Lindemann Law Research Articles

Role of lattice structure on the Lindemann fusion theory of metals

A simple atomistic model for the fusion process of metals is derived from a combination of the atomic model of the lattices and the simple theory of harmonic vibration of the atoms of A1 (FCC), A2 (BCC) and A3 (HCP) type crystals. The lattice factors Lj of these metals and a new physical quantity rho j identical to Lj delta j= alpha /Lj with j=1, 2 and 3 for the fusion transitions BCC to liquid, FCC to liquid and HCP to liquid, involved with the vibrational instability fusion hypothesis of Lindemann, are defined. The derived structure-dependent simple fusion equation is related to a similar equation obtained from the combination of the Debye-Waller-Lindemann (DWL) formulae through a numerical factor f=(3/2)12/. Vibration amplitudes of atoms Aj (root-mean-square displacements Aj (DWL)=((Ui2))12/), Lindemann parameters delta j and the physical quantities rho j of 54 metals are calculated through the simple and DWL sets of equations above by using the empirical Lindemann constants Lj. The L1.2 and rho 1.2 are constants, L3( gamma ) and rho 3( gamma ) for HCP metals are smoothly varying functions of the axial ratio gamma =c/a, although L3 (HCP) and delta 3 (HCP) are not. A firm conclusion is made that the Lindemann constants Lj and Lindemann parameters delta j for the BCC, FCC and HCP metals are significantly different and the Lindemann law holds for each structure separately including the HCP metals.

Studies of Ordered Monodisperse Latexes. IV. Mechanism of Ordering

The order-disorder phase transition of aqueous suspensions of charged polystyrene spheres was studied both experimentally and theoretically. Phase diagrams of four kinds of samples with different surface charge numbers were determined from the observation of disappearance of the characteristic iridescence in the ordered state. The ordered phase appeared in the large volume fraction and low salt concentration regions. The ordered region became larger according to the surface charge number increase. On the basis of Lindemann's law, phase diagrams were also obtained theoretically. The theory reproduced quantitatively the experimental phase diagrams of all the samples used. The applicability of the Debye-Hückel approximation was discussed and the evaluation of the static rigidity reported in a previous paper [Jpn. J. Appl. Phys.: 19 (1980) 439] was improved.

Classical Wigner crystallization, liquid structure factor peak height and freezing of liquid alkalis

It is pointed out that, in studying crystallization, one should focus on the first reciprocal lattice vector which has its analogue in the first peak of the liquid structure factor. A freezing criterion of a classical one-component plasma emerging from the structure factor at the first peak, which transcends the Lindemann's Law, is then applied to the nearly-free-electron liquids Na and K and compared with experiments.

Dynamic properties of concentrated suspensions of charged polystyrene spheres

Ordered structure formation of charged polystyrene spheres was studied by measuring the order-disorder phase diagrams as well as the mechanical properties. The phase diagrams indicate that the ordering of polystyrene spheres obeys Lindemann's law of crystal melting, in which the Lindemann's parameter is about 5%. The rigidity of about 10(3) dyn/cm(2) was observed in the ordered suspension of polystyrene spheres as measured by a torsional quartz crystal method. The steady flow properties of suspensions of polystyrene spheres showed a remarkable change from a Bingham body to a Newtonian liquid at the transition point. The limit of elasticity in the ordered phase was about 1 dyn/cm(2). The viscosity in the disordered phase was well explained by the free volume theory of liquids. It is concluded from these facts that the ordered phase of polystyrene spheres is a real "crystal" whereas the disordered phase is a "liquid". Properties of ordered structures in biological systems are also discussed.

Self-consistent Debye-Waller factors of the electron solid on liquid helium

Based on the self-consistent field formalism we propose a shear-mode self-consistency for the high-frequency Debye-Waller factors (HFDWF) of the electron solid bound on a free surface of liquid helium. Our results are qualitatively in agreement with the empirical DW factor determined by Fisher, Halperin, and Platzman with the experiment of Grimes and Adams. We also report the analysis of the HFDWF according to the Lindemann law.

