Diamond cubic lattice

Figure 5.1. Illustration of a unit cell of the diamond cubic lattice. The arrows designate the [001], [010], and [100] directions. Both silicon and germanium crystallize into the diamond cubic lattice structure, where each atom is bonded to four neighboring atoms in a tetrahedral geometry. Figure reproduced from Ref. [30] with permission.

Figure 5.4. The highest occupied molecular orbital of a Si911,2 dimer cluster. The top two silicon atoms comprise the surface dimer, and the remaining seven Si atoms contain three subsurface layers which are hydrogen terminated to preserve the sp3 hybridization of the bulk diamond cubic lattice. The up atom is nucleophilic and the down atom is electrophilic.

In its crystalline state, germanium, similar to silicon, is a covalent solid that crystallizes into a diamond cubic lattice structure. Like for Si, both the (100) and (111)... 

The element crystallizes in a diamond cubic lattice. It is brittle, and has a bright metalhc luster. Ge can absorb H2,02,... 

In some structures, several planes and directions may be equivalent by symmetry. For example, this is the case for the (100), (010), (001), (100), (010), and (OOl) planes in the diamond cubic structure. Equivalent directions are denoted concisely as a group by using angular brackets. Thus, the (100) directions in a diamond cubic lattice include all of the directions that are perpendicular to the six planes noted above. The Miller index notation thus provides a concise designation for describing the surfaces of semiconductor crystals. 

The starting point is a supercell of the diamond-cubic lattice (FC-2) consisting of N = 8n atoms, where n is the number of FC-2 cells along one dimension. Typically, the supercells have contained from 216 to 4096 atoms. These cells are randomized, roughly speaking melted, by progressively and randomly introducing bond switches as illustrated in Fig. 1. 

Silicon has a diamond cubic lattice structure with an atomic density of 5 x 1022 atoms cnT3, an atomic weight of 28.09 and a density of 2.33 g cnT3 (a) What are silicon s lattice parameter and atomic volume, 2V1... 

Hornstra [10] discussed several possible structures of dislocations in the diamond-cubic lattice. Starting from the fact that an arbitrary direction in the crystal may be considered as the sum of steps in <110) directions, he considered that the only simple dislocations that need to be studied are those along <110) directions. Then, it appears that the dislocation segments can be of type screw, edge, or 60° [Figs 3(a) and 3(b)]. 

In a class of reahstic lattice models, hydrocarbon chains are placed on a diamond lattice in order to imitate the zigzag structure of the carbon backbones and the trans and gauche bonds. Such models have been used early on to study micelle structures [104], monolayers [105], and bilayers [106]. Levine and coworkers have introduced an even more sophisticated model, which allows one to consider unsaturated C=C bonds and stiffer molecules such as cholesterol a monomer occupies several lattice sites on a cubic lattice, the saturated bonds between monomers are taken from a given set of allowed bonds with length /5, and torsional potentials are introduced to distinguish between trans and "gauche conformations [107,108]. 

Figure 4.3.2 The diamond crystalline lattice structure composed of two interpenetrating face-centered cubic lattices.

For nuclei that have perfect cubic site symmetry (e.g., those in an ideal rock salt, diamond, or ZB lattice) the EFG is zero by symmetry. However, defects, either charged or uncharged, can lead to non-zero EFG values in nominally cubic lattices. The gradient resulting from a defect having a point charge (e.g., a substitutional defect not isovalent with the host lattice) is not simply the quantity calculated from simple electrostatics, however. It is effectively amplified by factors up to 100 or more by the Sternheimer antishielding factor [25],... 

