Born-von Karman Model Research Articles

Vibrational dynamics in 2D crystal lattices of borophene

Vibration dynamics of crystalline borophene is considered in the framework of the Born–von Karman model. The vibrations perpendicular to the plane of 2D borophen lattice (flexural modes) are studied.

Measurements of nonequilibrium interatomic forces using time-domain x-ray scattering

We determine experimentally the excited-state interatomic forces in photoexcited bismuth. The forces are obtained by a constrained least-squares fit of the excited-state dispersion obtained by femtosecond time-resolved x-ray diffuse scattering to a fifteen-nearest neighbor Born-von Karman model. We find that the observed softening of the zone-center $A_{1g}$ optical mode and transverse acoustic modes with photoexcitation are primarily due to a weakening of three nearest neighbor forces along the bonding direction. This provides a more complete picture of what drives the partial reversal of the Peierls distortion previously observed in photoexcited bismuth.

The quasi-low temperature behaviour of specific heat.

A new mathematical approach to condensed matter physics, based on the finite temperature field theory, was recently proposed. The field theory is a scale-free formalism; thus, it denies absolute values of thermodynamic temperature and uses dimensionless thermal variables, which are obtained with the group velocities of sound and the interatomic distance. This formalism was previously applied to the specific heat of condensed matter and predicted its fourth power of temperature behaviour at sufficiently low temperatures, which was tested by experimental data for diamond lattice materials. The range of temperatures with the quartic law varies for different materials; therefore, it is called the quasi-low temperature regime. The quasi-low temperature behaviour of specific heat is verified here with experimental data for the fcc lattice materials, silver chloride and lithium iodide. The conjecture that the fourth order behaviour is universal for all condensed matter systems has also supported the data for glassy matter: vitreous silica. This law is long known to hold for the bcc solid helium-4. The characteristic temperatures of the threshold of the quasi-low temperature regime are found for the studied materials. The scaling in the specific heat of condensed matter is expressed by the dimensionless parameter, which is explored with the data for several glasses. The explanation of the correlation of the ‘boson peak’ temperature with the shear velocity is proposed. The critique of the Debye theory of specific heat and the Born–von Karman model of the lattice dynamics is given.

The calculation of radial and tangential force constants for nickel using an experimental phonon spectrum

The possibility of calculating radial and tangential force constants, as well as elastic constants of face-centered cubic (FCC) metals, has been demonstrated for the case study of nickel on the basis of the experimentally measured phonon spectrum, which is theoretically described by the DeLaunay model in terms of the dynamic matrix D(q). The proposed calculation procedure is an alternative to the Born–von Karman model, which is widely applied for the description of the phonon spectrum in terms of force constants for interatomic interactions and in calculations of elastic constants using the phonon spectrum. The reliability of the obtained force constants for phonon spectrum was verified by comparison between the experimental and calculated data for the phonon spectrum and elastic constants for nickel. The obtained values of force constants for nickel have been compared with the theoretically calculated values using the pseudopotential method.

Calculation of the radial and tangential force constants of cesium from the experimental phonon spectrum

By the example of cesium, a procedure for calculating the radial and tangential force constants of the interatomic interaction of metals with the body-centered cubic (BCC) structure from the experimentally measured phonon spectrum, which can be theoretically described using elements of the dynamic matrix D(q), is proposed. This technique is an alternative to the Born–von Karman model, which is still widely applied in describing the phonon spectrum in terms of the force constants of interatomic interaction and in calculation of the elastic constants of the phonon spectrum. The reliability of the procedure and obtained values of the force constants is verified by calculating the phonon spectrum and elastic constants of cesium and by comparing the results with experimental data.

Specific heat of twisted bilayer graphene: Engineering phonons by atomic plane rotations

We have studied the phonon specific heat in single-layer, bilayer, and twisted bilayer graphene. The calculations were performed using the Born-von Karman model of lattice dynamics for intralayer atomic interactions and spherically symmetric interatomic potential for interlayer interactions. We found that at temperature T < 15 K, specific heat varies with temperature as Tn, where n = 1 for graphene, n = 1.6 for bilayer graphene, and n = 1.3 for the twisted bilayer graphene. The phonon specific heat reveals an intriguing dependence on the twist angle in bilayer graphene, which is particularly pronounced at low temperature. The results suggest a possibility of phonon engineering of thermal properties of layered materials by twisting the atomic planes.

