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Elastic Constants of LiF from 4.2°K to 300°K by Ultrasonic Methods
Abstract
The values of the elastic constants of LiF at 4.2°K are found to be c11=12.46×1011, c44=6.49×1011, and c12=4.24×1011 dynes/cm2. At 300°K c11=11.12×1011, c44=6.28×1011, and c12=4.20×1011 dynes/cm2. A Debye characteristic temperature, thetaD, has been calculated from these values by using the Houston method of averaging the velocity over different directions in the crystal. At 4.2°K, thetaD=734°K in good agreement with the specific heat work of Martin. At 300°K, thetaD=708°K which is not in good agreement with values derived from specific heat measurements. A model has been suggested in which a combination of Debye and Einstein terms is used in the specific heat theory.
... Consistent with weak interactions between Debye bosons and phonons is a relatively weak temperature dependence of sound velocity vT(T) (compare Figure 3 and Figure 5). The initial slope of the sine function agrees perfectly with the measured sound velocity [14]. Note that sound velocity and phonon dispersions are measured by completely different experimental methods on different samples. ...
... In fact, in the low wave vector range a fairly linear function of wave vector can be seen. The slope of this line agrees reasonably with the measured sound velocity [14]. Note that sound velocities (and elastic constants) depend somewhat on the crystal perfection of the investigated sample. ...
... Similar analytical crossover events can be identified in the magnon dispersions of magnetic materials [15]. Figure 2 compiles calculated dispersion data of the Debye bosons with transversal polarization from different sound velocity data sets (dots [14], triangles [7]). ...
A method is described that allows one to construct the dispersion of the Debye bosons (sound waves) from the known temperature dependence of the sound velocities. The method simply assumes that the sound velocity measured at temperature T gives the slope of the dispersion relation at excitation energy E = kBT. The associated reduced wave vector is set equal to q/q0 = a0kBT/hvL/T. In this way the dispersion of the Debye bosons can be constructed for all thermal energies for which sound velocities vL/T(T) are known. This can be up to melting temperature. Surprisingly, for metals and for insulators the dispersion of the Debye bosons can continue up to wave vector values of several times the zone boundary. At melting temperature the wavelength of the Debye bosons is of the order of the atomic diameters. The sources of the Debye bosons therefore must have atomic dimensions. Spontaneous generation of Debye bosons by individual atoms is, however, a completely unexplored process. Interactions with the atomistic background of phonons or lattice defects provide damping to the Debye bosons and make the material specific sound velocity vL/T(T) additionally sample and temperature dependent. For practically all solids it is observed that elastic constants and vL/T(T) decrease as a function of increasing temperature. For high energies the dispersion relation of the Debye bosons therefore becomes visibly lower than linear. Interactions between Debye bosons and phonons can modify the dispersion of the acoustic phonons appreciably. Because of their different symmetries the dispersion relations of Debye bosons and acoustic phonons can attract each other. It is observed that for low wave vector values the dispersion of the acoustic phonons can assume the linear wave vector dependence of the Debye bosons. At the end of the linear section a functional crossover to a sine-like function of wave vector occurs.
... The refractive index n for LiF at 400 nm (25,000 cm −1 ) is 1.4 [41]. We can calculate the sound velocity of the substrate u = 6657 m/s, which is in good agreement with the reported value (6750 m/s at 160 K) [42]. ...
... Figure 2f illustrates that the frequency of the predominant LA phonon generated within the LiF substrate exhibits a linear increase from 294 to 50 K, followed by a nearly constant behavior from 50 to 4 K. This observed trend aligns well with the temperature-dependent behavior of C 33 in LiF [42]. In Figure 2g, it can be observed that the LA phonon mode frequency and the photoinduced ellipticity amplitude of the 9 nm thin film exhibit a state of relative stability from 294 K to 160 K, followed by an increase commencing at ∼160 K. Notably, a significant and rapid change is observed in the temperature range of 110 to 80 K, with subsequent cooling resulting in a diminishing rate of change. ...
We obtain the through-thickness elastic stiffness coefficient (C33) in nominal 9 nm and 60 nm BaFe2As2 (Ba-122) thin films by using picosecond ultrasonics. Particularly, we reveal the increase in elastic stiffness as film thickness decreases from bulk value down to 9 nm, which we attribute to the increase in intrinsic strain near the film-substrate interface. Our density functional theory (DFT) calculations reproduce the observed acoustic oscillation frequencies well. In addition, temperature dependence of longitudinal acoustic (LA) phonon mode frequency for 9 nm Ba-122 thin film is reported. The frequency change is attributed to the change in Ba-122 orthorhombicity (a−b)/(a+b). This conclusion can be corroborated by our previous ultrafast ellipticity measurements in 9 nm Ba-122 thin film, which exhibit strong temperature dependence and indicate the structural phase transition temperature Ts.
... By fitting the experimental data for pristine LiF presented earlier by Pohl [2] and Berman [3], the parameters of Umklapp process are then defined as = 3.2 which is very close to = 3 reported previously in [41]. The fitting parameters of the Umklapp process were a 0 = 1.033 ×10 28 The longitudinal and transverse phonon group velocities are taken as = v l 6602 and = v t 4938 m/s, respectively [45]. Then, the temperature is fixed at 300 K at which the corresponding thermal conductivity k = 11.3 ± 1.5 W m K , which is close to the values reported previously [46]. ...