Applications of liquid state physics to the Earth's core

By use of the modern theory of liquids and some guidance from the hard-sphere model of liquid structure, the following new results have been derived for application to the Earth's outer core. (1) d K/ dP ⋍ 5 − 5. 6P / K, where K is the incompressibility and P the pressure. This is valid for a high-pressure liquid near its melting point, provided that the pressure is derived primarily from a strongly repulsive pair potential φ. This result is consistent with seismic data, except possibly in the lowermost region of the outer core, and demonstrates the approximate universality of d K/d P proposed by Birch (1939) and Bullen (1949). (2) dln T M/dln ρ = ( γC V − 1)/( C V − 3 2 ), where T M is the melting point, ρ the density, γ the atomic thermodynamic Grüneisen parameter and C V the atomic contribution to the specific heat in units of Boltzmann's constant per atom. This reduces to Lindemann's law for C V = 3 and provides further support for the approximate validity of this law. (3) It follows that the “core paradox” of Higgins and Kennedy can only occur if γ < 2 3 . However, it is shown that γ < 2 3 ⇔ ∫ ∞ 0 (∂g/∂T) ρ r( d/ dr)(r 2 φ) dr > 0 , which cannot be achieved for any strongly repulsive pair potential φ and the corresponding pair distribution function g. It is concluded that γ > 2 3 and that the core paradox is almost certainly impossible for any conceivable core composition. Approximate calculations suggest that γ ∼ 1.3–1.5 in the core. Further work on the thermodynamics of the liquid core must await development of a physically realistic pair potential, since existing pair potentials may be unsatisfactory.

The dependence of the melting temperatures of iron upon the choice of the interatomic potential

Summary The melting curve (melting temperature versus pressure) of iron is fundamental to an understanding of the physics of the Earth's core. Since experimental data are available for pressures a few orders of magnitude smaller than the core pressures, the problem is to find the correct ways of extrapolating them. Here we employ Ross’ melting criterion, a reformulation of Lindemann's law in the framework of statistical mechanics, because all the proposed empirical melting laws can be deduced from it; it also allows us to take into account the interatomic potential and the various crystal structures which iron may possess. The results of our calculations lead us to the conclusion that the melting temperatures of iron at the core pressures are extremely uncertain since no definite choice among possible potential functions is allowed. The melting temperature at the mantle–core boundary may range between 3500 and 4300 K, while at the inner core–outer core boundary it may range between 4500 and 7000K. The average melting temperature gradient is shown to be between 0.6 and 1.1 deg/km. The problem of establishing if the core is made of fcc or hcp iron is definitely secondary to that of the choice of the potential function.

A simple model for the melting of fine particles

Abstract The size dependence of the melting temperature of fine particles is discussed, on the basis of the Debye model and the Lindemann law, by phenomenologically taking into account surface phonon softening.

Studies of Ordered Monodisperse Latexes. II. Theory of Mechanical Properties

The electrostatic potential around a particle in an ordered latex under a deformation is numerically calculated form the nonlinear Poisson-Boltzmann equation by an eccentric cell model. The static rigidity of an ordered latex and the order-disorder Lindemann's law of the crystal melting are obtained from the calculated potential. The calculated phase diagram agrees with the experimental phase diagram of Hachisu et al. The Debye-Hückel approximation turns out to be invalid at low ionic concentrations. On the basis of a two-continua model and the calculated static rigidity, propagation of shear waves in an ordered latex is studied and the existence of two kinds of modes is shown. The dynamical complex rigidity is calculated from the mechanical impedance at an oscillating plate. The results agree with the experiment reported in a previous paper.

A thermal model of the earth

A thermal model, consistent as far as possible with the parameterised earth model of Dziewonski et al. and with thermodynamic principles and relevant equations of state, is tabulated. This is made more secure by two recent developments, an experimental study of the FeS eutectic to 100 kbar by Usselman and a calculation by Bukowinski which reveals an electronic phase collapse of potassium in the 200–300 kbar range and explains the core heat source. Use is made of the Vashchenko-Zubarev formulation of the Grüneisen ratio, and Lindemann's melting law, both of which have been shown recently to have particular relevance at very high pressures. Values of electronic specific heat and the Grüneisen ratio, which contribute significantly to core properties are calculated from the electron equation of state of Zharkov and Kalinin.

Applications of thermodynamics to fundamental earth physics

New fundamental thermodynamic relationships of complete generality and absolute rigour of derivation are not to be expected, because the subject has such a secure and complete basis in classical physics. There is, however, still scope for original, fundamental work based on recognised assumptions and approximations which may be obviously acceptable in particular situations. Clarification of relationships between thermodynamic parameters for materials within the Earth is particularly important because there is so little possibility of measuring them individually. This survey first summarises the established relationships in a very condensed form and then concentrates on some recent developments which have direct bearing on the thermal and mechanical states of the Earth's mantle and core. Considerable use is made of the thermodynamic Gruneisen parameter, which is a dimensionless quantity of order unity for almost all materials, solid, liquid and gaseous, and is directly related to the pressure dependences of elastic constants. This allows its value to be estimated for the different regions of the Earth from seismological data. The thermodynamic (heat engine) efficiency of convection in a homogeneous medium, driving tectonic activity or the geomagnetic dynamo, is found to be the ideal (Carnot) efficiency corresponding to adiabatic temperature differences between the heat source and sink, within the assumption that the thermal expansion coefficient is not strongly temperature dependent. The use of this conclusion to infer tectonic stresses is indicated. The thermodynamic basis for Lindemann's melting law is restated and the reasons for supposing it to be valid for materials at megabar pressures reaffirmed. Application to the inner core boundary gives a ‘fixed point’ on the Earth's temperature profile. Use of thermodynamic relationships in the interpretation of shock wave compressions is indicated.

