No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Lattice Dynamics of Lithium Fluoride
Abstract
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.
... Core excitons in halite crystals like LiF and MgO have been determined to couple most strongly to X-point longitudinal optical phonons ω LO [63]. The phonon energies of LiF (ω LO = 80 meV [67,68]) and MgO (ω LO = 60 meV [69]) are of similar magnitude, as are the lifetimes of their core exciton signals. These similarities suggest that a phonon-mediated dephasing mechanism may also drive the rapid coherence decay of the LiF core exciton, as suggested in other studies. ...
... B4. Here, M i are constants describing the coupling between the bright exciton states X i and the phonons, N is the thermal phonon population, and ω LO = 80 meV is the Li X-point optical phonon energy [67,68]. ...
The ability to control absorption by modifying the polarization of light presents an exciting opportunity to experimentally determine the orbital alignment of absorption features. Here, attosecond extreme ultraviolet (XUV) transient absorption spectroscopy is used to investigate the polarization dependence of core exciton dynamics in LiF thin films at the Li+ K edge. XUV pulses excite electrons from the Li 1s core level into the conduction band, allowing for the formation of a p-orbital-like core exciton, aligned along the XUV light polarization axis. A sub-5 fs near-infrared (NIR) probe pulse then arrives at variable time delays, perturbing the XUV-excited states and allowing the coherence decay of the core exciton to be mapped. The coherence lifetimes are found to be ~2.4 +- 0.4 fs, which is attributed to a phonon-mediated dephasing mechanism as in previous core exciton studies. The differential absorption features are also shown to be sensitive to the relative polarization of the XUV and NIR fields. The parallel NIR probe induces couplings between the initial XUV-excited p-like bright exciton and s-like dark excitons. When crossed pump and probe polarizations are used, the coupling between the bright and dark states is no longer dipole-allowed, and the transient absorption signal associated with the coupling is suppressed by approximately 90%. This interpretation is supported by simulations of a few-level model system, as well as analysis of the calculated band structure. The results indicate that laser polarization can serve as a powerful experimental tool for exploring the orbital alignment of core excitonic states in solid-state materials.
... This can be concluded from the fact that magnons can be observed far above the magnetic ordering temperature [33,41]. [29][30][31][32]. The initial slopes agree quantitatively with the sound velocities (straight lines) measured at the same temperature as the phonon dispersions [12]. ...
... At the Debye cut-off energy at ~15.3 THz the dispersion is continuous. The dispersion of the LA-phonons, measured at T=298 K is given by circles [30]. The fitted sine-function includes a phase shift. ...
For non-magnetic solids the two experimental signatures of a non-negligible and decisive interaction between the Debye bosons (sound waves) and the (acoustic) phonons are discussed: 1.) for large thermal energies the dispersion of the mass-less Debye bosons is a weaker than linear function of wave vector, and 2.) for many cubic materials the dispersion of the acoustic phonons along [ζ 0 0] direction follows a perfect sine function of wave vector, which is known to be the dispersion of the linear atomic chain. Only the absolute phonon energies are due to the inter-atomic interactions. It is argued that the sine-function originates in a relatively weak Debye boson-phonon interaction. For a strong Debye boson-phonon interaction, the dispersion of the acoustic phonons assumes initially over a large q-range the linear dispersion of the Debye bosons, followed by an analytical crossover to the sine-function. As a consequence of the boson-controlled wave-vector dependence of the phonons, the temperature dependence of the heat capacity of the phonon system is also determined by the Debye bosons, and exhibits universal power functions of absolute temperature. Quantitative analyses of the dispersion relations of the mass-less Debye bosons (sound waves) of cubic materials along [ζ 0 0] direction show that the dispersion is a linear function of wave vector only for low energies. When all phonon modes are excited, that is, for thermal energies of larger than corresponds to the Debye temperature (ΘD), the dispersion of the Debye bosons follows a power function of wave-vector ~q x. For the exponent x the rational values of x=0, 1/4, 1/3, 1/2, 2/3 and 3/4 could firmly be established experimentally. The discrete values of x show that there are distinct modes of interaction with the phonons only. Quantitative analyses show that the temperature dependence of the heat capacity can be described accurately over a large temperature range by the expression cp=c0-B‧T-ε. The constants c0 and B are material specific and define the absolute value of the heat capacity. However, for the exponent ε the same rational value can be observed for materials with different chemical compositions and lattice structures. The finite temperature range of the cp=c0-B‧T-ε function and the rational exponents ε are the typical characteristics of a boson determined universal behavior. This universality must, however, be considered as a non-intrinsic dynamic property of the atomistic phonon system, arising from the Debye boson-phonon interaction. Safely identified values for ε are ε=1, 5/4 and 4/3. The discrete modes of the boson-phonon interaction are essential for the different universality classes of the heat capacity, i.e. for the different exponents ε. The fit values for c0 are generally larger than the theoretical Dulong-Petit value. Universal exponents are identified also in the temperature dependence of the coefficient of the linear thermal expansion, α(T). Since the universal power functions in the α(T) dependence are functions of absolute temperature and hold for the same thermal energies (temperatures) as the ~q x functions in the dispersion of the Debye bosons, it can be concluded that the Debye bosons determine also the temperature dependence of α(T). Our results show that the Debye bosons dominate the dynamics of the atomic lattice of the non-magnetic solids for all temperatures. The atomistic models restricting on the inter-atomic interactions therefore are neither sufficient to explain the phonon dispersion relations nor the detailed temperature dependence of the heat capacity.
... Often the acoustic phonons with transverse polarization follow perfect sine function of wave vector. As an example Figure 1 shows neutron scattering data of the acoustic phonons for LiF measured along cube edge at room temperature [13]. For transverse polarization (dots) phonon dispersion is excellently described by sine function of wave vector (solid curve). ...
... Dispersions of the acoustic phonons with transverse (T) and longitudinal (L) polarization of LiF measured along cube axis at T=298 K[13]. For transverse polarization (dots) phonon dispersion is perfectly described by sine function of wave vector. ...
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.
... but the atomic positions are given by: [105][106][107][108] and to the values calculated in [8]. ...
Describing electron-phonon interactions in a solid requires knowledge of the electron-phonon matrix elements in the Hamiltonian. State-of-the-art first-principles calculations for the electron-phonon interaction are limited to the 1-electron-1-phonon matrix element, which is suitable for harmonic materials. However, there is no first-principles theory for 1-electron-2-phonon interactions, which occur in anharmonic materials with significant electron-phonon interaction such as halide perovskites and quantum paraelectrics. Here, we derive an analytical expression for the long-range part of the 1-electron-2-phonon matrix element, written in terms of microscopic quantities that can be calculated from first principles. We show that the long-range 1-electron-2-phonon interaction is described by the derivative of the phonon dynamical matrix with respect to an external electric field. We calculate the quasiparticle energy of a large polaron including 1-electron-2-phonon interaction, and show that it can be written in terms of a 1-electron-2-phonon spectral function . We demonstrate how to calculate this spectral function for the benchmark material LiF. The first-principles framework developed in this article is general, paving the way for future calculations of 1-electron-2-phonon interactions from first principles.
... Also using classical MD, Young 18 studied ion damage of LiF crystals. Related to thermal properties of LiF, Nüsslein and Schröder 19 calculated the dispersion and phonon density of states (phDOS) via polarizable model of the inter-atomic interactions of LiF at 0 K. Dolling et al. 20 also calculated the phDOS of LiF via lattice dynamics and compared it to dispersion data derived from slow neutron inelastic scattering. In their work, the crystal has phonon content up to 20 THz with most of the low frequency content attributed to the F ion. ...
Given the unique optical properties of LiF, it is often used as an observation window in high-temperature and pressure experiments; and, hence, estimates of its transmission properties are necessary to interpret observations. Since direct measurements of the thermal conductivity of LiF at the appropriate conditions are difficult, we resort to molecular simulation methods. Using the Belonoshko et al. (2000) empirical potential validated against ab initio phonon density of states, we estimate the thermal conductivity of LiF at high temperatures (1000-4000K) and pressures (100-400 GPa) with the Green-Kubo method. We also compare these estimates to those derived directly from ab initio data. To ascertain the correct phase of LiF at these extreme conditions we calculate the (relative) phase stability of the B1 and B2 structures using a quasiharmonic ab initio model of the free energy. We also estimate the thermal conductivity of LiF in an uniaxial loading state that emulates initial stages of compression in high-stress ramp loading experiments and show the degree of anisotropy induced in the conductivity due to deformation.