We present thermal conductivity measurements of lithium fluoride single crystals irradiated by 710 MeV Bi ions at fluences ranging between 10¹⁰ and 10¹³ cm⁻². The thermal transport degradation due to irradiation damage was examined across two depths regions, on nanometer and micrometer scales, by picosecond time-domain thermoreflectance and modulated continuum wave thermoreflectance methods, respectively. The proliferation of swift heavy ion-induced structural defects (color centers) was characterized using optical absorption and photoluminescence spectroscopies. Klemens thermal model was applied to correlate the concentrations of color center defects to thermal conductivity reduction in irradiated LiF crystals.
... Superconductivity, one of the most intriguing material properties, has sparked countless studies since its discovery in 1911. [1] Superconductors are categorized as either conventional (if their behaviour can be explained by the Bardeen-Cooper-Schrieffer (BCS) theory [2] or its derivatives) or unconventional (otherwise). Based on BCS theory, materials with light elements are especially favourable for achieving superconductivity because these elements provide high frequencies in the phonon spectrum. ...
Hydrogen-rich compounds are promising candidates for high- T c or even room-temperature superconductors. The search for high- T c hydrides poses a major experimental challenge because there are many known hydrides and even more unknown hydrides with unusual stoichiometries under high pressure. The combination of crystal structure prediction and first-principles calculations has played an important role in the search for high- T c hydrides, especially in guiding experimental synthesis. Crystal structure AnaLYsis by Particle Swarm Optimization (CALYPSO) is one of the most efficient methods for predicting stable or metastable structures from the chemical composition alone. This review summarizes the superconducting hydrides predicted using CALYPSO. We focus on two breakthroughs toward room-temperature superconductors initiated by CALYPSO: the prediction of high- T c superconductivity in compressed hydrogen sulfide and lanthanum hydrides, both of which have been confirmed experimentally and have set new record T c values. We also address the challenges and outlook in this field.
... We also compare properties for a small set of materials with experiment, and these results are provided in the Supplemental Material [69] (Table S3). The mean absolute error in individual elastic constants could be up to 15 GPa [71][72][73][74][75][76][77][78][79][80][81][82]. ...
In this work we present a high-throughput first-principles study of elastic properties of bulk and monolayer materials mainly using the vdW-DF-optB88 functional. We discuss the trends on the elastic response with respect to changes in dimensionality. We identify a relation between exfoliation energy and elastic constants for layered materials that can help to guide the search for vdW bonding in materials. We also predicted a few novel materials with auxetic behavior. The uncertainty in structural and elastic properties due to the inclusion of vdW interactions is discussed. We investigated 11 067 bulk and 257 monolayer materials. Lastly, we found that the trends in elastic constants for bulk and their monolayer counterparts can be very different. All the computational results are made publicly available at easy-to-use websites: https://www.ctcms.nist.gov/∼knc6/JVASP.html and https://jarvis.nist.gov/. Our dataset can be used to identify stiff and flexible materials for industrial applications.
... From these investigations, it is known that LiF has a high melting temperature (∼3000 K at 1 Mbar) and remains in the B1 structural phase up to at least 1 Mbar, which makes LiF an excellent window material in shock wave experiments. Additionally, ultrasonic measurements [41][42][43] have been used to measure elastic moduli. ...
We perform first-principles path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) calculations to explore warm dense matter states of LiF. Our simulations cover a wide density-temperature range of ~gcm and ~K. Since PIMC and DFT-MD accurately treat effects of atomic shell structure, we find a pronounced compression maximum and a shoulder on the principal Hugoniot curve attributed to K-shell and L-shell ionization. The results provide a benchmark for widely-used EOS tables, such as SESAME, LEOS, and models. In addition, we compute pair-correlation functions that reveal an evolving plasma structure and ionization process that is driven by thermal and pressure ionization. Finally, we compute electronic density of states of liquid LiF from DFT-MD simulations and find that the electronic gap can remain open with increasing density and temperature to at least 15.7 gcm.
... The temperature distributions in the photoexcited regions were expressed by Gaussian functions. We used the density (ρ = 2.6 gcm −3 ), thermal expansion coefficient (β = 3.7 × 10 −5 K −1 ), and elastic constants (C 11 = 111 GPa, C 23 = 63 GPa and C 44 = 42 GPa) of a LiF single crystal [18]. ...
... The temperature distributions in the photoexcited regions were expressed by Gaussian functions. We used the density (ρ = 2.6 gcm −3 ), thermal expansion coefficient (β = 3.7 × 10 −5 K −1 ), and elastic constants (C 11 = 111 GPa, C 23 = 63 GPa and C 44 = 42 GPa) of a LiF single crystal [18]. ...
Crack formations inside a LiF single crystal after femtosecond laser irradiation at multiple points were investigated. In the case of sequential laser irradiation at three points, the propagations of some cracks were prevented by the dislocation bands generated by the previous laser irradiation. On the other hand, in the case of simultaneous laser irradiation at three points with a spatial light modulator, cracks in all the <100> directions from the photoexcited regions were generated clearly, but the length of one crack depended on the distribution of laser irradiation positions. The simulation of elastic dynamics after fs laser irradiation at three points elucidated that the interference of laser induced stress waves depended on the distributions of the irradiation positions. We found that the constructive interference of stress waves at a crack tip should have prevented the crack from propagating further and the tensile stress by destructive interference of stress waves along a crack should have facilitated the propagation of the crack.
