Lindemann Criterion Research Articles

Estimates for the thermodynamic signatures of vortex-lattice melting in conventional superconductors

The first-order nature of the vortex-lattice melting transition in copper-based layered high-Tc superconductors is well established. The associated discontinuities in magnetization have been extensively studied, for example, in YBa2Cu3O7[1,2] and Bi2Sr2CaCu2O8[3], while the respective latent heats have been systematically investigated only in YBa2Cu3O7 and related compounds [4–13]. The apparent absence of such signatures in conventional superconductors such as Nb raises the question whether or not the concept of vortex-lattice melting is applicable at all in such materials [14]. Based on available literature to describe the vortex-state and using the Lindemann criterion, we estimate quantitatively the order of magnitude for the expected latent heats of melting and the associated discontinuities in magnetization, respectively, as functions of a few known material parameters. It turns out that both thermodynamic quantities are not strictly vanishing even in isotropic materials as long as κ>1/2, but they are small and may often be beyond the available experimental resolution.

Melting of monatomic glass with free surfaces

Melting of monatomic glass with free surfaces has been studied by molecular dynamics simulations in models with Lennard-Jones-Gauss interatomic potential. Models have been heated up from a glassy state toward a normal liquid state. Atomic mechanism of melting has been analyzed via monitoring spatio-temporal arrangements of liquid-like atoms occurred during heating process. Liquid-like atoms are detected via the Lindemann criterion of melting. It is clear that the transition from glass into supercooled liquid of our "ordinary" glass with free surfaces exhibits a non-heterogeneous behavior, i.e., although liquid-like atoms initiate/grow mainly in the surface shell, significant amount of liquid-like atoms also initiates/grows simultaneously in the interior during heating process. We found three characteristic temperatures of melting of glass with a free surface. Temperature dependence of structure and various thermodynamic quantities of the system upon heating is also presented and discussed.

Atomic mechanism of homogeneous melting of bcc Fe at the limit of superheating

Atomic mechanism of homogeneous melting of bcc Fe is studied via monitoring spatiotemporal arrangements of the liquid-like atoms, which are detected by the Lindemann criterion of melting, during the heating process. Calculations are performed by molecular dynamics (MD) simulations. Calculations show that liquid-like atoms occur randomly in the crystalline matrix at temperature far below the melting point due to local instability of the crystalline lattice. Number of liquid-like atoms increases with increasing temperature and they have a tendency to form clusters. Subsequently, a single percolated liquid-like cluster is formed in the crystalline model and at the melting point 99% atoms in the model become liquid-like to form a liquid phase. Melting is also accompanied by the sudden changes in various static and thermodynamic quantities. However, total melting is reached just at the point above the melting one. Three characteristic temperatures of the homogeneous melting of bcc Fe are determined.

Effect of the dimension of the phonon spectrum on the stability of condensed media states

The effect of dimension index df of the phonon spectrum, which is a structural characteristic in continual models, on the stability of states of condensed media is considered in the Einstein and Debye approximations. The estimate of the phase state stability is based on the Lindemann criterion generalized to arbitrary values of 0 ≤ df ≤ ∞. The problem of variation of physical characteristic of a substance by controlling the structure of its phonon spectrum is considered by analyzing the possibility of obtaining molecular hydrogen in the superfluid state. The Einstein and Debye models as applied to the problem on the dynamics of atomic oscillations are compared, and the divergence of the latter model for fractal dimensions df < 2 of the phonon spectrum is demonstrated, as well as the incompatibility of the Debye model at high temperatures and the model of a classical oscillator for all dimensions except df → ∞.

Effect of doping on structural and superconducting properties in Ca1−xNaxFe2As2single crystals (x=0.5, 0.6, 0.75)

We study the correlation between crystalline structure and superconducting properties in Na-doped Ca${}_{1\ensuremath{-}x}$Na${}_{x}$Fe${}_{2}$As${}_{2}$ single crystals for three chemical compositions ($x$ = 0.5, 0.6, 0.75). We find the maximum superconducting transition temperature ${T}_{c}$ \ensuremath{\sim} 33.4 K at $x$ \ensuremath{\sim} 0.75. The Na substitution causes the decrease of the a-b crystallographic axes and the increase of the $c$ axis in the tetragonal phase. The single crystals show perfect diamagnetism, indicating full superconducting volume. The anisotropy ratio for the upper critical field near the superconducting transition temperature is \ensuremath{\gamma} = 1.85 \ifmmode\pm\else\textpm\fi{} 0.05, independently of the Na content. A narrow vortex liquid phase was detected in the sample with highest ${T}_{c}$ ($x$ = 0.75), consistent with the expectations based on a Lindemann criterion. The analysis of the critical currents shows no evidence of correlated pinning and indicates that the pinning arises from a combination of several mechanisms. At low fields, pinning by random nanoparticles dominates. At higher fields, a small and field independent ${J}_{\mathrm{c}}$ in the optimally doped crystal may originate in the simultaneous presence of sparse large nanoparticles and a much denser distribution of smaller particles, with the sparse pins producing a caging effect that constrains the volume of the vortex bundle associated with the denser and weaker defects.