... In the other insulators, the only acoustic phonons are active in the corresponding regime. [44], (b) diamond [45], (c) Si [46], (d) LiF [47], and (e) hcp 4 He [48]. Shaded region represents the energy-range corresponding to the Ziman regime which is between the conductivity maximum temperature and A. ...
When an ideal insulator is cooled, four regimes of thermal conductivity are expected to emerge one after another. Two of these, the Ziman and the Poiseuille, are hydrodynamic regimes in which collision among phonons are mostly Normal. It has been difficult to observe them, save for a few insulators with high levels of isotopic and chemical purity. Our thermal transport measurements, covering four decades of temperatures between 0.1 K and 900 K, reveal that sapphire displays all four regimes, despite its isotopic impurity. In the Ziman regime, the thermal conductivity exponentially increases attaining an amplitude as large as 35,000 W/Km. We show that the peak thermal conductivity of ultra-pure, simple insulators, including diamond, silicon and solid helium, is set by a universal scaling depending on isotropic purity. The thermal conductivity of sapphire is an order of magnitude higher than what is expected by this scaling. We argue that this may be caused by the proximity of optical and acoustic phonon modes, as a consequence of the large number of atoms in the primitive cell.
... Further, although the magnitude of scattering rates for the emission processes is practically insensitive to isotopes, they are largely suppressed due to the high energies of the LO phonons. In addition to LiH, we offer general guidelines for evaluating isotope effects on carrier transport in different materials [13,54]. Our work provides a physical mechanism for engineering the intrinsic carrier mobility limited by electron-phonon interactions, which can potentially facilitate various applications such as high-speed electronics and thermoelectric energy conversion. ...
Isotope effects on phonon properties and transport have been predicted and observed for decades. However, despite the crucial impact of electron-phonon interactions, the effect of isotopes on electron transport remains largely unexplored. Here, by using first-principles calculations, we theoretically predict that the electron mobility of lithium hydride (LiH) can increase by up to ∼100% as H3 is replaced with H1. This remarkable phenomenon is primarily attributed to the isotope engineering of the Fröhlich interaction by the mass-induced line shift of the longitudinal optical (LO) phonons. Notably, the isotope-dependent absorption of LO phonons dominates while the isotope-insensitive emission process is mostly suppressed due to energy conservation. We further propose general guidelines for evaluating isotope effects on carrier transport in different materials.
... Among these potentials, the empirical Tosi-Fumi/Born-Mayer-Huggins potential developed by Belonoshko et al. 36 is the only one that has been tested for simulations of both phonons and dislocations. The phonon dispersion relations for LiF based on the potential are calculated using GULP 37 and presented in Fig. 1, which shows good agreement with the experimental measurements conducted by Dolling et al. 38 There are six phonon branches: a slow transverse acoustic (STA) phonon branch, a fast transverse acoustic (FTA) phonon branch, a longitudinal acoustic (LA) phonon branch, two transverse optical (TO1 and TO2) phonon branches, and a longitudinal optical (LO) phonon branch. This potential has been used to investigate the thermal conductivity of LiF at high temperatures (1000-4000 K) and high pressures (100-400 GPa). ...
The dynamic interaction between phonons and dislocations in LiF has been studied using molecular dynamics simulations. The simulations have captured the strong dynamic interactions between low-frequency slow transverse acoustic phonons and dislocations that were observed in experiments. Simulation results reveal that the strong dynamic interaction is attributed to resonant interactions between dislocations and slow transverse acoustic phonons. Each dislocation segment is found to possess a set of resonant modes characterized by large-amplitude out-of-phase vibrations of atoms on both sides of the dislocation slip plane. The resonant frequencies associated with these modes exhibit a nearly linear distribution with respect to the mode order. Contrary to previous beliefs, the resonant frequencies of dislocations exhibit only a weak correlation with the dislocation length. Additionally, each dislocation exhibits a dominant resonant mode that corresponds to the strongest vibration mode in response to phonons. This dominant resonant mode is not always the first resonant mode with the lowest frequency. Its specific order depends on the dislocation length. Simulation results have also demonstrated that the resonant modes of dislocations can be influenced by the interactions from neighboring dislocations.
... Moreover, strength enhancement of solid helium under pressure has been documented (44), and analogous results were observed in hydrogen (45). Experimental studies of other low-Z systems, such as Li, LiH, and LiF, document significant contributions of both thermal (e.g., Debye-Waller) effects and quantum zero-point effects on their lattice dynamics (46,47). For phase III of hydrogen, anharmonic finitetemperature effects have been shown to reduce the electronic band gaps (24). ...
Solid molecular hydrogen has been predicted to be metallic and high-temperature superconducting at ultrahigh hydrostatic pressures that push current experimental limits. Meanwhile, little is known about the influence of nonhydrostatic conditions on its electronic properties at extreme pressures where anisotropic stresses are inevitably present and may also be intentionally introduced. Here we show by first-principles calculations that solid molecular hydrogen compressed to multimegabar pressures can sustain large anisotropic compressive or shear stresses that, in turn, cause major crystal symmetry reduction and charge redistribution that accelerate bandgap closure and promote superconductivity relative to pure hydrostatic compression. Our findings highlight a hitherto largely unexplored mechanism for creating superconducting dense hydrogen, with implications for exploring similar phenomena in hydrogen-rich compounds and other molecular crystals.
... Li-F vibration bands significantly weaken, with the 450 cm -1 band (assigned to F 2 molecules) also dropping. Then, the absorption at 500-550 cm -1 due to vibrations of Li-Li bonds in aggregates is retained in agreement with LiF lattice dynamics and density-ofstates function [13]. ...
Electron microscopy and optical vibrational spectroscopy techniques have been used to study Li nanoparticles formed on the cleaved surface of lithium fluoride (LiF) crystals irradiated by 4-MeV electron beam to a total fluence of 5 × 10¹⁶ cm–2. It is suggested that ionization accompanying this intense irradiation leads to radiolysis of the target and removal of fluorine from interblock boundaries, where lithium cations are collected and mutually oriented to form microwires. This functional surface may be of practical interest.
... Из-за радиолиза (вылета фтора) при температуре облучения выше 100 • C и плотности тока пучка электронов 1 µA/cm 2 число связей Li−F в приповерхностном слое 20 µm резко уменьшается, соответственно обе полосы Li−F резко ослабляются, полоса 450 cm −1 от колебаний молекул F 2 (междоузельные ионы фтора) тоже ослабляется. Тогда остается полоса 500−550 cm −1 от колебаний связей Li−Li в агрегатах, что согласуется с динамикой решетки и функцией плотности состояний LiF [13]. Это объясняет и подтверждает появление чешуек и нитей Li на микроснимках облученной поверхности и высокую нанотвердость сформированных тонких слоев на поверхности кристаллов LiF, облученных бета-частицами [9,10]. ...
Electron microscopy and optical vibrational spectroscopy techniques have been used to study Li nanoparticles formed on the cleaved surface of lithium fluoride (LiF) crystals irradiated by 4-MeV electron beam to a total fluence of 5 × 10^16 cm^–2. It is suggested that ionization accompanying this intense irradiation leads to radiolysis of the target and removal of fluorine from interblock boundaries, where lithium cations are collected and mutually oriented to form microwires. This functional surface may be of practical interest.
... Also using classical MD, Young 18 studied ion damage of LiF crystals. Related to thermal properties of LiF, Nüsslein and Schröder 19 calculated the dispersion and phonon density of states (phDOS) via polarizable model of the inter-atomic interactions of LiF at 0 K. Dolling et al. 20 also calculated the phDOS of LiF via lattice dynamics and compared it to dispersion data derived from slow neutron inelastic scattering. In their work, the crystal has phonon content up to 20 THz with most of the low frequency content attributed to the F ion. ...
