Lattices, space

The external development of smooth faces on a crystal arises from some regularity in the internal arrangement of the constituent ions, atoms or molecules. Any account of the crystalline state, therefore, should include some reference to the internal structure of crystals. It is beyond the scope of this book to deal in any detail with this large topic, but a brief description will be given of the concept of the space lattice. For further information reference should be made to the specialized works listed in the Bibliography. 

It is well known that some crystals can be split by cleavage into smaller crystals which bear a distinct resemblance in shape to the parent body. While there is clearly a mechanical limit to the number of times that this process can be repeated, eighteenth century investigators, Hooke and Haiiy in particular, were led to the conclusion that all crystals are built up from a large number of minute units, each shaped like the larger crystal. This hypothesis constituted a very important step forward in the science of crystallography because its logical extension led to the modern concept of the space lattice. 

A space lattice is a regular arrangement of points in three dimensions, each point representing a structural unit, e.g. an ion, atom or a molecule. The whole 

Type of symmetry Lattice Corresponding crystal system 

Cubic Cube Body-centred cube Face-centred cube Regular 

Although the exact morphology (the collection of particular crystal forms, their shapes, and texture) adopted by a specimen is under kinetic control, the number of possible crystal forms is restricted by symmetry, as stated earlier. The number of faces that belong to a form is determined by the symmetry of the point group of the lattice. Put another way, the symmetry of the internal structure causes the symmetry of the external forms. In this section we consider that internal symmetry in more detail. 

Matter is composed of spherical-like atoms. No two atomic cores—the nuclei plus inner shell electrons—can occupy the same volume of space, and it is impossible for spheres to fill all space completely. Consequently, spherical atoms coalesce into a solid with void spaces called interstices. A mathematical construct known as a space lattice may be envisioned, which is comprised of equidistant lattice points representing the geometric centers of structural motifs. The lattice points are equidistant since a lattice possesses translational invariance. A motif may be a single atom, a collection of atoms, an entire molecule, some fraction of a molecule, or an assembly of molecules. The motif is also referred to as the basis or, sometimes, the asymmetric unit, since it has no symmetry of its own. For example, in rock salt a sodium and chloride ion pair constitutes the asymmetric unit. This ion pair is repeated systematically, using point symmetry and translational symmetry operations, to form the space lattice of the crystal. 

The atoms in a crystalline substance occupy positions in space that can be referenced to lattice points, which crystallographers refer to as the asymmetric unit (physicists call it the basis). Lattice points represent the smallest repeating unit, or chemical point group. For example, in NaCl, each Na and Cl pair may be represented by a lattice point. In structures that are more complex, a lattice point may represent several atoms (e.g., polyhedra) or entire molecules. The repetition of lattice points by translations in space forms a space lattice, representing the extended crystal structure. 

It is important to study two-dimensional (2D) symmetry because of its applicability to lattice planes and the surfaces of three-dimensional (3D) solids. In two dimensions, a lattice point must belong to one of the 10 point groups listed in Table 1.3 (by the international symbols) along with their symmetry elements. This group, called the two-dimensional crystallographic plane group, consists of combinations of a single rotation axis perpendicular to the lattice plane with or 

TABLE 1.3 The 10 Two-Dimensional Crystallographic Plane Point Groups and Their Symmetry Elements (International Symbols)  

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Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

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 ... 

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

Iridium is not attacked by any of the acids nor by aqua regia, but is attacked by molten salts, such as NaCl and NaCN. The specific gravity of iridium is only very slightly lower than osmium, which is generally credited as the heaviest known element. Calculations of the densities of iridium and osmium from the space lattices give values of 22.65 and 22.61 g/cm 3, respectively. These values may be more reliable than actual physical measurements. At present, therefore, we know that either iridium or osmium is the densest known element, but the data do not yet allow selection between the two. 

Morphology. A crystal is highly organized, and constituent units, which can be atoms, molecules, or ions, are positioned in a three-dimensional periodic pattern called a space lattice. A characteristic crystal shape results from the regular internal stmcture of the soHd with crystal surfaces forming parallel to planes formed by the constituent units. The surfaces (faces) of a crystal may exhibit varying degrees of development, with a concomitant variation in macroscopic appearance. 

Figure 3 (a) Real-space lattice and reciprocal-space mesh for the GaAs (110) plane, (b)... 

Yet when Max von Laue, in 1943, commemorated the centenary of Groth .s birth, he praised him for keeping alive the hypothesis of the space lattice which was languishing everywhere else in Germany, and added that without this hypothesis it w ould have been unlikely that X-ray diffraction would have been discovered and even if it had been, it would have been quite impossible to make sense of it. 

Crystal growth is a diffusion and integration process, modified by the effect of the solid surfaces on which it occurs (Figure 5.3). Solute molecules/ions reach the growing faces of a crystal by diffusion through the liquid phase. At the surface, they must become organized into the space lattice through an... 