Lithium halides are the important members of alkali halides with important industrial application. Here, we study the structural, elastic, electronic and optical properties of lithium halides, including LiF, LiCl, LiBr, and LiI, by using the first-principles calculations. The results of elastic constants and phonon spectra show that the calculated lithium halides are all mechanical and dynamical stability under ambient condition. Using the Voigt-Reuss-Hill method, we calculate shear moduli, bulk moduli and Young's modulus of LiF, LiCl, LiBr, and LiI compounds and they are all in the order of LiF > LiCl > LiBr > LiI. The calculated structural and mechanical parameters of LiF, LiCl, LiBr, and LiI compounds correspond well with the previous reported theoretical and experimental results. The electronic properties suggest that all the calculated lithium halides have wide direct gap with strong covalent bonding. The calculated optical properties indicate that LiI is the most suitable for the reflection of ultraviolet light.
Prediction of ionic-solid properties presents a great challenge to Kohn-Sham conventional density functional theory, due to the fact that there is strong van der Waals interaction between ion cores. In this work, we apply the Tao-Mo (TM) semilocal density functional to calculate lattice constants, bulk moduli, and cohesive energies for ionic solids consisting of alkali halides, silver halides, and other ionic solids. Our calculation shows that the TM functional yields accurate lattice constants, with a mean absolute error (MAE) of 0.087 Å, which is obviously smaller than commonly-used semilocal functionals such as the local spin-density approximation (LSDA) and Perdew-Burke-Ernzerhof generalized gradient approximation (PBE GGA). It is even more accurate than dispersion-corrected density functional PBE + D3 (MAE = 0.116 Å). This is unexpected success. For bulk moduli, TM functional can also yield very good accuracy, with an MAE of 3.02 GPA. Finally, we evaluate the cohesive energies of ionic solids. We find that this functional produces an MAE of 0.13 eV/atom, which is more accurate than the LSDA and PBE GGA, but less accurate than PBE + D3. We have also made a comparison of TM with other semilocal density functional theory (DFT) methods on several other ionic solids. It appears that TM also gives an overall improvement upon other popular semilocal DFT methods such as PBE for solids and Tao-Perdew-Staroverov-Scuseria.
The influence of the anisotropy of elastic energy on the phonon propagation and phonon transport in single crystal nanofilms with different types of anisotropy of elastic energy in a Knudsen flow of a phonon gas is studied. The angular distribution of phonon mean free paths in the planes of the films and in their cross section is analyzed. The physical reasons leading to the dependence of the thermal conductivity on the orientation of the film planes and the directions of the heat flux relative to the crystal axes are studied. An analysis of the effect of focusing on the phonon propagation made it possible to explain the qualitative difference between the anisotropy of phonon mean free paths in films of cubic nanocrystals of various types having different orientations of the planes.
The motion of atoms in materials is generally treated within the classical framework of Newton's laws and molecular dynamics. In this limit, atoms move with mean kinetic energy proportional to the temperature of the system, as in the case of a perfect gas of noninteracting particles. For systems with more quantized degrees of freedom, such as intramolecular motions, or in the descriptions of the influence of a finite kinetic energy of the atoms on the properties of solids, atomic dynamics may be described in terms of a normal-mode, harmonic oscillator picture, or treated with rigid constraints. However, there are a number of important examples where atoms should be treated as fully quantum objects, with their dynamics derived from a Schrödinger equation based on a well-defined interatomic interaction potential. Such an approach is mandatory in order to describe nuclear quantum effects, such as zero-point energies. The quantum nature of atoms can be quantitatively described by nuclear momentum distributions and mean kinetic energies of atoms. These observables are accessible using deep inelastic neutron scattering, a technique based on the use of epithermal neutrons available at instruments at spallation neutron sources. VESUVIO is the most successful example of this type of instrument and it is based at the ISIS Facility, UK. A description of deep inelastic neutron scattering and the VESUVIO spectrometer is presented here, together with the conceptual framework and a number of case studies to emphasize the role of nuclear quantum effects in condensed matter systems.
Summary This document is part of Subvolume A ‘Second and Higher Order Elastic Constants’ of Volume 29 ‘Low Frequency Properties of Dielectric Crystals’ of Landolt-Börnstein - Group III Condensed Matter.
Summary This document is part of Subvolume A ‘Second and Higher Order Elastic Constants’ of Volume 29 ‘Low Frequency Properties of Dielectric Crystals’ of Landolt-Börnstein - Group III Condensed Matter.
The influence of elastic energy anisotropy on phonon density of states (PDOS) and phonon mean free paths for acoustic modes in nanowires of cubic crystals with positive (Ge, Si, LiF, GaSb, GaAs, and GaN) and negative (CaF2, SrF2, and PbS) anisotropy of the second order elastic moduli has been investigated. The proposed calculation method allows one to determine angular distributions of the PDOS of fast and slow quasi-transverse modes in elastically anisotropic crystals for phonon focusing and defocusing regions. It has been shown that the PDOS and the phonon mean free paths in the cubic crystals take maximum and minimum values in the directions of phonon focusing and defocusing, respectively. Moreover, the anisotropy of the crystals differs qualitatively: the maxima values for the type-I crystals correspond to the minima for the type-II crystals. The heat flux directions that provide the maximum and minimum phonon thermal conductivity in the nanowires have been identified.