Given the unique optical properties of LiF, it is often used as an observation window in high-temperature and pressure experiments; and, hence, estimates of its transmission properties are necessary to interpret observations. Since direct measurements of the thermal conductivity of LiF at the appropriate conditions are difficult, we resort to molecular simulation methods. Using the Belonoshko et al. (2000) empirical potential validated against ab initio phonon density of states, we estimate the thermal conductivity of LiF at high temperatures (1000-4000K) and pressures (100-400 GPa) with the Green-Kubo method. We also compare these estimates to those derived directly from ab initio data. To ascertain the correct phase of LiF at these extreme conditions we calculate the (relative) phase stability of the B1 and B2 structures using a quasiharmonic ab initio model of the free energy. We also estimate the thermal conductivity of LiF in an uniaxial loading state that emulates initial stages of compression in high-stress ramp loading experiments and show the degree of anisotropy induced in the conductivity due to deformation.
... The Li-F stretch in BaLiF 3 , where the Li-F distance is nearly the same as in LiF (2.02 Å in BaLiF 3 versus 2.02 Å in LiF [19]), occurs at 491 cm −1 for BaLiF 3 . However, the LiF frequency is 660 cm −1 [57]. The Li-H stretching frequency is sensitive to the Li-H distance in BaLiH 3 and SrLiH 3 . ...
The structural, mechanical, dynamical and thermal properties of CsCaF3 and CsCdF3 are presented by using an ab initio pseudopotential method and a linear response scheme, within the generalized gradient approximation. The obtained structural and mechanical properties are in good agreement with other available theoretical and experimental studies. The calculated elastic constants of these materials obey the cubic stability conditions. It has been found that CsCaF3 is brittle whereas CsCdF3 has ductile manner. The full phonon dispersion curves of these materials are reported for the first time in the literature.Wehave found that calculated phonon modes are positive along the all symmetry directions, indicating that these materials are dynamically stable at the cubic structure. The obtained zone-center phonon modes for CsCaF3 (IR data) are found to be 83 (98) cm-1, 104 (115) cm-1, 120 cm-1, 180 (192) cm-1, 231 (250.5) cm-1, 361 (374) cm-1, 446 (449) cm-1. Also, we have calculated internal energy, Helmholtz free energy, constant-volume specific heat, entropy and Debye temperature as function of temperature. At the 300 K, specific heats are calculated to be 113.36 J mol-1 K-1 and 115.58 J mol-1 K-1 for CsCaF3 and CsCdF3,respectively, which are lower than Doulong-Petit limit (12 472 J mol-1 K-1).
... The Li-F stretch in BaLiF 3 , where the Li-F distance is nearly the same as in LiF (2.02 Å in BaLiF 3 versus 2.02 Å in LiF [19]), occurs at 491 cm −1 for BaLiF 3 . However, the LiF frequency is 660 cm −1 [57]. The Li-H stretching frequency is sensitive to the Li-H distance in BaLiH 3 and SrLiH 3 . ...
The structural, mechanical, electronic, optical, and dynamical properties of BaLiF3, BaLiH3, and SrLiH3 cubic perovskite materials are theoretically investigated by using first principles calculations. Obtained results are in reasonable agreement with other available theoretical and experimental studies. The considered materials are found to be mechanically stable in the cubic structure. We found that all materials are brittle. The modified Becke–Johnson (mBJ) exchange potential has been used here to obtain an accurate band order. The calculated band-gap energy value of BaLiF3 (8.26 eV) within the mBJ potential agrees very well with the experimentally reported value of 8.41 eV. In order to have a deeper understanding of the bonding mechanism and the effect of atomic relaxation on the electronic band structure, the total and partial density of states have also been calculated. We have investigated the fundamental optical properties, such as the real ε 1(ω) and imaginary ε 2(ω) parts of the dielectric function, absorption coefficient α(ω), reflectivity R(ω), and refractive index n(ω) in the energy range from 0 to 40 eV within the mBJ potential. The band-gap energy obtained from the absorption spectrum is around 8.76, 3.99, and 3.31 eV for BaLiF3, BaLiH3, and SrLiH3 crystals, respectively. It should be noted that BaLiF3 could be a strong potential candidate as a laser material for the development of a vacuum-ultraviolet light emitting diode once direct transition is confirmed by experimental studies. Finally, we have calculated the lattice dynamical properties of BaLiF3, BaLiH3, SrLiH3, and SrLiF3 crystals. The full phonon dispersion curves of these materials are reported for the first time. Our results clearly indicate that the materials are dynamically stable, except for SrLiF3, in the cubic structure. The obtained zone-center phonon frequencies of BaLiF3, BaLiH3, and SrLiH3 accord very well with previous experimental measurements.
... where ν * is the typical phonon frequency (e.g., ν * ≈ 10 13 Hz along [110] direction [63]), x is the net travel distance in each hop, g is the geometric factor, f (f = [64], where Q j is the jumping possibility along angle θ j ) is the correlation factor, and E m (i,q) is the diffusion barrier of the defect (i,q). For 1D diffusion approximation (i.e., V − Li diffuses along [110] direction), z j =1 Q j cosθ j = 1 2 [cos(0) + cos(π )] = 0 and g = 1 2 . ...
Understanding the ionic conduction in solid electrolytes in contact with electrodes is vitally important to many applications, such as lithium ion batteries. The problem is complex because both the internal properties of the materials (e.g., electronic structure) and the characteristics of the externally contacting phases (e.g., voltage of the electrode) affect defect formation and transport. In this paper, we developed a method based on density functional theory to study the physics of defects in a solid electrolyte in equilibrium with an external environment. This method was then applied to predict the ionic conduction in lithium fluoride (LiF), in contact with different electrodes which serve as reservoirs with adjustable Li chemical potential ({\ensuremath{\mu}}_{\mathrm{Li}}) for defect formation. LiF was chosen because it is a major component in the solid electrolyte interphase (SEI) formed on lithium ion battery electrodes. Seventeen possible native defects with their relevant charge states in LiF were investigated to determine the dominant defect types on various electrodes. The diffusion barrier of dominant defects was calculated by the climbed nudged elastic band method. The ionic conductivity was then obtained from the concentration and mobility of defects using the Nernst-Einstein relationship. Three regions for defect formation were identified as a function of {\ensuremath{\mu}}_{\mathrm{Li}}: (1) intrinsic, (2) transitional, and (3) p-type region. In the intrinsic region (high {\ensuremath{\mu}}_{\mathrm{Li}}, typical for LiF on the negative electrode), the main defects are Schottky pairs and in the p-type region (low {\ensuremath{\mu}}_{\mathrm{Li}}, typical for LiF on the positive electrode) are Li ion vacancies. The ionic conductivity is calculated to be approximately {10}^{\ensuremath{-}31}\phantom{\rule{4pt}{0ex}}\text{S}\phantom{\rule{0.16em}{0ex}}{\text{cm}}^{\ensuremath{-}1} when LiF is in contact with a negative electrode but it can increase to {10}^{\ensuremath{-}12}\phantom{\rule{4pt}{0ex}}\text{S}\phantom{\rule{0.16em}{0ex}}{\text{cm}}^{\ensuremath{-}1} on a positive electrode. This insight suggests that divalent cation (e.g., ) doping is necessary to improve Li ion transport through the engineered LiF coating, especially for LiF on negative electrodes. Our results provide an understanding of the influence of the environment on defect formation and demonstrate a linkage between defect concentration in a solid electrolyte and the voltage of the electrode.
... Considering the calculating speed and precision reason, 3 * 3 * 2 supercell is implemented and 5 * 5 * 5 k-mesh point is selected to confine the Brillion zone. Comparison of curves between the two diagrams of shows that acoustic mode involves the motion mainly of F − ion, with very little contribution from Li + , and vice versa for the optic mode, which has the same trend as in Dolling et al.'s study [15]. Note that the room temperature is chosen in this calculation. ...
The influences of thermal neutron scattering data for BeF2 and LiF crystals on molten salt reactor physics are investigated in this work. Based on the structure parameters of BeF2 and LiF, the coherent scattering for both crystals is added to NJOY source code. The ENDF6 format thermal neutron scattering sub-libraries for both crystals are evaluated with their phonon spectra using LEAPR; the ACE format data are produced by NJOY subsequently. Finally, the effect of thermal neutron scattering of BeF2 and LiF crystals on k eff and spectrum are investigated. The result shows that thermal neutron scattering for bound state of BeF2 and LiF influence k eff and spectrum obviously. The elastic scattering cross section for bound state of crystals is smaller than free atom; it makes k eff decrease (1%–2%) and spectra be hardened. The higher temperature the bound state has, the smaller coherent elastic scattering cross section it gets; therefore, k eff decreases with temperature. It is suggested that the thermal neutron data of LiF and BeF2 should be taken into account for molten salt reactor.
