Thermal expansion, crystals

The single-crystal thermal expansion coefficients for mullite are... 

Systematic underestimation of the high-temperature Cp, cai T) data is a shortcoming of the quasi-harmonic approximation used to describe Ciat(T ). Westrum (1979) analyzed methods for including the contribution of crystal thermal expansion. It follows from the representation of Cd(T) in the form of the dependence... 

The dynamics of atoms in solids is responsible for many phenomena which cannot be explained within the static lattice model. Examples are the specific heat of crystals, thermal expansion, thermal conductivity, displacive ferroelectric phase transitions, piezoelectricity, melting, transmission of sound, certain optical and dielectric properties and certain aspects of the interaction of radiation such as X-rays and neutrons with crystals. The theory of lattice vibrations, often called lattice dynamiosy and its implications for many of the above mentioned phenomena is the subject of this two-volume book. 

Easier V, Schweiss P, Meingast C, Obst B, Wuhl H, Rykov A I and Tajima S 1998 3D-XY critical fluctuations of the thermal expansivity in detwinned YBa2Cu30y g single crystals near optimal doping Phys. Rev. Lett. 81 1094-7... 

The glass-ceramic phase assemblage, ie, the types of crystals and the proportion of crystals to glass, is responsible for many of the physical and chemical properties, such as thermal and electrical characteristics, chemical durabiUty, elastic modulus, and hardness. In many cases these properties are additive for example, a phase assemblage comprising high and low expansion crystals has a bulk thermal expansion proportional to the amounts of each of these crystals. 

Metal Crystal 22° C stmeture 1000° c Melting point, °C Density, g/cm Thermal expansion coefficient at RT, ioV°c Thermal conductivity at RT, W/(m-K)" Young s modulus, GPa "... 

The physical properties of tellurium are generally anistropic. This is so for compressibility, thermal expansion, reflectivity, infrared absorption, and electronic transport. Owing to its weak lateral atomic bonds, crystal imperfections readily occur in single crystals as dislocations and point defects. 

A summary of physical and chemical constants for beryUium is compUed ia Table 1 (3—7). One of the more important characteristics of beryUium is its pronounced anisotropy resulting from the close-packed hexagonal crystal stmcture. This factor must be considered for any property that is known or suspected to be stmcture sensitive. As an example, the thermal expansion coefficient at 273 K of siagle-crystal beryUium was measured (8) as 10.6 x 10 paraUel to the i -axis and 7.7 x 10 paraUel to the i -axis. The actual expansion of polycrystalline metal then becomes a function of the degree of preferred orientation present and the direction of measurement ia wrought beryUium. 

Coefficient of Thermal Expansion (GTE). The volumetric thermal expansion (VTE) of manufactured graphite expressed ia equation 1 is anomalously low when compared to that of the graphite single crystal, where wg designates with-grain and eg, cross-grain. 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

At each temperature one can determine the equilibrium lattice constant aQ for the minimum of F. This leads to the thermal expansion of the alloy lattice. At equilibrium the probability f(.p,6=0) of finding an atom away from the reference lattice point is of a Gaussian shape, as shown in Fig. 1. In Fig.2, we present the temperature dependence of lattice constants of pure 2D square and FCC crystals, calculated by the present continuous displacement treatment of CVM. One can see in Fig.2 that the lattice expansion coefficient of 2D lattice is much larger than that of FCC lattice, with the use of the identical Lennard-Lones (LJ) potential. It is understood that the close packing makes thermal expansion smaller. 

The precursive anisotropy in the thermal expansion for an In-26.5at%Tl alloy is shown in Figure 1 taken from reference 7, where for curve I the measurement direction becomes a c axis in the transformed crystal and for curve II it becomes an a axis. 

---