Recently, Tao and Mo (TM) proposed an accurate all-purpose nonempirical meta-generalized gradient approximation (meta-GGA). The exchange part was derived from the density matrix approximation, while the correlation part is based on a modification of TPSS correlation in the low-density or strong-interaction limit. To further understand this density functional, we combine the TM exchange part with the original TPSS correlation and make a comprehensive assessment of this combination, which we call TMTPSS functional, on solids and solid surfaces. Our test includes 22 lattice constants and bulk moduli, 30 band gaps of semiconductors, 7 cohesive energies, and surface exchange-correlation energies for rs ranging from 2 to 3 bohr. Our calculations show that TMTPSS functional is quite competitive to the TM meta-GGA functional, improving upon the nonempirical functionals LSDA, PBE GGA, and TPSS meta-GGA for most properties considered. In particular, it significantly improves the surface exchange-correlation energy calculation, with a mean absolute error of only 1 erg/cm2.
Lattice energies of LiCl, NaCl, KCl, RbCl, LiBr, NaBr, KBr, RbBr, CsBr, NaI, KI, RbI and CsI at different temperatures have been evaluated making use of the data of elastic constants of the single crystals and employing Kudriavtsev's theory. In all these alkali halides, the lattice energies are found to decrease with increase in temperature. At any given temperature, the lattice energy of LiCl > NaCl > KCl > RbCl, LiBr > NaBr > KBr > RbBr > CsBr, NaI > KI > RbI > CsI and RbF > RbCl > RbBr > RbI indicating that as the size of the alkali metal ion or halide ion increases, the lattice energy of the alkali halide crystal decreases. The results are discussed in the light of mutual interaction of the ions and increase in amplitude of lattice vibrations with temperatures.
The second-order elastic constants(SOECs)and third-order elastic constants(TOECs) of the alkalihalides(LiF, NaF, KF, LiCl, NaCl and KCl)with NaCl structure is presented using the density functionalthe-ory(DFT) and homogeneous deformation method. The k-points and the cutoff energy are given reliable verification while TOECs arecon-cerned. Fromthe nonlinearleast-square fitting, the elastic constants are extracted from a polynomial fit to the calculated strain-energydata. Both the SOEC sand TOECs are in agreement with the available experimental and previous results. The Lagrangian stress is changing linearly for small Lagrangian strain and it is found that the nonlinear effects play an impor-tant role while the finite strains are larger than approximately 0.025. The pressure derivatives of SOECs are calculated directly using the TOECs, which is used to discuss roughly the phase transition pressure of alkalihalide. The Grüneisen constants of long-wave length acoustic modes char-acterizing the anharmonic properties of alkali halide are also presented.
Summary This document is part of Subvolume A ‘Second and Higher Order Elastic Constants’ of Volume 29 ‘Low Frequency Properties of Dielectric Crystals’ of Landolt-Börnstein - Group III Condensed Matter.
Strain and stress dynamics inside MgO and LiF single crystals after photoexcitation by a focused femtosecond laser pulse were investigated by the observation of transient distributions of birefringence around the photoexcited region using a time-resolved polarization microscope. Both in MgO and LiF, propagation of two stress waves, which were attributed to quasi-longitudinal and quasi-transverse elastic waves, were observed, but crack propagation was observed only in LiF. Inside MgO, the observed strain distributions could be reproduced by elastic simulation, whereas inside LiF the strain distributions during crack propagation were largely different from the simulated ones; strain was widely distributed between cracks and the 〈110〉 regions in a stress wave, the strained region around the photoexcited region was smaller, and the strains in the 〈110〉 region and near the crack tips were larger than those by the simulation. The amplitudes of strain and stress in stress waves and temperature change in the photoexcited region were evaluated, and the origins of strain distribution change due to crack generation were discussed based on the differences between MgO and LiF.
This chapter discusses the elastic constants of crystals. The elastic properties of solid matter hold interest for both technology and basic research. In the first field applied elasticity is an important discipline for those fundamental considerations of engineering design which are usually included under the designation “strength of materials.” The treatment of structural materials requires semiempirical methods, because their compositions are complex and prior treatment has a pronounced effect. On the other hand, basic research into elastic properties is usually concerned with work on specimens in the simplest state which can be obtained reproducibly—for example, annealed single crystals. The complete tabulation of elastic constants for such specimens is valuable, not only for itself, but also because the data can be correlated with other physical measurements and thereby provide possible insight into the nature of the atomic forces in solid matter. The research aspect of elastic studies is of primary interest for this chapter. The case of the isotropic medium is important both for the chronological development of the subject of elasticity and also for its applicability to polycrystals and glasses.
Thermal expansion is the dimensional change that occurs with change in temperature. It is clearly related to dilatation, which may result from altering other parameters such as pressure, magnetic field, etc., and it reveals something about the dependence on volume of the energies of various interaction processes in solids. Glasses and diamond-structure solids present an interesting problem as they display negative expansion coefficients at moderately low temperatures. At least one of the diamond-structure solids shows a return to positive behavior in the low temperature limit and others seem likely to do the same thing, but whether glasses will also is unknown. Equally uncertain are which vibrational modes in glass are responsible for the negative behavior. Finally, magnetic materials and superconductors present a host of problems but existing data show clearly that accurate measurements, with and without magnetic fields, made on samples of suitable purity, can give useful information concerning the volume dependence of the interaction energies.