A univocal confirmation of the development of crystalline aggregation in the fiber is the occurrence of layer reflexes Oil, HI, ill, and 101 on the textural x-ray diffraction pattern. The details of organization of the space lattice are defined by the parameters of the unit cell and the number of polymers felling into one cell. The data, established by different authors, are presented in Table 2. Daubenny and Bunn s [8] pioneer findings are considered the most probable for space lattices occurring in PET fibers. 

A specific attribute of unit cell building is the inclination of the axis of macromolecule chains, in relation to the normal, to the plane of the base of the cell (ab). According to Yamashita [11] this inclination is within the range of 25-35° (Fig. 4). Against the background of space lattices of other types of fibers, the lattice of crystalline regions in PET fibers is characterized by a number of specific features. These are ... 

The space lattice does not undergo polymorphous transformation. As with other kinds of fibers, no transformation of the space lattice under the effect of any physical or chemical treatment of PET fibers has yet been found. 

The parallelization of crystallites, occurring as a result of fiber drawing, which consists in assuming by crystallite axes-positions more or less mutually parallel, leads to the development of texture within the fiber. In the case of PET fibers, this is a specific texture, different from that of other kinds of chemical fibers. It is called axial-tilted texture. The occurrence of such a texture is proved by the displacement of x-ray reflexes of paratropic lattice planes in relation to the equator of the texture dif-fractogram and by the deviation from the rectilinear arrangement of oblique diffraction planes. With the preservation of the principle of rotational symmetry, the inclination of all the crystallites axes in relation to the fiber axis is a characteristic of such a type of texture. The angle formed by the axes of particular crystallites (the translation direction of space lattice [c]) and the... 

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

Fig. 6.10 (a) Billiard balls of radius r = l/ /2 traveling on a unit-spaced lattice balls B and... 

Compound or mineral Approximate formula Symmetry Space Lattice constants Interlayer -distance (pm) Reference... 

For more precise work, micro hot stage methods under a microscope are used. For all compds, except those which are isotropic or become so on heating, the mp can best be observed by means of a polarizing microscope, since the temp at which color disappears arid the space lattice is ruptured is the true mp. Among numerous models of micro-hot stages, the Kofler micro-hot stage has attained widespread use and is commercially available (Refs 3 4)... 

Madelimg s proof of the hypothesis of space lattices was an indirect one. The direct proof was made in 1912 by Max von Lane, who used two conjectrrres as a starting point for his experiment. The first conjectrrre concerned the newly discovered x-rays, whose wave length was estimated in the range between 12 rrm and 5 pm. The other conjecture concerned the distance between the lattice planes. Based on these two conjectures he birilt the hypothesis that the interaction x-rays with crystal lattices should lead to interference, what he coitld show in experimerrts. 

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

Thermally activated mixed metal hydroxides, made from naturally occurring minerals, especially hydrotalcites, may contain small or trace amounts of metal impurities besides the magnesium and aluminum components, which are particularly useful for activation [946]. Mixed hydroxides of bivalent and trivalent metals with a three-dimensional spaced-lattice structure of the garnet type (Ca3Al2[OH]i2) have been described [275,1279]. 

One of the concepts in use to specify crystal structures the space lattice or Bravais lattice. There are in all fourteen possible space (or Bravais) lattices. 

Space lattices and crystal systems provide only a partial description of the crystal structure of a crystalline material. If the structure is to be fully specified, it is also necessary to take into account the symmetry elements and ultimately determine the pertinent space group. There are in all two hundred and thirty space groups. When the space group as well as the interatomic distances are known, the crystal structure is completely determined. 

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

Let a , a2, a3 be the real space lattice vectors and b, b2, b3 be the reciprocal space lattice vectors. We have then the following relations ... 

We now introduce a Fourier transform procedure analogous to that employed in the solution theory, s 62 For the purposes of the present section a more detailed specification of defect positions than that so far employed must be introduced. Thus, defects i and j are in unit cells l and m respectively, the origins of the unit cells being specified by vectors R and Rm relative to the origin of the space lattice. The vectors from the origin of the unit cell to the defects i and j, which occupy positions number x and y within the cell, will be denoted X 0 and X for example, the sodium chloride lattice is built from a unit cell containing one cation site (0, 0, 0) and one anion site (a/2, 0, 0), and the translation group is that of the face-centred-cubic lattice. However, if we wish to specify the interstitial sites of the lattice, e.g. for a discussion of Frenkel disorder, then we must add two interstitial sites to the basis at (a/4, a/4, a]4) and (3a/4, a/4, a/4). (Note that there are twice as many interstitial sites as anion-cation pairs but that all interstitial sites have an identical environment.) In our present notation the distance between defects i and j is... 

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