The melting temperature of magnesium under shock loading

Temperature data from optical intensity measurements through anvils at high shock pressures are reported for magnesium samples. When account is taken for imperfect gaps at the anvil interface, the temperature measurements are shown to determine the melting line of magnesium at pressures from 40 to 50 GPa (400 to 500 kbar). The resulting high-pressure melting line for magnesium is in good agreement with the Lindemann law, with reasonable high pressure values for the Grüneisen coefficient γG.

Hard cores of molecules of simple fluids and the Lindemann law for hard spheres

Hard cores of molecules of simple fluids and the Lindemann law for hard spheres

Theory of Melting: Thermodynamic Basis of Lindemann's Law

A melting equation closely resembling the differential form of Lindemann's melting law, which relates the melting point to the pressure in terms of the thermal Griineisen parameter and incompressibility, is derived from the Clausius-Clapeyron relation with the assumption that the MieGriineisen equation can be adapted directly to describe the pressure change associated with melting at constant volume. This assumption implies that melting is only a minor perturbation of the crystal structure, such that the atomic coordination is almost unaffected and that atomic bonds are merely stretched or compressed. It appears that 'normal' melting complies with this assumption reasonably closely but that departures from Lindemann's law occur when materials undergo major changes in coordination during melting. A particular merit of Lindemann's law is that it allows the extrapolation of melting points to the pressures of the Earth's deep interior. Extrapolation of data on the ironsulphur eutectic suggests a temperature of 4160 K at the boundary of the Earth's inner (solid) and outer (liquid) cores. Adiabatic extrapolation to the core-mantle boundary gives 2940 K at the base of the mantle.

Hard-sphere results and atomic transport properties of liquid-metals

Experimental data on the shear viscosity of liquid metals have been used in combination with the results of hard-sphere model calculations in order to determine a hard-core size. By using the known density at the melting point and the result of applying Lindemann's law it turns out that the shear viscosity of high-melting refractory metals can be predicted. From temperature- dependent viscosity data interpreted in terms of hard-sphere results if follows that application of a corresponding-states principle to atomic transport in liquid metals is questionable.

Hard core size at the melting point and Lindemann law

Hard core size at the melting point and Lindemann law

On the Lindemann law of melting of solids

The Lindemann law of melting has been investigated by lattice dynamics. It is found that the law is not universal so far as the Lindemann parameter is concerned, and is structure and interaction dependent. This holds separately for each class of solids having similar structure and inter-particle interaction; i.e. the parameter has one value for all solids possessing the same structure as well as interparticle interaction.

Melting of lead and zinc to 60 kbar

The melting curves for lead and zinc were determined to 60 kbar. Our melting data for lead is in good agreement with that of Millet up to 22 kbar, beyond which Millet's values are significantly higher than ours. The melting curve for zinc is almost linear to 60 kbar and our values are lower than the values reported by other workers. The melting relationship proposed by Kennedy as well as the Lindemann law have been examined in the light of the new melting data for these two metals. A straight line can be fitted through the T m vs Δ V/V 0 plots for zinc within the limits of experimental precision, but the data for lead shows a departure from the straight line fit. The lead melting curve is concave toward the T m axis, as predicted by the Lindermann law and, in this respect, resembles previously studied Van der Waals solids.

Lindemann law for ideal solids

The applicability of the Lindemann law of melting is investigated for the simple solids like neon, argon, krypton, and xenon by lattice dynamics, and is found to hold good for the class. It is observed that the law is structure as well as interaction dependent, and holds separately for each class of solids having similar structure and interparticle interactions.

Short-range order and melting anomalies in thin films

Recent experiments show that high-coverage4He monolayers adsorbed on graphite have low-temperature heat capacities corresponding to two-dimensional solids. At higher temperatures the solid phase appears to “melt” by a continuous process, suggesting a connection with the theoretical prediction that there can be no long-range crystalline order at finiteT in two-dimensional systems governed by typical interatomic forces. This paper explores the theory in greater detail, with attention to long-range and short-range order and their experimental implications. Several models are studied: two-dimensional harmonic solids with “Debye” phonon spectra, lattices with Van Hove singularities, two-dimensional quantum solids, and effects of periodic potentials due to substrate structure. The theory is discussed in the light of current understanding of melting, and the following new melting hypothesis is proposed. Applied to two-dimensional systems, the hypothesis predicts a continuous transition with a heat-capacity contribution similar in its temperature dependence to the anomalies seen in4He monolayers and N2 multilayers. Applied to three-dimensional Debye solids, the same hypothesis predicts an abrupt first-order process, with a melting temperature obeying the Lindemann law.