This chapter describes the lattice theory of the anharmonic effects. This theory contains an extension of the generally used harmonic approximation. The chapter discusses the calculation the thermal and caloric data of an ideal single crystal and to indicate the relationships among them. The methods of the harmonic theory are no longer applicable here, because the anharmonic terms cause a coupling between the different modes of oscillations. In the case of sound waves, there is a superposition of mechanically excited vibrations and thermal ones, which are already present. The anharmonicity causes an exchange of energy between thermal and mechanical vibrations. The mechanical sound waves therefore also loose energy to the thermal vibrations. Macroscopically, this leads to damping effects. The chapter describes the theory of the anharmonic effects that is as comprehensive as possible. The symmetry and invariance properties of the potential energy are discussed.
The effect of phonon focusing on the phonon transport in single-crystal nanofilms and nanowires is studied in the boundary scattering regime. The dependences of the thermal conductivity and the free path of phonons on the geometric parameters of nanostructures with various elastic energy anisotropies are analyzed for diffuse phonon scattering by boundaries. It is shown that the anisotropies of thermal conductivity for nanostructures made of cubic crystals with positive (LiF, GaAs, Ge, Si, diamond, YAG) and negative (CaF2, NaCl, YIG) anisotropies of the second-order elastic moduli are qualitatively different for both nanofilms and nanowires. The single-crystal film plane orientations and the heat flow directions that ensure the maximum or minimum thermal conductivity in a film plane are determined for the crystals of both types. The thermal conductivity of nanowires with a square cross section mainly depends on a heat flow direction, and the thermal conductivity of sufficiently wide nanofilms is substantially determined by a film plane orientation.
Density functional theory (DFT) is the work–horse of modern materials modeling techniques, but scattered
evidence indicates it often fails for important surface properties. This thesis investigates how DFT
estimates of the surface energy (σ) and molecular adsorption energies of ionic systems are affected by
the choice of exchange–correlation (xc) functional. Accurate diffusion Monte–Carlo (DMC) and quantum
chemistry (QC) calculations are presented for these quantities showing marked improvement over
DFT and agreement of much better than chemical accuracy.
DFT estimates of σ are presented for the (001) surfaces of LiH, LiF, NaF and MgO. Five xc functionals,
LDA, PBE, RPBE, Wu–Cohen and PW91 are used. A clear xc functional bias is demonstrated
with σLDA > σWC > σPBE > σPW91 > σRPBE.
To improve the picture detailed pseudopotential DMC calculations are presented for LiH and LiF.
The lattice parameters and cohesive energies agree with experiment to better than 0.2 % and 30 meV
respectively. For LiH novel all–electron DMC calculations are also presented showing significant improvement
over pseudopotential DMC.
Accurate all–electron Hartree–Fock calculations of σ for LiH(001) and LiF(001) are presented
along with calculations of the LiF bulk using specially adapted Gaussian basis–sets. Combined with
existing QC correlation estimates the bulk and surface properties of LiH and LiF show excellent agreement
to both experiment and DMC and allow a longstanding disagreement between two experimental
estimates for σLiF to be resolved.
Finally the adsorption energy curve for water on LiH(001) is obtained by both DMC and incremental
QC techniques leading to agreement of better than 10 meV. DFT and dispersion corrected DFT
estimates are also presented highlighting the large xc functional dependence.
Thus we demonstrate that is possible and necessary to obtain agreement between higher levels of
theory and produce benchmark values beyond DFT.
After the photoexcitation by a femtosecond laser pulse inside a LiF single crystal, four cracks appear in the <110> directions of the crystal from the photoexcited region. In our previous study, we found that a femtosecond laser-induced stress wave is responsible for generation and elongation of cracks inside a LiF single crystal. This finding suggests that we can control laser-induced cracks by modulating laser-induced stress waves. In this study, we applied parallel fs laser irradiation with a spatial light modulator to generate multiple stress waves at the same time, and found the modulation of crack formation; one crack became thinner and shorter than any other cracks. By a pump-probe imaging of dynamics of crack generation, we showed that the constructive interference of stress waves at a crack tip could compress the crack, which results in a thinner and shorter crack.
Lattice-dynamical calculations, using both the modified
deformation-dipole model and the breathing-shell model, of the lithium
halides are presented. Using the phonon frequencies and eigenvectors
generated from these calculations, the frequency and temperature
dependence of the Hermitean and anti-Hermitean parts of the q~0
transverse-optic-mode self-energy are calculated. The results of the
anharmonic calculations reveal that the zero-frequency self-energy shift
is dominated at low temperatures for all four halides by the first-order
quartic anharmonicity, but that, as the temperature is increased, the
second-order cubic and quartic anharmonic contributions become
increasingly more dominant, causing a changeover in sign for the
zero-frequency self-energy shift, a result observed experimentally for
other alkali halides crystalizing in the NaCl structure. This same
phenomenon is also found for the far-infrared self-energy shift for LiBr
and LiI, but for LiF and LiCl it appears that the second-order cubic and
quartic anharmonicity provides the dominant contribution to the
far-infrared self-energy shift at all temperatures. For all four
materials the second-order cubic anharmonicity provides the dominant
contribution to the inverse lifetime at low temperatures, but as the
temperature is increased the second-order quartic contribution becomes
increasingly important and finally provides the dominant contribution.