An extreme ultraviolet pump and visible-light probe transmission experiment in crystalline LiF, carried out at the Free Electron Laser facility FERMI, revealed an oscillating time dependence of the plasmon mode excited in the high-density high-temperature electron plasma. The effect is interpreted as a fingerprint of the electron-ion interaction: the ion motion, shaped by the electron dynamic screening, induces, in turn, electron density fluctuations that cause the oscillation of the plasmon frequency at the timescale of the ion dynamics. Fitting the high resolution transmission data with an RPA model for the temperature-dependent dielectric function, which includes electron self-energy and electron-ion coupling, confirms the interpretation of the time modulation of the plasmon mode.
We develop a theoretical and computational framework to study polarons in semiconductors and insulators from first principles. Our approach provides the formation energy, excitation energy, and wavefunction of both electron and hole polarons, and takes into account the coupling of the electron or hole to all phonons. An important feature of the present method is that it does not require supercell calculations, and relies exclusively on electron band structures, phonon dispersions, and electron-phonon matrix elements obtained from calculations in the crystal unit cell. Starting from the Kohn-Sham (KS) equations of density-functional theory, we formulate the polaron problem as a variational minimization, and we obtain a nonlinear eigenvalue problem in the basis of KS states and phonon eigenmodes. In our formalism the electronic component of the polaron is expressed as a coherent superposition of KS states, in close analogy with the solution of the Bethe-Salpeter equation for the calculation of excitons. We demonstrate the power of the methodology by studying polarons in LiF and Li2O2. We show that our method describes both small and large polarons, and seamlessly captures Frohlich-type polar electron-phonon coupling and non-Frohlich coupling to acoustic and optical phonons. To analyze in quantitative terms the electron-phonon coupling mechanisms leading to the formation of polarons, we introduce spectral decompositions similar to the Eliashberg spectral function. We validate our theory using both analytical results and direct calculations on large supercells. This study constitutes a first step toward complete ab initio many-body calculations of polarons in real materials.
We develop a theoretical and computational framework to study polarons in semiconductors and insulators from first principles. Our approach provides the formation energy, excitation energy, and wave function of both electron and hole polarons, and takes into account the coupling of the electron or hole to all phonons. An important feature of the present method is that it does not require supercell calculations, and relies exclusively on electron band structures, phonon dispersions, and electron-phonon matrix elements obtained from calculations in the crystal unit cell. Starting from the Kohn-Sham (KS) equations of density-functional theory, we formulate the polaron problem as a variational minimization, and we obtain a nonlinear eigenvalue problem in the basis of KS states and phonon eigenmodes. In our formalism, the electronic component of the polaron is expressed as a coherent superposition of KS states, in close analogy with the solution of the Bethe-Salpeter equation for the calculation of excitons. We demonstrate the power of the methodology by studying polarons in LiF and Li2O2. We show that our method describes both small and large polarons, and seamlessly captures Fröhlich-type polar electron-phonon coupling and non-Fröhlich coupling to acoustic and optical phonons. To analyze in quantitative terms, the electron-phonon coupling mechanisms leading to the formation of polarons, we introduce spectral decompositions similar to the Eliashberg spectral function. We validate our theory using both analytical results and direct calculations on large supercells. This study constitutes a first step toward complete ab initio many-body calculations of polarons in real materials.
The conventional notion of elastic, coherent atom-surface scattering originates from the scattering particles acting as a quantum-mechanical matter wave, which coherently interfere to produce distinct Bragg peaks which persist at finite temperature. If we introduce inelastic scattering to this scenario, the result is that the surface particles become displaced by the scattering atoms, resulting in emission or absorption of phonons that shift the final energy and momentum of the scatterer. As the lowest-lying phonons are gapless excitations, the ability to measure these phonons is very difficult and this difficulty is exacerbated by the roughly 1-eV resolution found in high-energy helium scattering experiments. Even though the surface has in effect measured the presence of the scatterer which decoheres the particle, we retain the diffraction spots which are referred to as coherent scattering. How do we reconcile these disparate viewpoints? We propose an experiment to more precisely examine the question of coherence in atom-surface scattering. We begin with an initially coherent superposition of helium particles with equal probabilities of interacting with the surface or not interacting with the surface. The beams are directed so that after the scattering event, the atoms are recombined so that we can observe the resulting interference pattern. The degree to which phonons are excited in the lattice by the scattering process dictates the fringe contrast of the interference pattern of the resulting beams. We use semiclassical techniques to simulate and test the viability of this experiment and show that for a wide range of conditions, despite the massive change in the momentum perpendicular to the surface, we can still expect to have coherent (in the superposition sense) scattering.
The knowledge of the vibrational properties of a material is of key importance to understand physical phenomena such as thermal conductivity, superconductivity, and ferroelectricity among others. However, detailed experimental phonon spectra are available only for a limited number of materials, which hinders the large-scale analysis of vibrational properties and their derived quantities. In this work, we perform ab initio calculations of the full phonon dispersion and vibrational density of states for 1521 semiconductor compounds in the harmonic approximation based on density functional perturbation theory. The data is collected along with derived dielectric and thermodynamic properties. We present the procedure used to obtain the results, the details of the provided database and a validation based on the comparison with experimental data.
The lattice thermal conductivity of lithium fluoride (LiF) is accurately computed from a first-principles approach based on an iterative solution of the Boltzmann transport equation. Real-space finite-difference supercell approach is employed to generate the second- and third-order interatomic force constants. The related physical quantities of LiF are calculated by the second- and third- order potential interactions at 30 K–1000 K. The calculated lattice thermal conductivity 13.89 W/(m K) for LiF at room temperature agrees well with the experimental value, demonstrating that the parameter-free approach can furnish precise descriptions of the lattice thermal conductivity for this material. Besides, the Born effective charges, dielectric constants and phonon spectrum of LiF accord well with the existing data. The lattice thermal conductivities for the iterative solution of BTE are also presented.
A method has been developed to derive the fundamental absorption spectra of solids directly from experimental infrared data without need for resorting to computational evaluation. These characteristic lattice absorption spectra represent the true energy absorption spectra of solids and are, through Kirchhoff’s Law, directly related to the emission spectra of solids. As opposed to the traditional absorption coefficient (k) versus frequency curve which is regarded as the absorption spectrum for solids in the region of anomalous dispersion, the spectra, as derived by the proposed method, clearly resolve the transverse and optical modes of vibration. They also exhibit the characteristics of anharmonicity and damping. The method uses the combined reflection and transmission data on single crystal and thin film specimens. Its validity has been verified on a wide variety of solids. The most recent data obtained from emission, Laser-Raman or cold neutron scattering techniques, is in complete agreement with the derived values.
A method is advanced also to calculate the various modes of vibration of solids directly from elastic constants. It is simultaneously used to render further support for the developed characteristic energy absorption spectra.
The importance of these spectra is discussed with regard to problems in solid state physics (particularly damping characteristics, temperature effects and non-linear effects of atomic binding in solids).
The basic features of some wellknown phenomenological force constant models are reviewed. Their interrelation with attempts for ab initio calculations of the phonon-spectrum is discussed. A formal derivation of the dispersion curves in the harmonic Born-Oppenheimer approximation and results of an application of the method to LiF are given.
Engineered isotope variation is a pathway toward modulating lattice thermal conductivity (κ) of a material through changes in phonon-isotope scattering. The effects of isotope variation on intrinsic thermal resistance is little explored, as varying isotopes have relatively small differences in mass and thus do not affect bulk phonon dispersions. However, for light elements, isotope mass variation can be relatively large (e.g., hydrogen and deuterium). Using a first principles Peierls-Boltzmann transport equation approach, the effects of isotope variance on lattice thermal transport in ultra-low-mass compound materials LiH and LiF are characterized. The isotope mass variance modifies the intrinsic thermal resistance via modulation of acoustic and optic phonon frequencies, while phonon-isotope scattering from mass disorder plays only a minor role. This leads to some unusual cases where κ values of isotopically pure systems (6LiH,7Li2H, and 6LiF) are lower than the values from their counterparts with naturally occurring isotopes and phonon-isotope scattering. However, these κ differences are relatively small. The effects of temperature-driven lattice expansion on phonon dispersions and calculated κ are also discussed. This paper provides insight into lattice thermal conductivity modulation with mass variation and the interplay of intrinsic phonon-phonon and phonon-isotope scattering in interesting light atom systems.
