Lattice points atoms

Orthorhombic crystals are similar to both tetragonal and cubic crystals because their coordinate axes are still orthogonal, but now all the lattice parameters are unequal. There are four types of orthorhombic space lattices simple orthorhombic, face-centered orthorhombic, body-centered orthorhombic, and a type we have not yet encountered, base-centered orthorhombic. The first three types are similar to those we have seen for the cubic and tetragonal systems. The base-centered orthorhombic space lattice has a lattice point (atom) at each comer, as well as a lattice point only on the top and bottom faces (called basal faces). All four orthorhombic space lattices are shown in Figure 1.20. 

For example, consider a simple crystal witii one atom per lattice point the total ionic potential can be written as... 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

By this discretization in the polar coordinates, one can allow an atom to be displaced to one of the 121 (=1+8+16+24+32+40) discrete points around each reference lattice point. 

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. 

In conclusion, we have presented a new formulation of the CVM which allows continuous atomic displacement from lattice point and applied the scheme to the calculations of the phase diagrams of binary alloy systems. For treating 3D systems, the memory space can be reduced by storing only point distribution function f(r), but not the pair distribution function g(r,r ). Therefore, continuous CVM scheme can be applicable for the calculations of phase diagrams of 3D alloy systems [6,7], with the use of the standard type of computers. 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

One of the first attempts to calculate the thermodynamic properties of an atomic solid assumed that the solid consists of an array of spheres occupying the lattice points in the crystal. Each atom is rattling around in a hole at the lattice site. Adding energy (usually as heat) increases the motion of the atom, giving it more kinetic energy. The heat capacity, which we know is a measure of the ability of the solid to absorb this heat, varies with temperature and with the substance.8 Figure 10.11, for example, shows how the heat capacity Cy.m for the atomic solids Ag and C(diamond) vary with temperature.dd ee The heat capacity starts at a value of zero at zero Kelvin, then increases rapidly with temperature, and levels out at a value of 3R (24.94 J-K -mol-1). The... 

The Bi external electronic configuration is (s2p3) it crystallizes in a rombohedral system, with two Bi atoms linked to each lattice point of the unit cell. The melting point is 544 K. 

A number of metal oxides are known to form nonstoicbiometric compounds, in which the ratios of atoms that make up the compound cannot be expressed in small whole numbers. In the crystal structure of a nonstoichiometric compound, some of the lattice points where one would have expected to find atoms are vacant. Transition metals most easily form nonstoichiometric compounds because of the number of oxidation states that they can have. For example, a titanium oxide with formula TiO, I( is known, (a) Calculate the average oxidation state of titanium in this compound. 

The nature of the structure found for this phase, with strong indication that there are thirty-two magnesium atoms per lattice point, suggests that the quaternary phase should be assigned the general formula Mg32(Al, Zn, Cu)49. 

On the right are the t5rpes of point defects that could occur for the same sized atoms in the lattice. That is, given an array of atoms in a three dimensional lattice, only these two types of lattice point defects could occur where the size of the atoms are the same. The term vacancy is self-explanatory but self-interstitial means that one atom has slipped into a space between the rows of atoms (ions). In a lattice where the atoms are all of the same size, such behavior is energetically very difficult unless a severe disruption of the lattice occurs (usually a "line-defect" results. This behavior is quite common in certain types of homogeneous solids. In a like manner, if the metal-atom were to have become misplaced in the lattice cuid were to have occupied one of the interstitial... 

Fig. 3 Difference in crystallization behavior between an atomic or short chain molecular system and a polymer system, a Atoms or short chain molecules can be independently rearranged on each lattice point, while b the order of the repeating units within a polymer chain is maintained during the rearranging process. Therefore, a chain should slide along its chain axis and disentangle for rearrangement onto the lattice points...

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

The importance of dislocations becomes evident when we consider the strain on the microstructure of a simple crystal. The atoms or ions in a crystal are in symmetric energy wells and so vibrate around their lattice site. When we track across a crystal plane, the potential energy increases and decreases in a regular fashion with the minima at the lattice points... 

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

The number of units in an MQW will be much more limited than the number of atomic planes sampled by the X-ray beam in a standard reflection. The intensity will be low, but also the MQW will behave as a thin crystal —the reciprocal lattice points will be extended into rods perpendicular to the crystal surface. This will broaden the reflection, and thus the width of each satellite peak is determined by the number of units in the MQW. It has even been possible, by... 

Figure 4.8 The 14 Bravais lattices. Black circles represent atoms or molecules. P cells contain only one lattice point, while C- and /-centred cells contain two and / -centred cells contain four.

It is clear from Eq. (2.2b) that the frequency to in Eq. (2.7) is a function of q, because q governs the relative displacement of two interacting atoms. The co(q) dependence on q (the dispersion relationships) is illustrated in Fig. 2.1 for the rock-salt structure. It can be shown that all normal modes can be represented in the first Brillouin zone, which extends from 0 to nja in the a direction of the rock-salt structure, or, more generally, is bounded by faces located halfway between the reciprocal lattice points in the space defined by1 = 27r<5fJ-. The... 

Careful measurements of the structure factors of vanadium (Ohba et al. 1981) and chromium (Ohba et al. 1982) up to sin 6/2 = 1.72 A / using AgKa radiation and small spherical crystals ( 0.2 mm diameter), have been reported. The bcc structure of these metals leads to pairs of reflections such as (330/441), (431/510), at identical values of sin 6/2, which have the same intensity for a structure with one spherical atom per lattice point. This is no longer true when the t2g and eg orbitals of the cubic site are no longer equally occupied. This is easiest seen as follows. 

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