Contrary to expectation, the longitudinal Doppler components are found to be less intense than the transverse Doppler eomponents.

Elastic constants of diamond

Employing the wedge method which has been developed by the authors, the elastic constants of diamond have been determined. The individual values are c 11 =9.5, c 12 =3.9, c 44 =4.3 x 10 12 dynes/cm. 2 . Force constants derived from the observed spectroscopic data in respect of diamond account very satisfactorily for the results.

Elastic Constants of Diamond

IN 1914 I published a paper1 on the lattice theory of diamond in which I obtained expressions for the elastic constants of this crystal in terms of two atomic force constants, one corresponding to the valency bond between next neighbour atoms, the other to the tetrahedral angle between the directions of these valencies.