Notation crystal planes

The selective oxidation of ra-butane to give maleic anhydride (MA) catalyzed by vanadium phosphorus oxides is an important commercial process (99). MA is subsequently used in catalytic processes to make tetrahydrofurans and agricultural chemicals. The active phase in the selective butane oxidation catalyst is identified as vanadyl pyrophosphate, (V0)2P207, referred to as VPO. The three-dimensional structure of orthorhombic VPO, consisting of vanadyl octahedra and phosphate tetrahedra, is shown in Fig. 17, with a= 1.6594 nm, b = 0.776 nm, and c = 0.958 nm (100), with (010) as the active plane (99). Conventional crystallographic notations of round brackets (), and triangular point brackets (), are used to denote a crystal plane and crystallographic directions in the VPO structure, respectively. The latter refers to symmetrically equivalent directions present in a crystal. 

Figure 3.23. Model of surface plane for the evaluation of the surface-molecule reorientation and its effects on the first substrate planes. The changes in the molecular orientations are replaced by a compression or dilation of a set of two planes connected elastically the missing forces are assumed to act only on the hatched planes. The parameters of the model are the distance d between two hatched planes and the distance a separating two nearest-neighbor hatched planes belonging to two different crystal planes. The interaction forces between planes of different "molecules" are indicated with the notation in the text [cf. (3.38)].

The notations (111), (110), (101)... are Miller indices and define the crystal planes in the metal lattice. 

There are several ways to handle mathematically this orientation g. One can define as a rotation matrix or specify a crystal plane (hkl) and a crystal direction [uvw] which are parallel to (A, B) and A, respectively. In the texture analysis field the most used description for g consists of the set of 3 angles, the Euler angles. The coincidence of the two co-ordinate systems is then achieved by three rotations, which is illustrated in the stereographic projection in Fig. 7. The notation g denotes then in fact three variables. 

Miller indices (h, k, 1 in Equation 6.10) are a way of denoting a family of crystal planes and are defined by the intersection of the crystal planes with the crystal axes. The crystal planes, shown in the twodimensional case in Figure 6.2, may be denoted as (9,4.5) and (10,5). Usually, we divide by 4.5 and 5, respectively, and obtain (2,1) in both cases that is the two lines drawn belong to the same family of planes (lines). The notation of a family may thus be obtained from either Miller indices or the reciprocal space. 

Crystallographic directions and planes are important in both the characteristics and the applications of semiconductor materials since different crystallographic planes can exhibit significantly different physical properties. For example, the surface density of atoms (atoms per square centimeter) can differ substantially on different crystal planes. A standardized notation (the so-called Miller indices) is used to define the crystallographic planes and directions normal to those planes. 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

If you look back at Fig. 4.3, you can imagine that one way to make the surfaces shown in that figure would be to physically cleave a bulk crystal. If you think about actually doing this with a real crystal, it may occur to you that there are many different planes along which you could cleave the crystal and that these different directions presumably reveal different kinds of surfaces in terms of the arrangement of atoms on the surface. This means that we need a notation to define exactly which surface we are considering. 

Zircon belongs to the tetragonal system and is a positive uniaxial. The typical form shows the ill and the 110 planes. The two orientations selected for luminescence polarization study were the (110) plane, parallel to the basal section and the [100] row. In such cases the axis perpendicular to the (110) plane will be called X. The orientation notation is made according to the so-called Porto notation (Porto et al. 1956). The Xi(ZX2)Xi orientation means that the laser light entered parallel to the Xi axis of the crystal and is polarized in the Z direction, while the emission is collected along the Xi axis with X2 polarization. By polarization spectroscopy with a high spectral resolution (less then 0.1 nm) six lines are observed for the Dq- Fi transition of the Eu-II center instead of the maximum three allowed for an unique site (Fig. 5.12). In Z(XX)Z geometry which corresponds to observation of a-polarized luminescence we... 

We can now complete our answer to the question, What information is conveyed when we read that the crystal structure of a substance is monodime P2JC7" The structure belongs to the monoclinic crystal system and has a primitive Bravais lattice. It also possesses a two-fold screw axis and a glide plane perpendicular to it. The existence of these two elements of symmetry requires that there also be a center of inversion. The latter is not specifically included in the space group notation as it would be redundant. 

