Perfect crystal, defining

Defining order in an amorphous solid is problematic at best. There are several qualitative concepts that can be used to describe disorder [7]. In figure Al.3.28 a perfect crystal is illustrated. A simple fonn of disorder involves crystals containing more than one type of atom. Suppose one considers an alloy consisting of two different atoms (A and B). In an ordered crystal one might consider each A surrounded by B and vice versa. 

The reason for this can be seen as follows. In a perfect crystal with the ions held fixed, a positive hole would move about like a free particle with a mass m depending on the nature of the crystal. In an applied electric field, the hole would be uniformly accelerated, and a mobility could not be defined. The existence of a mobility in a real crystal derives from the fact that the uniform acceleration is continually disturbed by deviations from a perfect lattice structure. Among such deviations, the thermal motions of the ions, and in particular, the longitudinal polarisation vibrations, are most important in obstructing the uniform acceleration of the hole. Since the amplitude of the lattice vibrations increases with temperature, we see how the mobility of a... 

Crystalline solids are built up of regular arrangements of atoms in three dimensions these arrangements can be represented by a repeat unit or motif called a unit cell. A unit cell is defined as the smallest repeating unit that shows the fuU symmetry of the crystal structure. A perfect crystal may be defined as one in which all the atoms are at rest on their correct lattice positions in the crystal structure. Such a perfect crystal can be obtained, hypothetically, only at absolute zero. At all real temperatures, crystalline solids generally depart from perfect order and contain several types of defects, which are responsible for many important solid-state phenomena, such as diffusion, electrical conduction, electrochemical reactions, and so on. Various schemes have been proposed for the classification of defects. Here the size and shape of the defect are used as a basis for classification. 

Let m be the set of occupation numbers of the perfect crystal and define a quantity v8l,... 

In a distorted crystal, where the atomic displacement from the perfect crystal is given by the vector u, we can define a local reciprocal lattice vector g by g =g-gxad(g.u) (8.19)... 

Chain Pair Modeling. In the following analysis, we assume that the chains are regular helices, i.e. that they have screw symmetry, with a repeat distance, t. In a perfect crystal, such chains must either be parallel or antiparallel. Four interhelical parameters are required to define the geometric orientation of chain A relative to chain B (see Figure 2). The parameters and their ranges are ... 

It seems quite natural to describe the extended part of a quantum particle not by wavepackets composed of infinite harmonic plane waves but instead by finite waves of a well-defined frequency. To a person used to the Fourier analysis, this assumption—that it is possible to have a finite wave with a well-defined frequency—may seem absurd. We are so familiar with the Fourier analysis that when we think about a finite pulse, we immediately try to decompose, to analyze it into the so-called pure frequencies of the harmonic plane waves. Still, in nature no one has ever seen a device able to produce harmonic plane waves. Indeed, this concept would imply real physical devices existing forever with no beginning or end. In this case it would be necessary to have a perfect circle with an endless constant motion whose projection of a point on the centered axis gives origin to the sine or cosine harmonic function. This would mean that we should return to the Ptolemaic model for the Havens, where the heavenly bodies localized on the perfect crystal balls turning in constant circular motion existed from continuously playing the eternal and ethereal harmonic music of the spheres. 

Bond orbitals are constructed ft om s/r hybrids for the simple covalent tetrahedral structure energies are written in terms of a eovalent energy V2 and a polar energy K3. There are matrix elements between bond orbitals that broaden the electron levels into bands. In a preliminary study of the bands for perfect crystals, the energies for all bands at k = 0 arc written in terms of matrix elements from the Solid State Tabic. For calculation of other properties, a Bond Orbital Approximation eliminates the need to find the bands themselves and permits the description of bonds in imperfect and noncrystalline solids. Errors in the Bond Orbital Approximation can be corrected by using perturbation theory to construct extended bond orbitals. Two major trends in covalent bonds over the periodic table, polarity and metallicity, arc both defined in terms of parameters from the Solid State Table. This representation of the electronic structure extends to covalent planar and filamentary structures. 

Screw dislocation. The simplest case to start with is that of a straight screw dislocation of Burgers vector b parallel to the surface of a thin parallel-sided crystal foil, as shown in Figure 5.10. Using the coordinate system defined there, the dislocation AB is parallel to y and at a depth z below the top surface. The dislocation causes a column CD of unit cells parallel to z in the perfect crystal to be deformed. If we assume that the atomic displacements around the dislocation are the same in the thin specimen as in an infinitely large, elastically isotropic crystal, then the components u, v, w of the deformation of the column along thex, y, and z directions will be... 

Since the interaction of hard X-rays with matter is small, the kinematical approximation of single scattering is valid in most cases, except for perfect crystals near Bragg scattering. The intensity scattered by a block-shaped crystal with N, q and N, unit cells along the three crystal axes defined by the vectors Uj, a and a, takes the form ... 

Although real crystals display beautiful symmetries to the eye, they are not perfect. As a practical matter, it is impossible to rid a crystal of all impurities or to ensure that it contains perfect periodic ordering. So, we describe real crystals as perfect crystals with defects, and define means to characterize these defects. If so many defects are present that crystalline order is destroyed, we describe the material as an amorphous solid. 

If a vacancy is created in an otherwise perfect lattice, the implication is that an additional atom must now occupy the surface of the crystal. The basic idea behind the vacancy formation energy is that it is a measure of the difference in energy between two states one being the perfect crystal and the other being that in which an atom has been plucked from the bulk of the crystal and attached to the surface. From a computational perspective, the vacancy formation energy may be defined as... 

Although LSMs can be employed, the layers within an LSM are amorphous and therefore not well-defined. It would be most desirable to be able to perform XSW studies on single-crystal surfaces without having to resort to perfect or nearly perfect crystals. 

The second statement of the third law (which bears Planck s name) is that as the temperature goes to zero, AS goes to zero for any process for which a reversible path could be imagined, provided the reactants and products are perfect crystals. Here, perfect crystals are defined as those which are non-degenerate, that is, they have only a single quantum state in which they can exist at absolute zero. This statement follows rigorously from Boltzmann s equation for entropy,... 

We also note that certain polymer melts crystallize partially upon cooling. The transition occurs at a well-defined temperature rf. Crystallization takes place only if the polymer has a perfect linear structure for instance, it must not contain any asymmetric carbon. However, this tacticity condition is not sufficient, since polydimethylsiloxane, which is perfectly periodical, does not crystallize under normal conditions. On the contrary, polyethylene and isotactic polyethylene crystallize easily. In general, these polymers contain a large amorphous fraction. This is why they are called semi-crystalline. In certain conditions, it is possible to prepare polymer samples that are perfect crystals, in particular by polymerization in situ of a crystal made of monomers (polydiacetylene and polyoxyethylene). 

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