Theoretical Investigation of Lindemann's Criterion in the Melting of Two-Dimensional Vortex-Lattice with Nano Defects

Theoretical Investigation of Lindemann's Criterion in the Melting of Two-Dimensional Vortex-Lattice with Nano Defects

Lattice dynamics and the phase diagram of sodium at high pressures and temperatures from ab initio calculations

The melting of sodium was investigated by two ab initio methods: (i) based on the calculations of the phonon spectra in the quasiharmonic approximation with the use of the Lindemann criterion and (ii) by moleculardynamics simulation. It is shown that the anomalous behavior of the melting curve Tm(p) for Na is explained well by the change in the phonon spectrum under compression; in particular, the decrease in T m (p) at p > 30 GPa is due to the strong softening of transversephonon frequencies. The results obtained within both approaches are in good agreement and yield a reasonable quantitative description of the experi� mental melting curve of sodium at pressures up to ~1 Mbar and temperatures from 300 to 1000 K. The good agreement between the two approaches indicates the smallness of anharmonicity effects in Na.

Numerical computer calculation of distribution of similar point charges at the dielectric surface

Results of numerical calculation of the distribution of similar point charges over a uniformly charged dielectric plane are presented for different surface charge densities and surface temperatures. It is shown that, in the case of the surface charge density ranging from 2 × 10−6 to 7 × 10−5 C/m2 and temperatures ranging from 300 to 800 K, the Lindemann criterion is fulfilled for the average deviation of the point charge from the equilibrium position, and the collection of point charges at the dielectric surface can be regarded as a 2D Coulomb crystal.

On the Lindemann Criterion for Quantum Clusters at Very Low Temperature

The Lindemann criterion to discern the solid-like or liquid-like nature of a quantum cluster at T = 0 is discussed. A critical analysis of current Lindemann parameters is presented and a new parameter is proposed that is appropriate to study quantum clusters made of identical particles. A simple model wave function is introduced to fix the range of variation of these parameters. The model presents two extreme limits that correspond to either a liquid-like or a solid-like system; besides, it fulfills the Bose symmetry and also permits evaluations without symmetrization. Variational and diffusion Monte Carlo calculations are also performed for clusters of spinless bosons interacting through Lennard-Jones potentials. It is shown that the liquid-like or solid-like character of quantum clusters at zero temperature cannot be simply established in terms of a single parameter.

Connection between slow and fast dynamics of molecular liquids around the glass transition

The mean-square displacement (MSD) was measured by neutron scattering at various temperatures and pressures for a number of molecular glass-forming liquids. The MSD is invariant along the glass-transition line at the pressure studied, thus establishing an "intrinsic" Lindemann criterion for any given liquid. A one-to-one connection between the MSD's temperature dependence and the liquid's fragility is found when the MSD is evaluated on a time scale of ∼4 ns , but does not hold when the MSD is evaluated at shorter times. The findings are discussed in terms of the elastic model and the role of relaxations, and the correlations between slow and fast dynamics are addressed.

Melting Temperature of Metals Based on the Nearly Free Electron Model

We propose a general formula for the melting temperature of metals in terms of electronic mass, electronic number, and nearest-neighbor lattice distance. We derive it from the instability of the transverse phonon in the solid phase, using the nearly free electron model. Including higher order terms of vibrations enhanced near the melting temperature, the electronic restoring force is reduced and the ionic one is negligible. This fact greatly brings down the melting temperature, bringing it close to experimental data in the range of 10 % for Cs, Cu, Au, and Ba. Also, this theory confirms the Lindemann criterion.

Effective interactions and melting of a one-dimensional defect lattice within a two-dimensional confined colloidal solid

We report Monte Carlo studies of a two-dimensional soft colloidal crystal confined in a strip geometry by parallel walls. The wall-particle interaction has corrugations along the length of the strip. Compressing the crystal by decreasing the distance between the walls induces a structural transition characterized by the sudden appearance of a one-dimensional array of extended defects each of which span several lattice parameters, a "soliton staircase." We obtain the effective interaction between the solitons. A Lindemann criterion shows that the reduction in dimensionality causes the finite soliton lattice to readily melt as the temperature is raised.

High-temperature phase transitions in rare-earth metals

The study into the relationship between the laws of structure formation in metallic melts and the properties of collective ion vibrations is continuing. The phase-transition temperatures are found from the Lindemann criterion, where the root-nean-square amplitude of ion vibrations about a center of equilibrium is compared to the properties of the distribution of such centers. In a liquid phase, the vibration amplitude can decrease with increasing temperature.

