Unit cell wurtzite

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. 

Fig. 4.12 (a) CdSe wurtzite unit cell (b) schematic illustration of a hexagonal (wurtzite) CdSe basal plane on a (111) section of the gold lattice, emphasizing the 2 3 lattice match. Note the [111] Au//(0001)CdSe orientation, with the CdSe a-directions aligned along the (llO)Au. The outlined rhombus indicates the projection of a CdSe unit cell. (Adapted from [112])... 

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... 

Aside from the superstructures mentioned, other superstructures with other enlargement factors for the unit cell are known, as well as superstructures of wurtzite. Defect structures,... 

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). 

In the sphalerite structure the anions form a cubic close packed array. The structure has a single adjustable parameter, the cubic cell edge. The 0 ions are too small for them to be in contact in this structure (see Fig. 6.4) so ZnO adopts the lower symmetry hexagonal wurtzite structure which has three adjustable parameters, the a and c unit cell lengths and the z coordinate of the 0 ion, allowing the environment around the Zn " ion to deviate from perfect tetrahedral symmetry. In the sphalerite structure the ZnX4 tetrahedron shares each of its faces with a vacant octahedral cavity (one is shown in Fig. 2.6(a)), while in the wurtzite structure one of these faces is shared with an empty tetrahedral cavity which places an anion directly over the shared face as seen in Fig. 2.6(b). The primary coordination number of Zn " in sphalerite is 4 and there are no tertiary bonds, but in wurtzite, which has the same primary coordination number, there is an additional tertiary bond with a flux of 0.02 vu through the face shared with the vacant tetrahedron. 

Wurtzite structure. Zinc sulfide can also crystallize in a hexagonal form called wurtzite that is formed slightly less exothermically than the cubic zinc blende (sphalerite) modification (Afff = —192.6 and —206.0 kJ mol-1, respectively) and hence is a high temperature polymorph of ZnS. The relationship between the two structures is best described in terms of close packing (Section 4.3) in zinc blende, the anions (or cations) form a cubic close-packed array, whereas in wurtzite they form hexagonal close-packed arrays. This relationship is illustrated in Fig. 4.13 note, however, that this does not represent the actual unit cell of either form. 

There are two polymorphic structures of ZnS, zinc blende (or sphalerite) (3 2PT) and wurtzite (2 2PT). In zinc blende there is a ccp arrangement of S atoms with Zn atoms filling one of the two T layers as shown in Figure 6.1. The diamond has the same structure, with the sites of P and one T layer filled by C atoms (Section 4.3.3). The structure of zinc blende has six (3 2) layers in the repeating unit. This structure is encountered for many binary compounds with significant covalent character as shown in Table 6.1. The space group for zinc blende is T%, F43m, and a0 = 5.4093 A, for the cubic unit cell. ... 

In the crystal structure of hexagonal zinc sulfide (wurtzite), the S atoms are arranged in hep, in which half of the tetrahedral interstices are filled with Zn atoms, and the space group is C v — P6imc. The positions of atoms in the hexagonal unit cell are... 

AIN exists in two types the hexagonal (wurtzite structure) and the cubic (zincblende structure). The former is more stable, and has been investigated in more detail. The wurtzitic AIN has two formula units per unit cell (4 atoms per cell) and 9 optical branches to the phonon dispersion curves [1] ... 

An enormous range of properties is found in oxides. The most successful and most widely used substrate for GaN to date is sapphire, AI2O3. Except for wurtzite materials, few of the unit cells in these materials match with GaN, but it is usually useful to think of these systems as having a close-packed nitrogen lattice in GaN matching to a near close-packed oxygen lattice in the oxide. 

Optical pumping experiments were first used to achieve lasing m GaN-based structures. Stimulated emission from GaN was observed as early as 1971 [17]. More recently, there have been a large number of reports on stimulated emission [18,19], without an intentionally formed cavity. This may partly be due to the well known difficulty of cleaving mirrors in the wurtzite nitrides grown on sapphire, due to the 30° tilt of the GaN unit cell with respect to the sapphire. 

The high temperature modification of ZnAl2S4 was described by Hahn and Frank (12) as being of the wurtzite type, with a disordered distribution of the three metal atoms and one gap on the four sites of the metal lattice. Recently, it has been shown (5) that this compound has a superstructure of wurtzite with an orthorhombic unit cell. The unit cell of ZnAl2S4 is actually monoclinic with a ft angle of 90°. Its space group is Pm—C1S. The parameters of the unit cell... 

Figure 1-28 Unit cells of the zinc blende and wurtzite structures. Black circles indicate metal cations, open circles anions. Note the similarity to the diamond structure, Fig. 1-26.

William B. Pearson (1921-2005) developed a shorthand system for denoting alloy and intermetallic structure types (Pearson, 1967). It is now widely used for ionic and covalent solids, as well. The Pearson symbol consists of a small letter that denotes the crystal system, followed by a capital letter to identify the space lattice. To these a number is added that is equal to the number of atoms in the unit cell. Thus, the Pearson symbol for wurtzite (hexagonal, space group PS mc), which has four atoms in the unit ceU, is hPA. Similarly, the symbol for sodium chloride (cubic, space group Fm3m), with eight atoms in the unit cell, is cF8. 