We demonstrate numerically through rigorous coupled wave analysis (RCWA) that replacing the prism in the Otto configuration with gratings enables us to excite and control different modes and field patterns of surface phonon polaritons (SPhPs) through the incident wavelength and height of the Otto spacing layer. This modified Otto configuration provides us the following multiple modes, namely, SPhP mode, Fabry-Pérot (FP) cavity resonance, dielectric waveguide grating resonance (DWGR) and hybridized between different combinations of the above mentioned modes. We show that this modified grating-coupled Otto configuration has a highly confined field pattern within the structure, making it more sensitive to local refractive index changes on the SiC surface. The hybridized surface phonon polariton modes also provide a stronger field enhancement compared to conventional pure mode excitation.
Alkali halides are the best investigated family of crystals. Experiments are usually done at room temperature or at that of liquid nitrogen. Since the model theory (mainly dipolar models such as the SM or DM) as well as the microscopic theory [4.4] are highly developed (for recent reviews see [2.41, 4.8]) it seems to be worthwhile to measure phonons at helium temperature in a few alkali halides such as LiD, NaI etc. This would provide a basis for a quantitative analysis of the anharmonic properties of these crystals which is still in its infancy.
Current methods for the experimental investigation of surface vibrations and for the calculation of surface phonon dispersion curves have been reviewed in the preceding contributions. For each method some characteristic results were shown. Our present knowledge of surface vibrations is, however, not restricted to the few examples presented in the preceding chapters. The surface vibrations of a large variety of solids have been studied, both experimentally and theoretically, during recent years and a large amount of data has been accumulated. In this last chapter the attempt is made to provide a systematic and complete survey of the surface phonon dispersion curves of clean surfaces.
How can a paper on the Green’s function method applied to surface lattice dynamics fit to a workshop on “Ab-initio calculation of phonon spectra”? We think that the answer comes out from the very essence of the method. All what happens at the free surface of a solid, such as the change of force constants, of atomic equilibrium positions, of electronic structure, is due to intrinsic properties of the bulk Hamiltonian manifesting themselves through the symmetry breaking. Thus a full knowledge of the dynamical structure of the bulk, regardless whether obtained ab-initio or in a phenomenological way, should be sufficient to account for all the surface intrinsic dynamical properties. In the Green’s function (GF) method applied to surface problems the bulk dynamical structure is the basic ingredient, whereas the perturbation of an intrinsic surface, induced by the symmetry breaking, turns out to be fully described by the corresponding change in the translational invariance (TI) and rotational invariance (RI) sum rules.
In the past very little was known about the dispersion curves of surface phonons and how they interact with gas atoms colliding with the surface. Recent molecular beam scattering experiments now provide the first detailed information on surface phonon dispersion curves and on the inelastic cross sections. In this brief review we first summarize some results from the theory of surface phonons and of inelastic cross sections. Next we describe the helium time of flight apparatus used to study the surface phonon dispersion curves of the alkali halides. Results are presented for LiF, NaF and KCl. Finally the influence of selective adsorption on the inelastic intensities is discussed.
A study of spectral and laser properties of the LiF : F2- crystal at low temperatures has revealed an electronic – vibrational interaction of electrons of the F2- centre with the local vibration of the centre, which occurs against the background of coupling between electrons of F2- centres and lattice phonons. The interaction of electrons with the local vibration manifests itself in spectra in the form of narrow lines superimposed on wide electron – phonon lines that are due to the electron – lattice interaction. An anomalous behaviour of spectral LiF : F2- laser lines is also found at liquid nitrogen temperature upon selective excitation; this behaviour is explained by the difference in the probabilities of the lattice and local interactions.
LiF is an important component in the mixtures of the coolant, fuel, and carrier for molten salt reactor. The studies of diffusion properties and viscous flowability of liquid LiF provide significant guidance to the development of modern molten salt reactor. In this work, the structural characteristics, diffusion properties and viscous flowability of liquid LiF are investigated by means of molecular dynamics simulations with the Born-Mayer potential. The distribution of coordination numbers of Li+ ions for liquid LiF is statistically calculated, and it is found that the Li+ ions of 3 and 4 coordination numbers are in the majority. The micro-structural analysis indicates that the Li+ ions and their nearest-neighbor coordination F- ions mainly form planar and tetrahedral configuration. The diffusion behaviors of Li+ and F- ions are simulated as the temperature ranged from 900 K to 1400 K. The results show that the diffusion coefficients increase with the increase of the temperature for both Li+ and F- ions, and the diffusion coefficients of Li+ ions are a little higher than F- ions'. In addition, the effects of pressure on the diffusion coefficients are discussed. The viscosity of liquid LiF is obtained at a temperature range of 1000-1400 K. The results confirm that the higher temperature, the lower viscosity, which is consistent with the experiential rule.
Using Fourier-transform infrared spectroscopic
ellipsometry, the infrared lattice absorption of LiF and NiO was studied in the reststrahlen region. The transverse optical (TO) and longitudinal optical phonon energies, broadenings, and amplitudes were determined. Both materials also show a weak two-phonon absorption, which modifies the shape of the reststrahlen bands. The authors did not find any evidence of a splitting of the TO phonon in NiO due to antiferromagnetic ordering and place an upper limit of 17 cm−1 on this splitting.
This chapter reviews the thermal conductivity of nonmetallic crystals at temperatures comparable to or higher than the Debye temperature. It deals with the intrinsic behavior of such pure crystals at high temperatures. In such crystals, the dominant carriers of thermal energy are phonons and the dominant scattering mechanism to be considered is the intrinsic phonon–phonon scattering. This is a small section of the much larger problem of the thermal conductivity of nonmetallic solids and clearly it neglects possible heat transport by photons, charge carriers, polarons, and magnons. It also neglects other possible phonon scattering mechanisms such as isotopes, impurities, vacancies, charge carriers, dislocations, grain boundaries, and crystal boundaries. It presents the absolute value of the thermal conductivity, K, as determined by phonon–phonon scattering, the temperature dependence of K, the volume dependence of K, the change in K upon melting, and the minimum value of K. The chapter discusses a composite curve for the thermal conductivity versus temperature of pure KCl measured at a constant pressure of, say, one atmosphere.
The mean-square amplitudes (MSA) and mean-square velocities (MSV) of thermal vibrations have been calculated for the (001) surface of LiF, MgO, NaF, NaCl, NaI, RbF, and RbCl, using shell models. The high-temperature maximum surface enhancement of the MSA, as expressed by the ratio 〈u2⊥(K)〉surf/〈u2⊥(K)〉bulk, ranges from a low of about 1.5 for O− in MgO to a high of about 1.9 for Cl− in RbCl. The calculated surface enhancement of MSA for F− in LiF is 1.65, in good agreement with the value 1.71 drawn from Goodman's analysis of the data on H/LiF scattering obtained by Hoinkes, Nahr, and Wilsch.
We report a classical trajectory study of the molecular desorption of a vibrationaly excited HF adsorbed on a corrugated LiF(001) surface. The desorption is induced by either a vibrational deexcitation of the adsorbed molecule (vibrational desorption) or an active motion of the surface ion (thermal desorption). A brief description of the calculation of the gas molecule/surface interaction potential is first presented. By applying this potential, Hamilton's equations of motion can be numerically integrated with the initial conditions selected, thus the dynamics is simulated. The result gives a clear indication about the features of the evolution of the vibrational, rotational and translational energies with increase of time. The angular momentum and translational energy distributions of the desorbed molecule are presented. Desorption probabilities for different initial states of the adsorbed molecule are also predicted, which are influenced by the polar‐angle‐dependent potential and the excitation of the initial vibrational level. The surface phonon mode is found to be strongly coupled to the vibrational, rotational, translational modes of the molecule and surface bond.