At low temperatures the calculations yield lifetimes which are much
longer than are actually observed, but for LiF, where experimental data
exist at temperatures up to 1100 K, the calculations are in reasonable
agreement for the range 200-1100 K.
The vibrational properties of crystals determine the photon infrared absorption, inelastic neutron scattering and, to a large extent, inelastic photon scattering by phonons, i.e., Raman scattering. The interpretation of infrared and Raman spectra requires, therefore, an understanding of the basic features of lattice dynamics The quantum theory of solids describes the crystal properties in terms of elementary excitations and their mutual interactions. Dynamic properties are represented by phonons (lattice vibrations) and their interactions mainly with other phonons (anharmonicity), electrons (electron-phonon coupling), and photons (interaction with radiation). This characterizes the scope of the article. Its emphasis is on the interrelation between theory and experiment, i.e., on the microscopic or model interpretation of experimental spectra.
Laser-induced deformation depends on the atomic structure of the
material. In amorphous materials, the deformation is random or
isotropic. On the other hand, in single crystals, anisotropic
deformations occur in the specific directions, which is because of their
regularly-arranged atomic structures. For example, when a fs laser pulse
is focused inside rock-salt type crystalline materials (MgO, LiF, etc.)
normal to the (001) plane, a void is formed in the photoexcited region
and highly concentrated dislocation bands and cleavages are formed in
the <110> and <100> directions, respectively. The directions
of the dislocation bands and cleavages are often explained by the slip
and cleavage planes of the crystals, however, stresses that induce these
modifications have not been elucidated. In this study, we observed the
dynamics of transient stress distributions after photoexcitation inside
various single crystals by a pump-probe polarization microscope. After a
femtosecond laser pulse was focused inside a MgO single crystal normal
to the (001) plane, a void appeared in the photoexcited region and two
stress waves (primary and secondary stress waves) were generated. In the
primary stress wave, which propagated faster than the secondary one, the
direction of the strain was identical to the propagation direction and
the stress direction depended on the direction from the photoexcited
region. In the secondary stress wave, there were tensile stresses normal
to the (100) planes, which were identical to the cleavage planes.
The three eleastic-constant changes, and the dilation, of LiF have been measured as a function of neutron flux. The elastic constants all decrease with flux, the change being linear for fluxes less than 1016 n/cm2. This is contrary to expectations based on considerations of Dienes and Nabarro, and consequently, it makes measurements of elastic constants less useful as a tool for distinguishing between interstitials and vacancies than had previously been expected. The ratio of the relative bulk modulus to the volume change is — 1.8, which is much smaller than some corresponding values previously reported for copper. A comparison of the experimental results with various available theories shows that a nonlinear elastic sphere-inhole calculation disagrees the least, the expectations from Zener's considerations come next, and the rest of the theoretical estimates differ in order of magnitude and even in sign from the experimental results.
The frequency wave-vector dispersion relation for the normal modes of vibration of Li7F at 298°K has been measured by means of slow-neutron inelastic scattering techniques. Triple-axis crystal spectrometers at two reactor facilities at the Oak Ridge National Laboratory were employed, mostly in the "constant-Q" mode of operation. The results can be satisfactorily fitted by a seven-parameter dipole approximation model involving nearest-neighbor and second-nearest-neighbor F--F- short-range forces, a variable ionic charge, and the polarizability of the F- ions only. The applicability of other force models to Li7F is also discussed. Certain inconsistencies have been found between the slopes of the acoustic branches near q=0 and the appropriate velocities of sound as measured by ultrasonic techniques. It is believed that these arise from anharmonic effects. The frequency distribution of the normal modes has been computed with high precision from the best-fit dipole approximation model, together with related quantities such as the lattice heat capacity, entropy, and Debye-Waller factors for each ion. Combined (two-phonon) density-of-states functions have also been calculated; comparisons are made between these one- and two-phonon distribution functions and certain features observed in optical and infrared absorption experiments on LiF.
A novel method, based on the Debye model of the density of the lattice vibration energy 1 and 2, is used to predict the thermal conductivity of insulator materials from room temperature up to the melting point. The model links the density of the lattice vibration energy and the mean free path of the phonons to the high temperature limit of the Debye temperature, θD¯(∞), and to the Grüneisen parameter, γ(∞). The phonon contribution to the thermal conductivity can be predicted from the knowledge of θD¯(∞) and γ(∞). The contribution of the present work is a new CALPHAD (CALculation of PHAse Diagrams) Method, based on physical models, where the heat capacity, the thermal expansion and the adiabatic bulk modulus are optimized simultaneously in order to calculate θD¯(∞) and γ(∞). In addition, a simple method to predict θD¯(∞) and γ(∞), and thus the thermal conductivity without any experimental data, is also presented. Results are given for the thermal conductivities of some typical insulator materials such as salts (halides), oxides and semiconductors. It is found that the agreement between the calculations and the available experimental data is excellent.