Excess of low frequency vibrational modes and glass transition: A molecular dynamics study for soft spheres at constant pressure

Using molecular dynamics at constant pressure, the relationship between the excess of low frequency vibrational modes (known as the boson peak) and the glass transition is investigated for a truncated Lennard-Jones potential. It is observed that the quadratic mean displacement is enhanced by such modes, as predicted using a harmonic Hamiltonian for metastable states. As a result, glasses loose mechanical stability at lower temperatures than the corresponding crystal, since the Lindemann criteria are observed, as is also deduced from density functional theory. Finally, we found that the average force and elastic constant are reduced in the glass due to such excess of modes. The ratio between average elastic constants can be approximated using the 2/3 rule between melting and glass transition temperatures.

Molecular simulation of gas hydrate nanoclusters in water shell: Structure and phase transitions

Gas hydrate nanoclusters surrounded by water shells are studied by the molecular dynamic method. Hydrates of methane (sI structures) and krypton (sII structures), as well an ice nanocluster in a supercooled water shell, are considered. The main attention was focused on studying the local structure and phase transitions. Variations in local partial densities with an increase in temperature are monitored. Melting points of nanosized samples of gas hydrates are determined using caloric curves. Additional information on the behavior of the considered systems is obtained from the temperature dependences of diffusion coefficients and the Lindemann criterion. Two-phase transitions are revealed for gas hydrate nanoclusters. The first phase transition at 210 K can be assigned to the melting of the ice shell. The second transition at 230–235 K is identified as the phase transition in the hydrate core. The melting of ice cluster is observed at 215 K, which corresponds to the melting point of bulk crystal upon the use of the SPC/E water model.

Two-Dimensional Melting of a Crystal of Ferrofluid Spikes

We report the observation of the transition from an ordered solidlike phase to a disordered liquidlike phase of a lattice of spikes on a ferrofluid surface submitted to horizontal sinusoidal vibrations. The melting transition occurs for a critical spike displacement which is experimentally found to follow the Lindemann criterion, for two different lattice topologies (hexagonal and square) and over a wide range of lattice wavelengths. An intermediate hexaticlike phase between the solid and isotropic liquid phases is also observed and characterized by standard correlation functions. This dissipative out-of-equilibrium system exhibits strong similarities with 2D melting in solid-state physics.

Lattice dynamics and melting features of Li and Na

The high-pressure melting of Li and Na has been studied using ab initio calculations of the lattice dynamics. It has been shown that the recently discovered anomalous melting of Na is adequately explained by the phonon spectrum behavior and, accordingly, the thermal vibration amplitudes under compression. In a simple approach using the Lindemann criterion, the nonmonotonic behavior of the melting curve Tm(p) of Na has been quantitatively described within very wide pressure and temperature ranges, and, in particular, the melting temperature drop at p ∼ 1 Mbar down to values lower than those at normal pressure. This approach leads to a nonphysical discontinuity of the melting curve Tm(p) of Li near the bcc-fcc-liquid triple point. This is due to the “softness” of the phonon spectrum of the bcc phase of Li that is the necessary condition for the existence of the high-temperature bcc phase. The melting of Na and Li is used as an example to determine why the Lindemann criterion is efficient in some situations and is inapplicable in the other cases.

Influence of the structure and boundaries of the phonon spectrum of a solid on the dynamics of atomic oscillations and the Lindemann criterion

It is shown that the oscillatory dynamic characteristics of a solid, the phonon spectrum of which has a structure strongly different from that according to the Debye theory, depend on the sample size and this dimensional effect is manifested until reaching a nanoscale level. A generalized expression for the Lindemann criterion is obtained, which takes into account the finite boundaries of the phonon spectrum for various values of its fractal dimension, including those corresponding to the infrared catastrophe phenomenon.

A First-Principle Study on Size-Dependent Thermodynamic Properties of Small TiO2 Nanoclusters

Abstract Thermodynamic properties of titanium dioxide (TiO 2 ) n ( n = 1–6) nanoparticles were studied by first-principle molecular dynamics (MD) simulation. The configurations for (TiO 2 ) n ( n = 1–6) nanoparticles with global minimum energies can be initially obtained by MD simulation combined with the fast inertial relaxation engine algorithm. These structures can be further refined by density functional theory simulation. The variation of internal energies and Lindemann criteria with the simulation temperature was used to indicate the occurring of phase change for (TiO 2 ) n ( n = 1–6) nanoparticles. The effect of size and geometry on the thermodynamic properties of TiO 2 nanoparticles was also found.

Transcending Lindemann's predictions for the melting temperatures of metals and for the long-wavelength limit of their liquid structure factors

Lindemann's melting criterion remains useful. However, one prediction it makes for liquid metals (our focus here) is that the long-wavelength limit of the structure factor S(q) at freezing, S T m (0), where T m is the melting temperature, is a universal constant. For 34 metals we have calculated S T m (0) from input data, which is essentially the measured T m and the surface thickness L, defined near freezing as the product of isothermal compressibility and surface tension. To complete the characterization of S T m (0) we fit to one metal, chosen as Rb, for which S T m (0) is well established experimentally. For a wide variety of metals considered, S T m (0) is then found to vary by a factor of 10.