The isotope effect on the lattice thermal conductivity for group IV and
group III-V semiconductors is calculated using the Debye-Callaway model
modified to include both transverse and longitudinal phonon modes
explicitly. The frequency and temperature dependences of the normal and
umklapp phonon-scattering rates are kept the same for all compounds. The
model requires as adjustable parameters only the longitudinal and
transverse phonon Grüneisen constants and the effective sample
diameter. The model can quantitatively account for the observed isotope
effect in diamond and germanium but not in silicon. The magnitude of the
isotope effect is predicted for silicon carbide, boron nitride, and
gallium nitride. In the case of boron nitride the predicted increase in
the room-temperature thermal conductivity with isotopic enrichment is in
excess of 100%. Finally, a more general method of estimating normal
phonon-scattering rate coefficients for other types of solids is
presented.
Surface phonon branches for the (001) face of LiF have been calculated
with a shell model a fifteen-layer film. The ``Lucas'' TO surface-mode
frequency at q = 0 is approximately equal to that of a small unexplained
dip in IR transmittance of a thin LiF film as observed by Berreman.
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.
Calculations of the frequency and temperature dependence of infrared
multiphonon absorption for the alkali halides and alkaline-earth
flourides are compared with previously published data and some new
results. The temperature dependence of the infrared absorption of pure
crystals of NaF, NaCl, and KCl at 10.6 μm from room temperature to
close to the melting point are reported. The calculations are based upon
a simple approach involving and interatomic Morse-potential function in
which lattice dispersion is introduced via a multiphonon frequency
distribution. Good agreement between experiment and theory is obtained
in which the "observed" lattice density-of-states and
thermal-expansion-coefficient data (which determine the degree of
anharmonicity) are the principal input to the calculations. Some
structure in the multiphonon spectrum is predicted for some compounds at
low temperatures, which has not yet been observed expermentally. Since
the Morse potential has an exact quantum-mechanical solution, the
anharmonicity is contained in the calculations without resorting to
perturbation theory.
The dispersion relation for the normal modes of vibration in AgCl at
78°K has been determined from neutron coherent inelastic scattering
data obtained on a triple-axis neutron spectrometer at the Oak Ridge
High Flux Isotope Reactor. Some difficulty in obtaining a satisfactory
fit to the results with a simple shell model was encountered. Only a
fair fit to the data is obtained with an 11-parameter model, which
includes next-nearest-neighbor interactions for both types of ions, a
variable ionic charge, and the electrical and mechanical
polarizabilities for both ions. A 13-parameter model, while providing an
improved fit to the data, gives values for several of the parameters
that have no obvious physical significance. The neutron data are in
reasonably good agreement with the results of low-temperature
measurements of the elastic constants and the infrared absorption. A
frequency-distribution function has been computed, together with the
lattice heat capacity and the mean-square displacements for each ion. A
combined density-of-states function has also been obtained for
comparison with the relevant optical data.
It is well known that the optical branches of the dispersion curves of ionic crystals
exhibit a polaritonic feature, i.e., a splitting about the electromagnetic dispersion line
ω = ck. This phenomenon is
considered to be due to the retardation of the electromagnetic forces among the ions.
However, the problem is usually discussed at a phenomenological level, through the
introduction of a macroscopic polarization field, so that a microscopic treatment is
apparently lacking. A microscopic first principles deduction is given here, in a classical
frame, for a model in which the ions are dealt with as point charges. At a qualitative
level it is made apparent that retardation is indeed responsible for the splitting. A
quantitative comparison with the empirical data for LiF is also given, showing a fairly
good agreement over the whole Brillouin zone.
With an eye on dust particles immersed into an ionized gas, we study the
effect of a negative charge on the scattering of light by a dielectric particle
with a strong transverse optical phonon resonance in the dielectric constant.
Surplus electrons alter the scattering behavior of the particle by their phonon
limited conductivity in the surface layer (negative electron affinity) or in
the bulk of the particle (positive electron affinity). We identify a
charge-dependent increase of the extinction efficiency for low frequencies, a
shift of the extinction resonance above the transverse optical phonon
frequency, and a rapid variation of the polarization angles over this
resonance. These effects could be used for non-invasive optical measurements of
the charge of the particle.
A three-body-force shell model (TSM) has been employed for a comprehensive and unified calculation of the crystal mechanics of some alkali halides. This includes the description of phonon dispersion, two-phonon Raman and infrared spectra, specific heats, cohesive energy, relative stability, phase-transition pressure and volume, dielectric constants, quasi- harmonic elastic and photoelastic properties of lithium fluoride, potassiumchloride and rubidium chloride. These predictions are reasonably good and have led to the conclusion that TSM is an appropriate model for lattice mechanical descriptions of ionic crystals.
Two simple methods, based on simple crystal models, for estimating the longitudinal optical surface vibration mode frequencies which lie in a LALO gap are examined. One method consists of deriving and solving a single-variable equation for the surface mode frequency, which depends only on the experimental bulk dispersion curves. The other method consists of fitting the known dispersion curves of a simple one-dimensional model, for which the optical surface mode frequency is given by a simple expression, to experimental bulk dispersion curves of more complex crystals. The results are compared with detailed theoretical surface mode calculations. The latter method is found to give better estimates.
Phonons in sodium fluoride have been studied at room temperature using inelastic neutron scattering techniques. Consistent results were obtained using a cold neutron time-of-flight apparatus and a triple-axis spectrometer. The time-of-flight results were interpolated on to symmetry directions from the observed scattering surfaces. The frequencies in units of 1012 sec-1 of some typical phonons are: TA(0,0,1), ν=4.39±0.04; LO(0,0,0.984), ν=8.52±0.15; LA(0,0,0.972), ν=7.94±0.18; TO(0.488, 0.488, 0.488), ν=6.19±0.07. The optical branches extrapolated to small wave vector are in agreement with the infrared absorption frequency and the Lyddane-Sachs-Teller relation. Hardy and Karo's deformation-dipole model is in agreement with the results to within 6%, but the rigid-ion model differs by as much as 19%. The results are well fitted by a shell model containing nine parameters in which the ionic charge is 0.91.
A new method of calculating absolute phonon frequency-distribution functions, which is an extension of the extrapolation method developed by Gilat and Dolling, is presented for cubic crystals. The method involves dividing the irreducible section of the first Brillouin zone into a cubic mesh and approximating the constant-frequency surfaces inside every small cube by a set of parallel planes. This method proves to be of high precision and resolution in obtaining fine details associated with a given model, and it requires relatively short computing time. Applications have been made to nickel, aluminum, and sodium, for which there exist satisfactory force-constant models. New critical points have been found for Al at ν=7.104±0.006 THz and for Na at ν=2.856±0.010 THz. Certain critical points associated with the longitudinal phonon band have been resolved more sharply than in earlier calculations.
If we introduce the isotropic deformation of ions as a new degree of freedom, we can show that the normal modes can be calculated from a few macroscopic parameters and the results agree with the neutron scattering measurements. The calculations are carried out for KBr and NaI.RésuméAm Beispeil von KBr und NaI wird gezeigt, daβ durch die Einfuhrung der isotropen Deformation als neuen Freiheitsgard die Normalschwingungen mit wenigen makroskopischen Parametern serh befriedigend beschrieben werden können, ohne daβ eine Anpassung an Neutronenbeugungsmessungen erforderlich ist.
The theory of the physical properties of an anharmonic crystal is
discussed by using the thermodynamic Green's functions for the phonons. A
penturbation procedure is developed to obtain the Green's functions and it is
shown that for some purposes a quasi-harmonic approximation is useful, in which
the frequencies of the normal modes are those determined by infra-red or neutron
spectrometry. The thermodynamic, elastic, dielectric, and scattering properties
of an anharmonic crystal are discussed in terms of the Green's functions, and
detailed expressions are given for the more important contributions. Detailed
numerical calculations are presented of the thermal expansion, dielectric
properties, and shapes of some of the inelastically scatiered neutron groups, for
sodium iodide and potassium bromide. The calculations, which give reasonable
agreement with experiment, show that even at quite low temperatures, the
lifetimes of some of the normal modes can be quite short. By using the quasi-
harmonic approximation it is shown that the large temperature dependence of the
normal modes in a ferroelectric crystal can be treated adequately. (auth)
It is shown that for a crystal, under the assumption of harmonicity for the interatomic forces and as a consequence of the periodic structure, the frequency distribution function of elastic vibrations has analytic singularities. In the general case, the nature of the singularities depends only on the number of dimensions of the crystal. For a two-dimensional crystal, the distribution function has logarithmically infinite peaks. In the three-dimensional case, the distribution function itself is continuous whereas its first derivative exhibits infinite discontinuities. These results are elementary consequences of a theorem of Morse on the existence of saddle points for functions defined on a torus.