Modeling of Carbon Nanotube Composites Based on Nonlocal Elasticity Approach

Nonlocal continuum theory is studied for modeling stress distributions in nanocomposites. The second-order approximation in nonlocal theory is considered since the first-order approximation leads to an unacceptable solution. A representative volume element (RVE) of CNT composite is utilized to derive unknown constants in the nonlocal theory model. Stress distributions in RVE using nonlocal theory, classical elasticity, and finite element method are obtained. All three approaches yield the same force, but classical elasticity gives an incorrect value of first moment. Wave propagation studies show that the dispersion curve obtained by nonlocal theory is quite close to the atomic Born-von Karman model.

Dynamic properties of graphene

The phonon spectrum of graphene has been studied with the minimum set of the nearest neighbors in the Born-von Karman model taking into account the electron-electron and electron-phonon interactions. The widths, both natural and owing to interactions with defects, of phonons have been estimated. Symmetry constraints imposed on force constants are taken into account. For symmetry reasons, vibrations with the polarization normal to the plane of the layer are not related to in-plane vibrations. The phonon frequencies at symmetry points and elastic moduli are expressed in terms of force constants.

Symmetry constraints on phonon dispersion in graphene

We calculate the phonon dispersion for graphene with interactions between the first, second, and third nearest neighbors in the framework of the Born–von Karman model taking into account the constraints imposed by the lattice symmetry. Analytical expressions give the nonzero sound velocity for the out-of-plane (bending) mode. The dispersion of four in-plane modes is determined by coupled equations. Values of the force constants are found in fitting with frequencies at critical points and elastic constants measured on graphite.

Phonons in aluminum at high temperatures studied by inelastic neutron scattering

Inelastic neutron scattering measurements on aluminum metal were performed at temperatures of 10, 150, 300, 525, and 775 K using direct-geometry Fermi chopper spectrometers. The temperature dependent phonon density of states (DOS) was determined from the scattering, and was used to fit Born–von Karman models of lattice dynamics. The shifts in the phonon frequencies with increasing temperature were largely explained by the softening of the longitudinal force constants out to third nearest neighbors. A significant broadening of the phonon spectra at high temperatures was also measured. The phonon DOS was used to determine the vibrational contributions to the entropy of aluminum as a function of temperature. All other contributions to the entropy of aluminum were calculated or assessed, and the total entropy was in excellent agreement with the NIST-JANAF compilation [M. W. Chase, J. Phys. Chem. Ref. Data Monogr. 9, 59 (1998)]. Anharmonic effects were attributed to phonon-phonon interactions. The quasiharmonic approximation was generally successful, but its weaknesses are discussed.

Phonon dispersions of Ni–Mn–Al shape memory alloy

Normal modes of vibration of a Ni–Mn–Al single crystal have been measured by inelastic neutron scattering. The force constants have been fitted to the Born–von Karman model using axially symmetric forces. Dispersion curves of both acoustical and optical phonons have been determined along the high symmetry [1 0 0], [1 1 0] and [1 1 1] directions. The temperature dependence of the normal modes revealed an anomalous softening of the TA 2[1 1 0] phonons observed in the range of 0.1–0.25 reciprocal lattice units in good agreement with recent ab initio calculations. Contrary to the acoustical TA 2[1 1 0] modes. The optical TO 2[1 1 0] modes with the same polarisation showed a normal behaviour with temperature, namely a decrease in frequency upon heating due to increasing anharmonicity. Elastic scattering performed along the [1 1 ¯ 0] direction did not reveal any significant elastic or diffuse scattering.

Neutron scattering measurements of phonons in nickel at elevated temperatures

Measurements of elastic and inelastic neutron scatterings from elemental nickel were made at 10, 300, 575, 875, and 1275 K. The phonon densities of states (DOSs) were calculated from the inelastic scattering and were fit with Born–von Karman models of the lattice dynamics. With ancillary data on thermal expansion and elastic moduli, we found a small, negative anharmonic contribution to the phonon entropy at high temperature. We used this to place bounds on the magnetic entropy of nickel. A significant broadening of the phonon DOS at elevated temperatures, another indication of anharmonicity, was also measured and quantified.

Diffuse mismatch model of thermal boundary conductance using exact phonon dispersion

The acoustic mismatch model (AMM) and the diffuse mismatch model (DMM) have been traditionally used to calculate the thermal boundary conductance of interfaces. In these calculations, the phonon dispersion relationship is usually approximated by a linear relationship (Debye approximation). This is accurate for wave vectors close to the zone center, but deviates significantly for wave vectors near the zone edges. Here, we present DMM calculations of the thermal conductance of Al–Si, Al–Ge, Cu–Si, and Cu–Ge interfaces by taking into account the full phonon dispersion relationship over the entire Brillouin zone obtained using the Born-von Karman model (BKM). The thermal boundary conductance thus calculated deviates significantly from DMM predictions obtained using the Debye model in all cases.