A quantum mechanical solution for the problem of inelastic scattering of atoms by solid surfaces is proposed. Only collisions with one-phonon exchange are considered; a criterion for the validity of this approximation is established. The problem is presented in its most general form; the one-phonon approximation is then introduced and solutions for the equations obtained under this assumption are given. The solution is not under the restriction of small moduli of matrix elements and is expressed in terms of a general potential. Absolute values for the differential cross-section of the He-LiF system are calculated. Results obtained are in good agreement with experimental values measured by other authors.
Anderson's theory has been used to explain the temperature dependence of the adiabatic bulk modulus of KCl, NaCl, LiF and TlBr in the temperature range up to the melting point of the material. In KCl and LiF, the value of δ which gives good agreement between theory and experiment is that which is obtained from the pressure dependence of Bs. In NaCl the value of δ for the best fit is the one obtained from the temperature coefficient of ln Bs. In TlBr Chang's relation δ = 2γ, however, gives a better agreement between the observed temperature dependence of Bs and the theoretical values.
Linear thermal expansions of eight alkali halides have been determined at liquid oxygen temperatures and at temperatures from 30 K down to 2 K. For temperatures T /20, where is the Debye temperature, the expansion coefficients are well represented by = AT + BT. Values are reported for the Gruneisen parameter = 3/C, where C/V is the heat capacity per unit volume and is the compressibility. For CsBr (b.c.c. structure) appears to be nearly independent of temperature, with a value of 2.0 but for the other crystals, which have the rock-salt structure, the parameter varies with temperature, chiefly between /10 and /5. At room temperature, lies between 1.45 and 1.7 but at low temperature this generally decreases to a value which is ca. -0.1 for RbI. +0.3 for KCl. KBr and KI and 1.0 for NaCl and NaI; LiF does not show this decrease, being 1.7. The values observed for are compared with those calculated from elastic constants and their pressure derivatives and the general behaviour of (T) is observed to conform qualitatively to the predictions of simple theoretical models of Born. Blackman and Barron.
The line width of the nuclear magnetic resonance in a single crystal of
LiF has been observed as 11 G for Li/sup 7/ and 12 G for F/sup 19/; these values
are constant with temperature between 1.4 and 80 deg K. The spin-lattice
relaxation time, T/sub 1/, has been measured by observing the exponential growth
of the absorption slgnai after saturation. The value of T/sub 1/ for Li/sup 7/
is 3.6 x 10/sup 4/ sec at 1.4 deg K and is 7.8 x 10/sup 2/ sec at 80 deg K; for F/
sup 19/ the value of T/sub 1/ is 1.8 x 10/sup 4/ sec at 1.4 deg K and is 3.0 x 10/
sup 2/ sec at 80 deg K. These values are for a steady field of 4 kG. (auth)
Lithium fluoride (LiF) is an important technological material. It is chemically stable, thermal insulator, having found application in the field of high pressure as a pressure transmitting medium. In the present paper, we report various thermophysical properties of rock salt structured LiF using plane wave pseudopotential density functional theory within local density approximation in conjunction with quasi harmonic Debye model. Ground state properties are studied using density functional theory. To include effect to ionic motion at finite temperatures, quasi harmonic Debye model is used. Calculated ground state and finite temperature/pressure thermophysical properties are found to be in good agreement with experimental and other theoretical results.
Effective models of the mechanical behavior of thermoelectric materials under device conditions require knowledge of temperature-dependent elastic properties. Between room temperature and 600 K, resonant ultrasound spectroscopy measurements of three skutterudite thermoelectric materials, i.e. n-type Co0.95Pd0.05Te0.05Sb3 (both with and without 0.1 at.% cerium dopant) and p-type Ce0.9Fe3.5Co0.5Sb12, showed that the Young's and shear moduli decreased linearly with temperature at a rate of −0.021 GPa/K to −0.032 GPa/K, and −0.011 GPa/K to −0.013 GPa/K, respectively. In contrast, the Poisson's ratio was approximately 0.22 for the three materials and was relatively insensitive to temperature. For temperatures >600 K, the elastic moduli decreased more rapidly and resonance peaks broadened, indicating the onset of viscoelastic behavior. The viscoelastic relaxation of the moduli was least for Ce-doped n-type material, for which grain boundary precipitates may inhibit grain boundary sliding which in turn has important implications concerning creep resistance. In addition, powder processing of the n- and p-type materials should be done cautiously since submicron-sized powders of both the n- and p-type powders were pyrophoric.
This paper discusses further the information that can be obtained about lattice harmonic frequency spectra from the experimental heat capacities reported in part I. A detailed study is made of sodium chloride, for which spectra of different analytic forms are fitted to the heat capacity within experimental error (0.2%), thus showing how closely the spectrum is determined by the heat capacity. The information obtained agrees with general features of the theoretical spectra of Kellerman, Karo and Hardy; but more important are the following general conclusions drawn from this example. The low-frequency expansions (alphanu^2+betanu^4+ldots) and moments mu_n derived in part II do not comprise all the information obtainable from the heat capacity, but suffice to determine a rough integral spectrum. This integral spectrum is determined a little more closely by the heat capacity itself, and the approximate shape of the differential spectrum can be determined not only at low but also at intermediate frequencies (hnu < 1/2kTheta_∞), provided that it is known from theory that at low frequencies the spectrum has the form alphanu^2+betanu^4+ldots if this expansion cannot be assumed, however, nothing can be deduced about the detailed spectral shape in any frequency range. Greater experimental accuracy would determine the spectrum more closely at low and intermediate frequencies, but (partly because of anharmonic effects) is still not likely to give much information at high frequencies. Further information must therefore be obtained from other types of experiment or from theoretical models. Particularly accurate information is seen to be contained in the negative moments, which are therefore tabulated for the five salts discussed in this series of papers: sodium chloride and iodide, and potassium chloride, bromide and iodide.