The simple classical theories of the dielectric constants and compressibility of ionic crystals lead to two relations among the experimental quantities from which arbitrary parameters have been eliminated, the Szigeti relations. Neither is satisfied by the data, indicating the inadequacy of these simple theories. The short-range repulsive interaction between ions with closed shell electron configurations is investigated, and an approximate interpretation of the Born-Mayer potential in terms of overlap integrals is developed. These results are applied to the interaction of model ions consisting of rigid charged shells bound to cores by harmonic restoring forces. Using this model, polarization mechanisms neglected in the simple dielectric constant theory, the "short range interaction polarization," and the "exchange charge polarization" are described. Both arise from charge redistributions occurring when the ions move with resulting changes in electron overlaps. Applied to a crystal, these ion models permit the derivation of generalizations of the Szigeti, Clausius-Mossotti, and Lorenz-Lorentz relations. The e*e of the second Szigeti relation can then be calculated and comparison with the e*e values derived from experimental data imply that the above polarization mechanisms must be at least in part responsible for the deviation of this parameter from unity. The failure of the first Szigeti relation is discussed and attributed to the inadequacy of the treatment of compressibility. The additivity feature of the simple theory and its absence in the refined theory are discussed in relation to the so-called vacuum and crystal ion polarizabilities.
Phonon frequencies for wave vectors along the principal symmetry directions in copper have been determined at 49 and 298\ifmmode^\circ\else\textdegree\fi{}K from neutron inelastic-scattering measurements. In general, the temperature dependences of the frequencies were found to be smaller for the higher-frequency modes. For the lower frequencies (\nu\lesssim{}3\ifmmode\times\else\texttimes\fi{}{10}^{12} cps), the frequency changes measured are consistent with the 3-4% changes estimated from the isothermal elastic constants. For higher frequencies the relative changes are much smaller, often being 1% or less. Axially symmetric force models, which included interactions to the sixth nearest neighbors, were fitted to the data and have been used to calculate a frequency distribution function g(\nu at each temperature. A comparison of the temperature dependences of the moments of these distributions with various Gr\"uneisen parameters leads to the conclusion that Cu does not satisfy the assumption of the quasiharmonic model. The Debye temperature {\Theta{}}_{C} versus temperature curve calculated with the 49\ifmmode^\circ\else\textdegree\fi{}K g(\nu is in excellent agreement with results from specific-heat measurements in the entire 0 to 298\ifmmode^\circ\else\textdegree\fi{}K range. A fairly strong temperature dependence for the widths of some well-focused phonons was observed.
The long wave-length, polar lattice vibrations of alkali halide crystals are discussed without making any specific assumptions about the detailed interactions between the ions. This is made possible by the introduction of the effective charge, e*, of an ion defined as follows: All of the positive ions in a crystal slab are displaced by an equal amount in a direction perpendicular to the faces of the slab and all of the negative ions in the opposite direction. Then e* is the ratio of the dipole moment per ion pair induced in the slab by this displacement to the relative displacement of the positive and the negative ions. Expressions are obtained for the frequency, omegal, of the longitudinal vibration and the frequency, omegat, of the transverse vibration in terms of the dielectric constant, k, of the crystal, the dielectric constant, k0, obtained by extrapolating the square of the index of refraction of the crystal from high frequencies to zero frequency, and e*. The ratio of the two frequencies is found to be independent of e* and given by omegalomegat=(kk0)12.
DOI:https://doi.org/10.1103/PhysRevLett.11.275
Measurements of the elastic constants of the alkali halogenides with improved methods are reported. Particular emphasis is placed upon the methods of preparation and the discussion of the sources of error.
In addition to 0-phonon lines associated with the R2 and N bands of a number of the alkali halides at low temperature, other subsidiary peaks often occur on the high-energy side of the 0-phonon line. We have observed such peaks on the R2 and N bands of LiF, NaCl, KCl, and KBr, and on the N bands only of NaF and KI. It is found with very few exceptions that energy separations between the 0-phonon lines and associated subsidiary peaks agree very well with theoretical normal lattice phonon energies near the zone boundary as determined by Karo and Hardy. It is felt that this is reasonable, as these are the lattice vibrations of highest probability density and of wavelength comparable to small lattice defects. The most common lattice coupling to the defects appears to be through transverse acoustical phonons propagating in 〈100〉 directions, although the results show that in most of the materials the defects are coupled with comparable strength to several phonon modes. Indeed, possible correlations with all normal modes are indicated in the data. The exceptions mentioned (on the N bands of NaCl, KCl, and KBr) are all considerably smaller in energy than any of the lattice phonons at the zone boundary. These may be associated with pseudolocalized modes due to resonance scattering. In addition to the structure listed above, 0-phonon lines alone have been observed on the R2 bands of NaF, KF, and KI.
The one-phonon differential scattering cross section for the coherent scattering of thermal neutrons by an anharmonic Bravais crystal is obtained correct to the lowest nonvanishing order in the anharmonic force constants. Cubic and quartic anharmonic terms are retained in the crystal's Hamiltonian. It is found that the delta-function peaks in the energy distribution of the scattered neutrons for a fixed momentum transfer (which occur at the unperturbed phonon energies) in the harmonic approximation are broadened and their positions are shifted in an anharmonic crystal. Some numerical results for the magnitudes of the phonon widths and shifts are obtained for a simple model of a face-centered cubic crystal.
Theoretical and experimental studies were made of the lattice dynamics ; of alkali halides. A theory of the lattice dynamics of ionic crystals is given ; based on replacement of a polarizable ion by a mcdel in which a rigid shell of ; electrons (taken to have zero mass) can move with respect to the massive ionic ; core. The dipolar approximation then makes the model exactly equivalent to a ; Born-von Karman crystal in which there are two "atoms" of differing charge at ; each lattice point, one of the "atoms" having zero mass. The model was ; specialized to the case of an alkali halide in which only one atom is ; polarizable, and computations of dispersion curves were carried out for Nal. The ; dispersion nu (q) relation of the lattice vibrations in the symmetric STA001!, ; STA110!, and STA111! directions of Nal at 110 deg K were determined by the ; methods of neutron spectrometry. The transverse acoustic, longitudinal ; acoustic, and transverse optic branches were determined completely with a ; probable error of about 3%. -ine dispersion relation for the longitudinal optic ; (LO) branch was determined for the STA001! directions with less accuracy. The ; agreement between the experimental results and the calculations based on the ; shell model, while not complete, is quite satisfactory. The neutron groups ; corresponding to phonons of the LO branch were anomalously energy broadened, ; especially for phonons of long wavelength, suggesting a remarkably short lifetime ; for the phonons of this branch. (auth);
The heat capacities of LiF and KI single crystals have been measured over the temperature range 2-7°K. An adiabatic calorimeter was employed making use of a mechanical contact "heat switch." A carbon resistor embedded directly in the specimen was used as thermometer. For LiF the heat capacity was found to be proportional to the cube of the temperature over the whole range of temperatures investigated, and the value of theta0 was found to be 722°K. For KI variations from the T3 dependence were found above 3°K. An extrapolation of a smooth curve through the data to zero degrees temperature indicates a value of theta0=128°K. These values of theta0 are compared to values of theta0 derived from elastic constants data.
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.
It is well known that if the interaction between electrons in a metal is neglected, the energy spectrum has a zonal structure. The problem of these "Brillouin zones" is treated here from the point of view of group theory. In this theory, a representation of the symmetry group of the underlying problem is associated with every energy value. The symmetry, in the present case, is the space group, and the main difference as compared with ordinary problems is that while in the latter the representations form a discrete manifold and can be characterized by integers (as e.g., the azimuthal quantum number), the representations of a space group form a continuous manifold, and must be characterized by continuously varying parameters. It can be shown that in the neighborhood of an energy value with a certain representation, there will be energy values with all the representations the parameters of which are close to the parameters of the original representation. This leads to the well-known result that the energy is a continuous function of the reduced wave vector (the components of which are parameters of the above-mentioned kind), but allows in addition to this a systematic treatment of the "sticking" together of Brillouin zones. The treatment is carried out for the simple cubic and the body-centered and face-centered cubic lattices, showing the different possible types of zones.
