Lattices definition

Within a crystal lattice, definite lines of cleavage are present along which a crystal may he split. This is an identifying property of ionic solids. 

Single crystals of vanadium oxide can be classified in their geometric structure, apart from their lattice definition by lattice and lattice basis vectors, by the occurrence of specific elementary VO building units. All single valency oxides are characterized by a common octahedral VOe unit. The different crystal lattices differ by the distortion of the octahedral units (as a result of actual V-0 distances and O-V-O angles) as well as by the connectivity of the units in the... 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

These two equations are a special case of the corresponding tliree-dimensional definition, connnon in XRD, with the surface nonnal n replacing the third lattice vector c. 

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

We assume that the mixture contains Ni solvent molecules, each of which occupies a single site in the lattice we propose to fill. The system also contains N2 polymer molecules, each of which occupies n lattice sites. The polymer molecule is thus defined to occupy a volume n times larger than the solvent molecules. Strictly speaking, this is the definition of n in the derivation which follows. We shall adopt the attitude that the repeat units in the polymer are equal to solvent molecules in volume, however, so a polymer of degree of... 

Fig. 2. Classification/nomenclature of host—guest type inclusion compounds, definitions and relations (/) coordinative interaction, (2) lattice barrier interaction, (J) monomolecular shielding interaction (I) coordination-type inclusion compound (inclusion complex), (II) lattice-type inclusion compound (multimolecular/extramolecular inclusion compound, clathrate), (III) cavitate-type inclusion compound (monomolecular/intramolecular inclusion...

The two-dimensional Bragg condition leads to the definition of reciprocal lattice vectors at and aj which fulfil the set of equations ... 

One now wonders whether these two phenomena are to be observed also for the whole two-dimensional surface of a crystal non-locking of the crystal surface in spite of lattice periodicity, and divergence of the fluctuation-induced thickening of the interface (or crystal surface), and in consequence the absence of facets. The last seems to contradict experience crystals almost by definition have their charm simply due to the beautifully shining facets which has made them jewelry objects since ancient times. 

Martensitic phase transformations are discussed for the last hundred years without loss of actuality. A concise definition of these structural phase transformations has been given by G.B. Olson stating that martensite is a diffusionless, lattice distortive, shear dominant transformation by nucleation and growth . In this work we present ab initio zero temperature calculations for two model systems, FeaNi and CuZn close in concentration to the martensitic region. Iron-nickel is a typical representative of the ferrous alloys with fee bet transition whereas the copper-zink alloy undergoes a transformation from the open to close packed structure. ... 

Secondly, the Llo structure must be stable with respect to both small and large strains. The former is assured by the positive definiteness of elastic moduli. Clearly, an exhaustive test of the stability with respect to large strains cannot be made and the following two types of tests were performed. First, the energy of the structure was calculated as a function of strains corresponding to 20% changes in the lattice parameters a and... 

Hydrogen has a very low solubility in the iron lattice, which makes direct observation of the location of the hydrogen atom in the lattice very difficult. The hydrogen definitely occupies an interstitial site in the bcc iron lattice. Two such sites are normally associated with interstitial solutes in bcc structures, the tetrahedral and the octahedral sites (see Fig. 8.39). Indirect evidence suggests that hydrogen occupies the tetrahedral site. 

Although it is obviously impossible to enumerate all possible configurations for infinite lattices, so long as the values of far separated sites are statistically independent, the average entropy per site can nonetheless be estimated by a limiting procedure. To this end, we first generalize the definitions for the spatial set and spatial measure entropies given above to their respective block-entropy forms. 

Block Entropies The definitions are the same as for finite lattices except that whereas before we could sum over all possible configurations, we now perform the same calculations for all possible sequences of sites within blocks of finite size B ... 

Fig. 8,20 First five iterations of an SDCA system starting from a 4-neighbor Euclidean lattice seeded with a single non-zero site at the center. The global transition rule F consists of totalistic-value and restricted totalistic topology rules C — (26,69648,32904)[3 3[ (see text for rule definitions). Solid sites have <7=1.

Lorentz-Invariance on a Lattice One of the most obvious shortcomings of a CA-based microphysics has to do with the lack of conventional symmetries. A lattice, by definition, has preferred directions and so is structurally anisotropic. How can we hope to generate symmetries where none fundamentally exist A strong hint comes from our discussion of lattice gases in chapter 9, where we saw that symmetries that do not exist on the microscopic lattice level often emerge on the macroscopic dyneimical level. For example, an appropriate set of microscopic LG rules can spawn circular wavefronts on anisotropic lattices. 

Observed first-order reflections from planes with one or two indices even, with the sum of all three indices even, and with the sum of any two indices even (Table II) require6 that the lattice underlying the structure be the simple orthorhombic lattice To- The types of prism planes giving first-order reflections (Table III) are such as to eliminate definitely all of the holohedral space groups7 VJ, to V, 6 (2Di—1 to 2Di—16) based on this lattice except V, V, Vj,3 and V 6. 

The presence of first-order reflections from all types of pyramidal planes (Table II) eliminates from consideration all space-groups based on any but the simple orthorhombic lattice r0. Of these the following are further definitely eliminated ) by the occurrence of first-order reflections from the prism planes given in Table III ... 

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