Equilibrium lattice constant

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. 

Table 1. Parameters of the interatomic potentials. Distances are given in as, densities in flg, charges in e and energies in Ry. ri4s and Vc have been set to 0.57 and 8.33 ag for iron. The corresponding values for nickel are 0.85 and 8.78 ag ao denotes the equilibrium lattice constant of the elements po is the electron density at equilibrium for the perfect lattices, i.e. 0.002776 ag and 0.003543 ag for iron and nickel respectively.

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

In this expression, Go is the equilibrium lattice constant, Vo is the equilibrium volume per atom, B0 is the bulk modulus at zero pressure, P 0, and B 0 = (dB/dP)r. To apply this equation to our data, we treat g0, Bq, and B 0 and Eq as fitting parameters. The results of fitting this equation of state to the full set of DFT data shown in Fig. 2.1 are shown in the figure with a dashed... 

It is relatively common for DFT calculations to not explicitly include electron spin, for the simple reason that this approximation makes calculations faster. In materials where spin effects may be important, however, it is crucial that spin is included. Fe, for example, is a metal that is well known for its magnetic properties. Figure 8.10 shows the energy of bulk Fe in the bcc crystal structure from calculations with no spin polarization and calculations with ferromagnetic spin ordering. The difference is striking electron spins lower the energy substantially and increase the predicted equilibrium lattice constant by 0.1 A. 

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

The model by Chesnut208>, the earliest one which appeared in the literature, simply explains the interaction term T by lattice strains. Suppose q s and qfts are equilibrium lattice constants of the pure LS and HS lattice, resp., and q the lattice constant of the real mixture of the spin isomers, then... 

Table III. Minus the total Si crystal valence electron energy per atom with relaxation energy and pseudopotential corrections included, along with the equilibrium lattice constant, bulk modulus, and cohesive energy calculated with four different exchange-correlation functionals (defined in the caption of Table I) are compared with experimental values. The experimental total energy is the sum of Acoh plus the four-fold ionization energy.

Table 4.24. The equilibrium lattice constant (a ) of CaO and some transition-metal monoxides. The experimental values are given in the second column. The theoretical values of in the nonmagnetic state are given in the third column. The theoretical values of are also calculated for the ferromagnetic and antiferromagnetic states...

As described briefly in Chapter 3, a promising new method of electronic structure calculation utilizing combined molecular-dynamics and density-functional theory has recently been developed by Car and Pari-nello (1985). This approach has recently been applied to cristobalite, yielding equilibrium lattice constants within 1% of experiment (Allan and Teter, 1987), as shown in Table 7.2. New oxygen nonlocal pseudopotentials were also an important part of this study. Such a method is a substantial advance upon density-functional pseudopotential band theory, since it can be efficiently applied both to amorphous systems and to systems at finite temperature. 

In the quasiharmonic approximation, the crystal potential is expanded as a power series in the displacement from the equilibrium lattice constant only through the displacement squared term. 

A brief review is given on the application of local density theory to the electronic structure of f-electron metals, including various ground state properties such as observed crystal structures, equilibrium lattice constants, Fermi surface topologies, and the electronic nature of known magnetic phases. A discussion is also given about the relation of calculated results to the unusual low energy excitations seen in many of these metals. 

The situation improves for the 5f elemental metals due to the greater delocalization of the 5f as compared to the 4f electrons. In fact, the equilibrium lattice constants of the 5f elemental metals are well reproduced even with the f electrons treated as band states, as long as spin polarization is taken into account (12). The spin polarization "turns on" at Am, and correlates with an inferred localization of the f electrons. Even for such strongly correlated metals as the heavy fermion superconductors UPt3 and UBe-ia, the equilibrium lattice constant and bulk modulus are well-given oy LDA f band calculations 0M4). 

Ab Initio Calculations of Structural and Elastic Properties. Equilibrium lattice constants, equilibrium volumes, as well as bulk and shear moduli can be assessed based on ab initio electron-structure calculations. They are obtained from the calculated total energies as a fimetion of volume in the bcc or fee crystal structure and from respective volume-conserving distortions of the lattice. In most cases, they agree well with experiments (Table 1.6). 

TABLE 1.6. Comparison of Equilibrium Lattice Constants and Bulk/Shear Moduli of bcc and fee... 

Table 4.8 Equilibrium lattice constant a, cohesive energy and bulk modulus B of FCC Al Exact exchange in comparison with LDA results.

Fig. 1. Total energy of fee thorium in dependenee of the lattiee eonstant, ealeulated with the relativistie FPLO method (RFPLO) using the Perdew-Wang 92 version of LDA [25]. The position of the minimum is indieated by the dashed line. Further, the experimental lattice constant is given by a box, where the width shows the scatter of the experimental data. Calculated equilibrium lattice constants with other relativistic band structure codes are denoted by arrows. Figure taken from Ref. [26].

Cohesive Energies and Equilibrium Lattice Constants for Solid Li and Be. 

The advanced methods allow one to calculate equilibrium lattice constants and cohesive energies of the noble gas crystals. 

In Table 15.5 the predicted equilibrium lattice constants and cohesive energies of the solid neon, argon, and krypton are compared with experiments. The results of the first-principle calculations are satisfactory, except for a value of cohesive energy for neon. As one can see, the theory overestimates the equilibrium lattice constant. Nevertheless, the updated theory [75] provides the adequate description of van der Waals bonded systems. 

Abstract Density functional theory (DFT) in various modifications provides the basis for studying the electronic structure of solids and surfaces by means of our WIEN2k code, which is based on the augmented plane wave (APW) method. Several properties, which can be obtained with this code, are summarized and the application of the code is illustrated with four selected examples focusing on very different aspects from electron-structure relations, complex surfaces or disordered layer compounds to the dependence of the equilibrium lattice constants on the DFT functionals. 

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