Lattice of concept

It is most important to note that the lattice of concepts given obviously allows reconstructing the context of Fig. 1. In fact, also in the general case, the lattice of concepts, better say the Hasse diagram of the lattice of concepts perfectly reflects and visualizes the information contained in the context in question For example, we can easily read off from the above diagram, that fire is supposed to be warm and dry (and neither humid nor cold). This is why Hasse Diagram Technique (HDT) is so helpful and important. 

The first electrodeposition of a compound superlattice appears to have been by Rajeshwar et al. [219], where layers of CdSe and ZnSe were alternately formed using codeposition in a flow system. That study was proof of concept, but resulted in a superlattice with a period significantly greater then would be expected to display quantum confinement effects. There have since been several reports of very thin superlattices formed using EC-ALE [152, 154, 163, 186], Surface enhanced Raman (SERS) was used to characterize a lattice formed from alternated layers of CdS and CdSe [163]. Photoelectrochemistry was used to characterize CdS/ZnS lattices [154, 186]. These EC-ALE formed superlattices were deposited by hand, the cycles involving manually dipping or rinsing the substrate in a sequence of solutions. 

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

A description which in some simple cases could be considered alternative to those exemplified in Table 3.2 is based on the lattice complex concept. Listing the symbols of the lattice complexes occupied by the different atoms in a structure (for instance, symbol P for the point 0, 0, 0 and its equivalent points), provides in fact... 

The structure of SWCNTs is characterized by the concept of chirality, which essentially describes the way the graphene layer is wrapped and is represented by a pair of indices (n, m). The integers n and m denote the number of unit vectors (a a2) along the two directions in the hexagonal crystal lattice of graphene that result in the chiral vector C (Fig. 1.1) ... 

A system of adsorbed particles is often treated as a two-dimensional gas covering the adsorbent surface. Such an approach is quite justified and fruitful, as long as we are dealing with physical adsorption when the influence of the adsorbent on the adsorbate can be regarded as a weak perturbation. In case of chemical adsorption (the most frequent in catalysis), the concept of a two-dimensional gas becomes untenable. In this case the adsorbed particles and the lattice of the adsorbent form a single quantum-mechanical system and must be regarded as a whole. In such a treatment the electrons of the crystal lattice are direct participants of the chemical processes on the surface of the crystal in some cases they even regulate these processes. 

We shall proceed from a concept which in a certain sense is contrary to that of the two-dimensional gas. We shall treat the chemisorbed particles as impurities of the crystal surface, in other words, as structural defects disturbing the strictly periodic structure of the surface. In such an approach, which we first developed in 1948 (I), the chemisorbed particles and the lattice of the adsorbent are treated as a single quantum-mechanical system, and the chemisorbed particles are automatically included in the electronic system of the lattice. We observe that this by no means denotes that the adsorbed particles are rigidly localized they retain to a greater or lesser degree the ability to move ( creep ) over the surface. 

It was therefore appropriate that the first attempt to produce lattice stabilities for non-allotropic elements dealt with Cu, Ag and Zn (Kaufman 1959b). It is also significant that, because of the unfamiliarity of the lattice stability concept, this paper did not appear as a mainstream publication although the work on Ti and Zr (Kaufman 1959a) was published virtually at the same time. It was also realised that Ae reliability of metastable melting points derived by extrapolation were best... 

Before interface energy was understood, the concept was explained by Friedel [2] in terms of a compound or twin lattice. This may be explained as follows (see Fig. 7.2). If the lattice of one individual crystal is extended to superpose that of the other, where both are projected onto the same plane, a new lattice consisting of common lattice points results. This lattice is called a twin lattice or compound lattice, and the twin index is defined by the number of multiples of the unit cell size of a single crystal. The smaller the twin index, the higher the probability that twinning will occur. 

The l index of every spot is thus obvious by mere inspection. The other two indices are best obtained by a graphical method. Just as all spots with the same l indices (in the present example) lie on definite lines, so all spots with the same hk values lie on definite curves. But these (hk curves have a form less simple than the l curves . The form of these curves is most readily determined by introducing a piece of mental scaffolding known as the reciprocal lattice —a conception which has proved to be a tool of the greatest value for the solution of all geometrical problems concerned with the directions of X-ray reflections from crystals. It was introduced by Ewald (1921). 