A traditional example of a Zintl phase is represented by NaTl which may be considered as a prototype of the Zintl rules. The structure of this compound (face centred cubic, cF16, a = 747.3 pm) can be described (see also 7.4.2.2.) as resulting from two interpenetrating diamond type lattices corresponding to the arrangements of the Na and T1 atoms respectively (Zintl and Dullenkopf 1932). Each T1 atom therefore is coordinated to other four T1 at a distance a)3/4 = 747.3)3/4 = 323.6pm which is shorter than that observed in elemental thallium (d = 341-346 pm in aTl, hP2-Mg type, CN = 6 + 6) and d = 336pm in /3 Tl, (cI2-W type, CN = 8). 

Diamond is crystallized in cubic form (O ) with tetrahedral coordination of C-C bonds around each carbon atom. The mononuclear nature of the diamond crystal lattice combined with its high symmetry determines the simplicity of the vibrational spectrum. Diamond does not have IR active vibrations, while its Raman spectrum is characterized by one fundamental vibration at 1,332 cm . It was found that in kimberlite diamonds of gem quality this Raman band is very strong and narrow, hi defect varieties the spectral position does not change, but the band is slightly broader (Reshetnyak and Ezerskii 1990). 

Structure tP4 (CuAu) is ordered with respect to an underlying face-centred cubic lattice, so that it takes the Jensen symbol 12/12. The CuAu lattice does show, however, a small tetragonal distortion since the ordering of the copper and gold atoms on alternate (100) layers breaks the cubic symmetry. Zinc blende (cF8(ZnS)) and wurtzite (hP4(ZnS)) are ordered structures with respect to underlying cubic and hexagonal diamond lattices respectively. Since both lattices are four-fold tetrahedrally coordinated, differing only in... 

This minimum is responsible for the diamond and graphite lattices with = 109° and 120° respectively having the smallest and second smallest values of the normalized fourth moment, and hence the shape parameter, s, in Fig. 8.7. This is reflected in the bimodal behaviour of their densities of states in Fig. 8.4 with a gap opening up for the case of the diamond cubic or hexagonal lattices. Hence, the diamond structure will be the most stable structure for half-full bands because it displays the most bimodal behaviour, whereas the dimer will be the most stable structure for nearly-full bands because it has the largest s value and hence the most unimodal behaviour of all the sp-valent lattices in Fig, 8.7, We expected to stabilize the graphitic structure as we move outwards from the half-full occupancy because this... 

We shall now discuss the method of crystal growth and the electronic properties of GaAs, a typical example of a III-V compound which is expected to become more useful than Si and Ge in the near future, concentrating on the relation between non-stoichiometry and physical properties. GaAs has a zinc blende type structure, which can be regarded as an interpenetration of two structures with face centred cubic lattices, as shown in Fig. 3.29. Disregarding the atomic species, the structure is the same as a diamond-type... 

C is equal to unity when each nodule has six first neighbors, by analogy with a simple cubic lattice29. Similarly Cis of the order of 0.87,1.09,1.12 when the geometric functionality is of the order of 4,8,12, respectively these values originate from calculations carried out on diamond-type, centered, and face-centered cubic lattices, which exhibit precisely these geometric functionalities (or coordination indices). In any case, Cis a constant for a given network, and its value is never very far from unity. 

Silicon crystallizes in the diamond structure,16 which consists of two interpenetrating face-centered cubic lattices displaced from each other by one quarter of the body diagonal. In zinc blende semiconductors such as GaAs, the Ga and As atoms lie on separate sublattices, and thus the inversion symmetry of Si is lost in III-V binary compounds. This difference in their crystal structures underlies the disparate electronic properties of Si and GaAs. The energy band structure in... 

The preparation of trivalent nitrogen was achieved in 2004 by Eremets et al. in a diamond cell at 1150 000 bar and 2000 K [47-49]. The crystallographic data for the trivalent nitrogen is cubic, lattice parameter a = 3.4542(9) A. A three-dimensional structure which consisted of trivalent nitrogen atoms (Fig. 9.5) was found. The N—N bond length at l.lMbar is 1.346 A, and the NNN angle is 108.8°. The nitrogen atoms form screws of trivalent atoms which are connected to form a three-dimensional network. 

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