The wurtzite structure is adopted by most of the remaining compounds comprised of elements from the same groups as zinc blende, but which do not take the zinc blende strucmre, for example, AIN, InN, CdSe. The strucmre seems to be able to accommodate larger electronegativity differences between the constiment atoms, as in BeO, GaN, and ZnO. For these more ionic compounds, the wurtzite unit cell must be more stable than that of zinc blende, to an extent governed by the specific bonding forces in each case. 

Figure 3.13. The hexagonal ZnS or wurtzite unit cell. Cations are the dark shaded circles.

Figure 3 Unit cells for (a) Rock salt, (b) Wurtzite, (c) Fluorite (d) Rutile...

Figure 21 Two complete hysteresis cycles for 4.5 mn CdSe NQDs presented as unit cell voliunes for the wurtzite sixfold-coordinated phase (triangles) and the rock-salt fourfold-coordinated phase (squares) versus pressure. Sohd arrows indicate the direction of pressure change, and dotted boxes indicate the mixed-phase regions. Unlike hulk phase transitions, the wurtzite to rock-salt transformation in nanocrystals is reversible and occurs without the formation of new high-energy defects, as indicated by overlapping hysteresis loops. The shape change that a sliding-plane transformation mechanism (see text) would induce is shown schematically on the right. (Reprinted figure with permission from J.N. Wickham, A.B. Herhold, and A.P. Alivisatos, Phys. Rev. Lett., 2000, 84, 923. 2000 by the American Physical Society)...

The crystal structure for this complex lattice (wurtzite structure) is shown in Figure 2.15. This is best described as an hep lattice of sulfide ions, with zinc ions occupying one-half of the available tetrahedral interstitial sites. For this lattice, there are two units of ZnS per unit cell ... 

Figure 2.15. Model of the wurtzite (ZnS) crystal structure. The framework is based oir air hep lattice of anions (yellow the unit cell consists of A and B ions) with zinc ions occupying tetrahedral interstitial sites (white, labeled as X and Y ions).

Fig. 7.3. The relationship between number of shared edges between cation tetra-hedra and both the electrostatic Madelung energy and the one-electron covalent band-structure energy for 2 real observed and 21 hypothetical polymorphs of BeO. The structure types plotted are the wurtzite (a-BeO) type (-I-), the p-BeO type (X), and the 21 hypothetical dipolar tetrahedral structures with (1,1) or (2,1) unit cells. The small numbers by some of the points show how many points are represented by the single symbol (after Burdett and McLarnan, 1984 reproduced with the publisher s permission).

Fig. 4.2 Unit cells of two zinc sulfiide (2 2) structures circles in order of decreasing sire are S and Zn (a) wurtzite. hexagonal, space group P6 mc (b> zinc blende, cubic, space group F43m. (Frooi Ladd, M. F. C. Structure and Bonding in Solid State Chemistiy, Wiley New York. 1979. Reproduced mth permission.]...

Wurtzite ZnO structure with four atoms in the unit cell has a total of 12 phonon modes (one longitudinal acoustic (LA), two transverse acoustic (TA), three longitudinal optical (LO), and six transverse optical (TO) branches). The optical phonons at the r point of the Brillouin zone in their irreducible representation belong to Ai and Ei branches that are both Raman and infrared active, the two nonpolar 2 branches are only Raman active, and the Bi branches are inactive (silent modes). Furthermore, the Ai and Ei modes are each spht into LO and TO components with different frequencies. For the Ai and Ei mode lattice vibrations, the atoms move parallel and perpendicular to the c-axis, respectively. On the other hand, 2 modes are due to the vibration of only the Zn sublattice ( 2-low) or O sublattice ( 2-high). The expected Raman peaks for bulk ZnO are at 101 cm ( 2-low), 380 cm (Ai-TO), 407 cm ( i-TO), 437 cm ( 2-high), and 583 cm ( j-LO). 

Figure 9. High-resolution transmission electron microscope image of most of the interior of an 4 nm diameter ZnS particle produced as the result of activity of sulfate-reducing bacteria. The image details show that the particle consists of a mixture of wurtzite and sphalerite-like regions. Unit cell axes are shown for the wurtzite region (Banfield et al., unpublished).

Other Metal Sulphides. A survey of lattice data and structure types of 40 compounds of the type Cu2ABS4, where A = Mn, Fe, Co, Ni, Zn, Cd, or Hg and B = Si, Ge, or Sn, has shown that three tetrahedral structure types, differing in symmetry and unit-cell size, exist.254 All of the compounds were found to adopt one of the following structure types the stannite structure, an orthorhombic superstructure of wurtzite, or a hitherto unknown structure based on slightly distorted sphalerite cells of tetragonal, orthorhombic, or monoclinic symmetry. 

Derive simplified expressions for F for the wurtzite form of ZnS, including the rules governing observed reflections. This crystal is hexagonal and contains 2 ZnS per unit cell, located in the following positions ... 

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