Lattice dynamics and the interaction with the crystal electric field inNdCu2

The lattice dynamics of NdCu2 has been investigated with inelastic neutron scattering measurements. An uniaxial Born–vonKarman model has been used to calculate the phonon dispersion relation. A verification ofa phonon–crystal electric field (CEF) interaction was found from temperature dependentmeasurements of phonons and CEF excitations. The impact of the phonon–CEFinteraction was found to be strongest at the Brillouin zone centre among the 7.5 meVAg-phononmode and the ε4 CEF level. This has been verified with theoretical model calculations.

Lattice dynamics and correlated atomic motion from the atomic pair distribution function

The mean-square relative displacements (MSRD) of atomic pair motions in crystals are studied as a function of pair distance and temperature using the atomic pair distribution function (PDF). The effects of the lattice vibrations on the PDF peak widths are modelled using both a multi-parameter Born von-Karman (BvK) force model and a single-parameter Debye model. These results are compared to experimentally determined PDFs. We find that the near-neighbor atomic motions are strongly correlated, and that the extent of this correlation depends both on the interatomic interactions and crystal structure. These results suggest that proper account of the lattice vibrational effects on the PDF peak width is important in extracting information on static disorder in a disordered system such as an alloy. Good agreement is obtained between the BvK model calculations of PDF peak widths and the experimentally determined peak widths. The Debye model successfully explains the average, though not detailed, natures of the MSRD of atomic pair motion with just one parameter. Also the temperature dependence of the Debye model largely agrees with the BvK model predictions. Therefore, the Debye model provides a simple description of the effects of lattice vibrations on the PDF peak widths.

Phonon density of states in nanocrystalline57Fe

The Born-von Karman model is used to calculate phonon density of states (DOS) of nanocrystalline bcc Fe. It is found that there is an anisotropic stiffening in the interatomic force constants and hence there is shrinking in the nearest-neighbour distances in the nanophase. This leads to additional vibrational modes above the bulk phonons near the bottom of the phonon band. It is found that the high energy phonon modes of nanophase Fe are the surface modes. The calculated phonon DOS closely agree with the experimental data except a peak at 37 meV. The calculated phonon dispersion relations are also compared with those of the bulk phonons and anomalous behaviour is discussed in detail. The specific heat in nanophase enhances as compared to bulk phase at low temperatures and the calculated Debye temperature ΘD agrees with the experimental results. It is predicted that the nanocrystalline Fe may consist of about 14 GPa pressure

Lattice dynamics of YCu 2

The lattice dynamics of the orthorhombic YCu 2 compound has been studied by inelastic neutron scattering on a single crystal. An axially symmetric Born–von Karman model was applied to fit the neutron data. Ab initio calculations of the lattice dynamics at the Γ-point provided important additional information for the Born–von Karman fit procedure. The calculated phonon density of states was compared with time-of-flight experiments on powdered YCu 2 sample material. The experimental studies were completed by Raman experiments and measurements of specific heat, thermal expansion, and compressibility.

Phonon dispersion in austenitic stainless steel Fe18Cr12Ni2Mo

The phonon dispersion of Fe18Cr12Ni2Mo austenitic stainless steel was measured along the symmetry directions [001], [110] and [111]. Data were analysed in the frame of the Born–von Karman model of lattice dynamics. The obtained force constants were used to evaluate the elastic constants and the engineering elastic moduli. Our results for the elastic constants confirm empirical relationships between the elastic constants found for FCC FeCrNi alloys.

Phonon density of states in MgB2.

We report inelastic neutron scattering measurements of the phonon density of states in Mg 11B2, which has a superconducting transition at 39.2 K. The acoustic phonons extend in energy to 36 meV, and there are highly dispersive optic branches peaking at 54, 78, 89, and 97 meV. A simple Born-von Kàrmàn model reproduces the mode energies, and provides an estimate of the electron-phonon coupling of lambda approximately 0.9. Furthermore, the estimated boron and magnesium contributions to the isotope effect are in qualitative agreement with experiment. The data confirm that a conventional phonon mechanism, with moderately strong electron-phonon coupling, can explain the observed superconductivity.

Lattice Dynamics of Li-ZrNCl: An Electron doped Layered Superconductor

We report recent results on structural and lattice dynamical investigations of Li.16ZrNCl and its non superconducting reference ZrNCl using the methods of elastic and inelastic neutron scattering as well as of inelastic light scattering. Comprehensive information is obtained by accompanying lattice dynamical calculations with a Born von Karman model. Significant spectral shifts which might be connected to strong electron-phonon coupling in the superconducting system are observed.