The frequencies of the normal modes of vibration of lead telluride propagating in certain symmetric directions have been determined by inelastic neutron scattering techniques. The results have been used to deduce the parameters of rigid ion models and of shell models for the interatomic forces. The latter models take account of the polarizability of the ions. Neither type of model is very successful, but the best shell model does provide a good fit to the results and has been used to calculate the frequency distribution and specific heat. Calculations of the thermal expansion, and of the lifetime of the transverse optic modes, show that anharmonic interactions in lead telluride are considerably more complex than in the simple alkali halides. A detailed account of the effect of doping on the lattice vibrations is given, in which it is shown that large effects are limited to the long wavelength regions of the dispersion curves. Detailed calculations are presented for the dispersion of the optical branches as a function of the dopant concentration. The importance of these effects for different kinds of experimental studies is briefly discussed.
The pressure derivatives of the elastic stiffness constants of LiF and NaF have been measured by the ultrasonic pulse echo technique. The values found are: View Within ArticleThe notation C = C44, , and has been used. These pressure results have been related to the temperature dependence of the elastic constants and to the thermal expansion. The data for the elastic constants and their pressure variations have also been analyzed in terms of the classical Born model. The theoretical Coulomb contributions have been subtracted from the data and the resulting short-range terms interpreted in terms of nearest and next-nearest neighbor interactions. Particularly useful in this analysis was the elastic stiffness C and its pressure derivative . The analysis indicates that next-nearest-neighbor interactions are definitely necessary in order to account for the observed experimental values for LiF but not for NaF.
Vibrational distribution functions are derived for a number of rocksalt-; structure alkali halides using a more refined treatment of the interionic forces ; than that provided by regarding them as rigid point charges. The dipole moment ; at any given ion site is calculated taking into account the contribution from the ; deformation of the electron distribution resulting from both polarization and ; overlap repulsion between nearest neighbors. In this way the dipole-dipole part ; of the Coulomb interaction is treated self-consistently. Both room-temperature ; and 0 deg K input parameters are used, and the derived specific heat data are ; compared with experimental results. The over-all agreement with experiment is ; significantly better than that obtained by treating the ions as rigid point ; charges. Sets of phonon dispersion curves are also given. For NaI they are in ; much better agreement with those determined experimentally by inelastic neutron ; scattering than are the rigid ion curves. There appears to be close agreement ; with the results of the shell model'' calculations. (auth);
The "shell model" of an alkali halide is extended to take into account short-range forces between both first- and second-nearest neighbor atoms in the crystal, the polarizability of both ions, and the possibility that the ionic charge may be less than one electronic charge. The arbitrary parameters of the model have been obtained by means of a least-squares fit to the measured dispersion relations for the lattice vibrations, the dielectric constants, and the elastic constants. For NaI and KBr, models have been found which give excellent agreement both with these measurements and with measurements of the specific heat. This good agreement is, however, obtained only when the simple shell-model concept of the ions is to some extent abandoned. The reasons for this, and the connection with work of other authors, are discussed.
Dispersion curves for the lattice vibrations propagating in the [00zeta], [zetazeta0], and [zetazetazeta] directions in NaI at 100°K and in KBr at 90°K have been measured using neutron spectrometry, and the results compared with calculations based on a simple shell model. Both substances obey the Lyddane-Sachs-Teller relation. In addition, some measurements were made on KBr at 400°K most frequencies showed a decrease of a few percent. At this temperature, the acoustical modes show no significant energy broadening, the transverse optical modes show slight broadening, and the longitudinal optical modes show very considerable broadening. The anomalous broadening of the LO modes is not yet understood and requires further study. It appears to be specimen-dependent as well as temperature-dependent.
The frequency/wave vector dispersion relation nuj(q) for the normal modes of vibration of pure potassium iodide at 90°K has been measured by inelastic neutron scattering techniques. The experiments were performed with both a time-of-flight machine and a triple-axis crystal spectrometer. The results obtained from the latter spectrometer have been analyzed in terms of various interionic-force models. A satisfactory fit to the results is provided by an eleven-parameter dipole approximation model, which allows for the polarizability of both ions. Comparison of normal-mode frequencies calculated from this model with those observed in the time-of-flight experiments shows generally good agreement. The frequency distribution function for the normal modes, computed from the best-fit model, displays a well-defined energy gap, from 8.65 to 11.85 meV, separating the acoustic and optic modes. The moments of this function are in excellent agreement with values derived from heat-capacity data. By assuming an exponential form for the nearest-neighbor short-range force constants, the temperature variation of the thermal expansion, and the frequency shift and inverse lifetime of the transverse optic mode of very long wavelength, have been calculated and compared with the available experimental results. The occurrence of localized impurity modes in the energy gap in the frequency distribution for potassium iodide doped with potassium nitrite is briefly discussed.
A general discussion is given for the angular and energy distribution of neutrons inelastically scattered by a crystal, with special emphasis on those features of the distribution in which the dynamical properties of the crystal manifest themselves most immediately. The direct relationship between the energy changes in scattering and the dispersion law of the crystal vibrations is analyzed. While for x-rays, due to the extremely small relative size of these energy changes, the dispersion law has to be inferred indirectly from intensity measurements, it is shown that the very much larger relative magnitude of energy transfers in the case of slow neutrons opens the possibility of direct determination of the frequency-wave vector relationship and the frequency-distribution function of the crystal vibrations by energy measurements on scattered neutrons. The general properties of the outgoing neutron distribution in momentum space which are relevant for this purpose are derived by first considering the particularly instructive limiting case of neutrons initially at rest and subsequently generalizing the results to incident neutrons of arbitrary energy.
Vibronic structure observed in the emission and absorption spectra of alkali halides containing either Sm2+, Eu2+, or Yb2+ is reported, and shown to result from pseudolocalized vibrations occurring at the rare-earth defect. A model for the defect is defined, and an analysis of its vibrational frequencies is given for the sample case of a RbCl host lattice. A Green's function formalism is used in the analysis. The calculation, in which a number of approximations are made, yields the qualitative features of the experimental results.
The theoretical lattice vibrational spectrum for white tin (body-centered tetragonal) is examined on the basis of the axially symmetric lattice dynamics model including fourth-nearest-neighbor interactions. The atomic force constants are derived using experimental data for the elastic constants, specific heat, Debye temperature, and the Debye-Waller factor at 100°K. The 6×6 dynamical matrix is diagonalized along the principal symmetry directions and the resulting dispersion curves are presented. The spectrum is characterized by low-lying optical modes which interact strongly with the acoustic modes, particularly near the surface of the first Brillouin zone. These optical modes give rise to a large density of states at intermediate acoustical frequencies and contribute significantly to both the specific heat and Debye-Waller factor at low temperature.
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.
The model for the lattice dynamics of alkali halides as given in a previous paper is generalized to include the case in which both ions are polarizable. Dispersion curves and density of states are calculated for LiF and NaCl at 0 °K. The Debye characteristic temperatures ΘD(T) for 0 °K and 300 °K are computed for these crystals, and good agreement with experimental values is obtained.In einer vorhergehenden Arbeit wurde ein neues Modell für die Gitterdynamik der Alkalihalogenide vorgeschlagen. Dieses Modell wird verallgemeinert für den Fall, daß beide Ionen polarisierbar sind. Anschließend werden die Dispersionskurven und Zustandsdichten für NaCl und LiF bei 0 °K angegeben. Die ebenfalls berechneten Debye-Temperaturen ΘD(T) dieser Kristalle für 0 und 300 °K stimmen gut mit experimentellen Ergebnissen überein.
One of the most serious problems encountered in the use of triple-axis crystal spectrometers for neutron inelastic scattering experiments is that of the order of contamination inherent in the Bragg-reflected neutrons from the monochromator and analyser crystals. An elegant solution to this problem is provided by the use of certain crystal reflecting planes for which the second and other higher order reflections are absent by symmetry, e.g. the (111) and (113) reflections from single crystals of germanium. High reflectivity germanium crystals having mosaic spreads in the range 0.3° to 0.5° have been produced by the general method of Barrett, Mueller and Heaton, and employed with considerable success in various crystal spectrometers at the NRX and NRU reactors at Chalk River. Details of the production method are given, and a comparison is made between aluminum and germanium analyser crystals in inelastic scattering experiments to determine the normal vibration frequencies of sodium nitrite and cobalt fluoride.