It is an unusual problem to construct a homogeneously distorted two-dimensional domain structure which allows us to cover the range of the hexagonal closely packed crystal up to a two-dimensional gas. For example, the concept of the paracrystal is based on the philosophy that every state of condensed matter has at least a micro-paracrystalline arrangement of segments within a distorted lattice of arbitrary symmetry with a defined coordination number. 

Cases of chemisorption are known in which at high coverages the net (two-dimensional lattice) of the adsorbate is not in registry with the lattice of the adsorbent. In such situations, the concept of sites of precise location and fixed number may not be applicable. Similar difficulties about the definition of sites will occur if surface reconstruction takes place upon interaction of adsorbate and adsorbent. 

Note that any mixing can hardly take place at either of the interfaces between reacting solid phases. Therefore, the basic assumptions of R. Pretorius et al.261,262 about a limiting element and its concentration at an interface seem to be somewhat artificial. In fact, in the reactions under consideration, there is no reaction volume, with all the interactions proceeding onto the phase surfaces exposed to each other. The distances between those surfaces scarcely exceed considerably the usual values of interplanar spacings in the crystal lattices of chemical compounds. Hence, the concept concentration of a limiting element at an interface does not appear to have any real physical meaning. 

Carbide s patent department had reservations because in their experience, composition of matter claims had to be drawn very narrowly to be valid and as such were frequently easy to circumvent. We convinced them that the unique properties that make A and X useful are singular results of their specific chemical composition and the arrangement of atoms in the crystal lattice of the zeolites. To our knowledge, the use of powder x-ray data as a finger print to uniquely identify a specific crystal structure was a new concept in patent protection. 

Thermal conductivity is the most difficult quantity to understand in terms of the electronic structure. Thermal energy can be stored in vibrational normal modes of the crystal, and one can transport thermal energy through the lattice of ions. These concepts seem to be macroscopic. Therefore, one can set up suitable wave packets to treat thermal conductivity as quantized matter. In particular, electron plus induced lattice polarization can be defined as polarons. For conduction electrons, the electrical conductivity and the thermal conductivity were first observed by Wiedemann and Franz as indicated in the following equation ... 

Note also that in a triclinic crystal a and a are not collinear in a monoclinic crystal (b unique setting) b is parallel to b, but a and c form the obtuse angle [>, while a and c form a smaller acute angle /T given by fJ = 180 — fi. The reciprocal lattice vectors and the direct lattice vectors are a ying-yang duo of concepts, as are position space and momentum space, or space domain and time domain. Fourier transformation helps us walk across from one space to other, as convenience dictates Some problems are easy in one space, others in the space dual to it this amphoterism is frequent in physics. The directions of the direct and reciprocal lattice vectors are shown as face normals in Fig. 7.22. 

Since polymers cannot be completely crystalline (i.e. cannot have a perfectly regular crystal lattice) the concept "crystallinity" has been introduced. The meaning of this concept is still disputed (see Chap. 2). According to the original micellar theory of polymer crystallisation the polymeric material consists of numerous small crystallites (ordered regions) randomly distributed and linked by intervening amorphous areas. The polymeric molecules are part of several crystallites and of amorphous regions. 

The conventional macroscopic Fourier conduction model violates this non-local feature of microscale heat transfer, and alternative approaches are necessary for analysis. The most suitable model to date is the concept of phonon. The thermal energy in a uniform solid material can be jntetpreied as the vibrations of a regular lattice of closely bound atoms inside. These atoms exhibit collective modes of sound waves (phonons) wliich transports energy at tlie speed of sound in a material. Following quantum mechanical principles, phonons exhibit paiticle-like properties of bosons with zero spin (wave-particle duality). Phonons play an important role in many of the physical properties of solids, such as the thermal and the electrical conductivities. In insulating solids, phonons are also (he primary mechanism by which heal conduction takes place. 

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