Measurements of the elastic constants from 4.2\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K have been made on single crystals of cesium bromide. The values of the elastic constants at 4.2\ifmmode^\circ\else\textdegree\fi{}K are {c}_{11}=3.350\ifmmode\pm\else\textpm\fi{}0.8%, {c}_{12}=1.025\ifmmode\pm\else\textpm\fi{}10%, and {c}_{44}=1.002\ifmmode\pm\else\textpm\fi{}0.8% in units of dynes/. The Debye temperature ({\theta{}}_{0}) at 0\ifmmode^\circ\else\textdegree\fi{}K as calculated from the elastic constants is 149.0\ifmmode^\circ\else\textdegree\fi{}K\ifmmode\pm\else\textpm\fi{}2\ifmmode^\circ\else\textdegree\fi{}K. The lattice energy at 0\ifmmode^\circ\else\textdegree\fi{}K is computed to be kcal/mole.
Interesting linear relationships have been found between second-order elastic stiffness constant C44 at room temperature and a crystallographic ratio. The rate of change between the two above-mentioned parameters is explained on the basis of lattice compressibility.
DOI:https://doi.org/10.1103/RevModPhys.31.675
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A pulse technique at 10 Mc/sec. has been used to measure acoustic velocity and attenuation of several alkali halides and Rochelle salt. Values for the elastic moduli of LiF, NaCl, BRr, and KI have been determined. The attenuation of the ultrasonic beam in these media is small. Corresponding measurements on the elastic moduli of Rochelle salt are reported. Under the assumption of plane wave transmission many of these moduli automatically correspond to those measured on foiled crystals by other methods. Attenuation is considerable in Rochelle salt, and appears to be excessively high for transmission involving the temperature-dependent c44 (foiled case). Where this modulus is involved marked effects have been observed from electric bias and mechanical pressure. A complete set of elastic moduli has been obtained and used to calculate the corresponding values for the moduli of compliance.
The velocities of propagation of 10-Mc/sec longitudinal and transverse sound waves in single crystals of rock salt have been measured over the temperature range from 60°K to 300°K by means of the ultrasonic pulse method. From these velocities the adiabatic elastic constants have been calculated, with the results that c11=4.828+/-0.015, c44=1.273+/-0.005, and c12=1.276+/-0.025 at 300°K in units of dyne/cm2×1011. The values of these elastic constants extrapolated to absolute zero temperature are in these units, 5.750, 1.327, and 0.986, respectively. The adiabatic compressibility has been calculated from c11 and c12 and is 4.07×10-12 cm2/dyne at 300°K. The Debye characteristic temperature is computed to be 292.6+/-0.5°K at T=0°K.
The pulse method has been applied to measure mechanical properties of NaCl, KBr, and KCl at 10 and 30 Mc/sec. The elastic constants of KBr have been measured in the low temperature region. The strain-optical constants have been measured by a traveling-wave method suggested by Mueller.
Houston's formula first derived for an approximate determination of the frequency spectrum of cubic crystals, essentially approximates the integral over the surface of a unit sphere of any function I(theta, varphi) which is invariant under the operations of the cubic symmetry group in terms of the values of I(theta, varphi) along the three directions (100), (110), and (111). In this paper approximate formulas are given for the integral in terms of the values of I(theta, varphi) along the above three and any or all of the (210), (211), and (221) directions. As an application, Debye temperatures Theta are calculated for nine cubic crystals. It appears that the Theta values calculated from the formula containing the values of I along all the above six directions may be expected to be correct to about 1% for crystals for which 0.25
Compressibility of eleven alkali halides and its variations with pressure and temperature have been determined by measurements by Bridgman's new method, up to 12,000 atm. for both 30° C and 75° C. The samples were all single simple cubic crystals each grown from the melt in a new way, described elsewhere. The error in the values of the compressibility at zero pressure, κ0, is probably less than one per cent; in the values for variation with pressure, ψ0, and temperature the error may be 5 and 20 per cent respectively. By extrapolation approximate values of κ0 for absolute zero are found. Periodic relations. Both κ0 and ψ0 when plotted against the alkali ion for a series of salts of the same halogen ion, or vice-versa, show similar behavior. The curves break sharply at the ion similar to argon (K or Cl) the rate of increase suddenly decreasing. This behavior is also shown by the grating space as measured by Davey, and tends to corroborate Bohr's theory of atomic structure according to which there is a discontinuity in atomic formation at argon, additional electrons going into inner shells.
The specific heats of lithium fluoride, sodium chloride and zinc sulphide have been measured within the temperature range 2°k to 30°k using a calorimeter of a novel type, upon which the crystal specimens were stuck with silicone grease. Ideally, these three substances have only a lattice contribution to their specific heats so that a direct comparison of the results with the theoretical predictions of the lattice theory of specific heats is possible. Good agreement between theory and experiment is found. The existence of a ‘T ’ region extending up to a temperature of about 20°k is shown clearly in the case of lithium fluoride.
- Clusius