Unit cell diamond

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

Crystal Structure. Diamonds prepared by the direct conversion of well-crystallized graphite, at pressures of about 13 GPa (130 kbar), show certain unusual reflections in the x-ray diffraction patterns (25). They could be explained by assuming a hexagonal diamond stmcture (related to wurtzite) with a = 0.252 and c = 0.412 nm, space group P63 /mmc — Dgj with four atoms per unit cell. The calculated density would be 3.51 g/cm, the same as for ordinary cubic diamond, and the distances between nearest neighbor carbon atoms would be the same in both hexagonal and cubic diamond, 0.154 nm. 

The ultimate covalent ceramic is diamond, widely used where wear resistance or very great strength are needed the diamond stylus of a pick-up, or the diamond anvils of an ultra-high pressure press. Its structure, shown in Fig. 16.3(a), shows the 4 coordinated arrangement of the atoms within the cubic unit cell each atom is at the centre of a tetrahedron with its four bonds directed to the four corners of the tetrahedron. It is not a close-packed structure (atoms in close-packed structures have 12, not four, neighbours) so its density is low. 

Figure 8.3 Structure of diamond showing the tetrahedral coordination of C the dashed lines indicate the cubic unit cell containing 8 C atoms.

Figure 2. Total energy for FeaNi along the Bain path with V = const (a). Binding energy versus volume of the unit cell for the ferromagnetic (FM) bet (circle) and fee (square) states. Diamonds results for the nonmagnetic (NM) fee phase.

X-ray diffraction experiments revealed a psendo-cubic orthorombic unit cell with cell dimensions similar to the expected cubic F centered arrangement of the predesigned diamond-like crystal. 

The unit cell of cubic diamond corresponds to a face-centered packing of carbon atoms. Aside from the four C atoms in the vertices and face centers, four more atoms are present in the centers of four of the eight octants of the unit cell. Since every octant is a cube having four of its eight vertices occupied by C atoms, an exact tetrahedral coordination results for the atom in the center of the octant. The same also applies to all other atoms — they are all symmetry-equivalent. In the center of every C-C bond there is an inversion center. As in alkanes the C-C bonds have a length of 154 pm and the bond angles are 109.47°. 

Structure of cubic (left) and hexagonal (right) diamond. Top row connected layers as in a-As. Central row the same layers in projection perpendicular to the layers. Bottom unit cells when the light and dark atoms are different, this corresponds to the structures of zinc blende (sphalerite) and wurtzite, respectively... 

The drawn cell corresponds to a unit cell of diamond (a-Sn Si-I) that has been strongly compressed in one direction. Right coordination about a tin (or Si) atom with bond lengths cf. the atom in the dashed octant... 

Of the numerous ternary and polynary diamond-like compounds we deal only with those that can be considered as superstructures of zinc blende. A superstructure is a structure that, while having the same structural principle, has an enlarged unit cell. When the unit cell of zinc blende is doubled in one direction (c axis), different kinds of atoms can occupy the doubled number of atomic positions. All the structure types listed in Fig. 12.8 have the tetrahedral coordination of all atoms in common, except for the variants with certain vacant positions. 

Left unit cell of NaTl. The plotted bonds of the thallium partial structure correspond to the C-C bonds in diamond. Right section of the structure of SrGa2 and MgB2 (A1B2 type)... 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

In the course of the PP calculations of these quantities for Si [10] and Ge [11], a characteristic local pattern which reflects position, shape and size of a specific atom in the crystal is observed on the contour map of the valence electron A(r)-function. The atom is one of the two atoms in the unit cell of diamond structure. It seems as... 

Figure 11.1 Comparison of the atomic structure of cristobalite (high temperature form of Si02) with that of silicon (diamond structure using tetrahedral unit cell).

Figure 7.5 (a) Artificial quantum dot architecture showing the confined electron spins, (b) A diamond unit cell showing a NV centre - a nitrogen defect and a carbon vacancy - with an S = 1 electronic spin... 

Figure 1.5 Diamond structure (a) unit cell and (b) viewed with [111] vertical.

Figure 4.15 Hollandite structure. The shaded diamonds represent chains of edge-shared MnC>6 octahedra and the shaded circles Ba2+. The light and heavy shading represent octahedra and atoms at two different heights. The unit cell is outlined.

Several methods are also available for determination of the isothermal compressibility of materials. High pressures and temperatures can for example be obtained through the use of diamond anvil cells in combination with X-ray diffraction techniques [10]. kt is obtained by fitting the unit cell volumes measured as a function of pressure to an equation of state. Very high pressures in excess of 100 GPa can be obtained, but the disadvantage is that the compressed sample volume is small and that both temperature and pressure gradients may be present across the sample. 

The diamond structure, see Fig. 7.14 below, is a 3D network in which every atom is surrounded tetrahedrally by four neighbours. The eight atoms in the unit cell may be considered as forming two interpenetrating face-centred cubic networks. If the two networks are occupied by different atoms, the derivative cF8-ZnS (sphalerite) type structure is obtained. As a further derivative structure, the tI16-FeCuS2 type structure can be mentioned. These are all examples of a family of tetrahedral structures which have been described by Parthe (1964). 

The otherl4th group elements, Si, Ge and oSn have the diamond-type structure. The tI4- 3Sn structure (observed for Si and Ge under high pressure) can be considered a very much distorted diamond-type structure. Each Sn has four close neighbours, two more at a slightly larger and another four at a considerable larger distance. Fig. 7.13 shows the (3Sn unit cell. Lead, at ambient pressure, has a face-centred cubic cF4-Cu type structure. 

Several superstructures and defect superstructures based on sphalerite and on wurtzite have been described. The tI16-FeCuS2 (chalcopyrite) type structure (tetragonal, a = 525 pm, c = 1032 pm, c/a = 1.966), for instance, is a superstructure of sphalerite in which the two metals adopt ordered positions. The superstructure cell corresponds to two sphalerite cells stacked in the c direction. The cfla ratio is nearly 1. The oP16-BeSiN2 type structure is another example which similarly corresponds to the wurtzite-type structure. The degenerate structures of sphalerite and wurtzite (when, for instance, both Zn and S are replaced by C) correspond to the previously described cF8-diamond-type structure and, respectively, to the hP4-hexagonal diamond or lonsdaleite, which is very rare compared with the cubic, more common, gem diamond. The unit cell dimensions of lonsdaleite (prepared at 13 GPa and 1000°C) are a = 252 pm, c = 412 pm, c/a = 1.635 (compare with ZnS wurtzite). 

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