Fig. 1.6 (a) Arrangement of X-ray source, sample, and detector, used in X-ray direction from powders, (b) Typical diffraction pattern, showing the X-ray scattering as a function of angle. (The notation 20 is conventionally used for the scattering angle, as this relates to the theoretical interpretation given in Fig. 1.7.) The different peaks in (b) come from crystals oriented at different angles, so as to satisfy Bragg s Law (eqn 1.10) for an appropriate set of atomic planes.

Let us start with the simple case of an ideal crystal with one atom per unit cell that is cut along a plane, and assume that the surface does not change. The resulting surface structure can then be described by specifying the bulk crystal structure and the relative orientation of the cutting plane. This ideal surface structure is called the substrate structure. The orientation of the cutting plane and thus of the surface is commonly notated by use of the so-called Miller indices. 

Miller indices are determined in the following way1 (Fig. 8.1) The intersections of the cutting plane with the three crystal axes are expressed in units of the lattice constants. Then the inverse values of these three numbers are taken. This usually leads to non-integer numbers. All numbers are multiplied by the same multiplicator to obtain the smallest possible triple of integer numbers. The triple of these three numbers h, k, and l is written as (hkl) to indicate the orientation of this plane and all parallel planes. Negative numbers are written as n instead of —n. The notation hkl is used specify the hkl) planes and all symmetrical equivalent planes. In a cubic crystal, for example, the (100), (010), and (001) are all equivalent and summarized as 100. ... 

Figure 8.1 Notation of a cutting plane by Miller indices. The three-dimensional crystal is described by the three-dimensional unit cell vectors di, 02, and 03. The indicated plane intersects the crystal axes at the coordinates (3,1,2). The inverse is (, j, ). The smallest possible multiplicator to obtain integers is 6. This leads to the Miller indices (263).

The crystal structure of antimony pentachloride, SbCls, is hexagonal (D (, P63 jmmc, a0 = 7.49, and c0 = 8.01 A) with two molecules in the cell. The molecules are hep giving the simple notation 2P(h). In Figure 4.26, we see that the trigonal bipyramidal molecules have their C3 axes parallel to c0, the packing direction. The axial Sb—Cl distance is 2.34 A and that in the equatorial plane is 2.29 A. The closest Cl—Cl distances (3.33 A) are between axial Cl atoms between adjacent molecules. 

In anisotropic materials, the electronic bonds may have different polarizabilities for different directions (you may think of different, orientation-dependent spring constants for the electronic harmonic oscillator). Remembering that only the E-vector of the light interacts with the electrons, we may use polarized light to test the polarizability of the material in different directions, lno is one of the most important electro-optic materials and we use it as an example. The common notations are shown in Figure 4.7. If the E-vector is in plane with the surface of the crystal, the wave is called a te wave. In this example, the te wave would experience the ordinary index na of LiNbOs (nG 2.20). If we rotate the polarization by 90°, the E-ve ctor will be vertical to the surface and the wave is called tm. In lno, it will experience the extraordinary index ne 2.29. Therefore these two differently polarized waves will propagate with different phase velocities v c/n. In the example of Figure 4.7, the te mode is faster than the tm mode. 

Figure 4.7 Explanation of the notations The light beam propagates to the right. The LiNbOs z-axis is vertical to the main surface of the crystal. This is a z-cut . If the E- vector of the light beam lies parallel to the surface, this is called TE polarization. If the /(. -vector is oriented normal to the surface plane, this is a TM polarization...

FIGURE 2 HREM image of partial dislocations and their schematic maps found in the low temperature GaN layer near the interface (a) interstitial Frank loop (the position of the extra half plane is indicated by arrows in the image) (b) isolated Frank partial dislocation (c) reaction between a dissociated edge dislocation and Shockley partial that is displaced by one basal plane (d) dislocation notation for hexagonal crystal used in (a) - (c). (From [3].)... 

By convention, surface superlattices are given names like 31/2 x 1 R 30° this is Wood s 20 notation of 1964 [8], [9]. This notation describes the symmetry of the superlattice, but does not identify its origin—that is, the exact point where the superlattice is anchored (even though this is a very desirable datum for aficionados of chemisorption). The 3m x 1 R 30° superlattice means that (1) the fls and b axes of the superlattice crystal lie in the plane of the surface of index hkl (whose axes we call abM and bbki), (2) the first superlattice axis as is 31/2 times longer than the first surface axis fl(3) the second superlattice axis bs is equal in length to the second surface axis b/ h (4) these two superlattice axes fls and bs are then rotated by 30° clockwise. An alternative notation uses... 

Construction of the (31/2 x 31/2) R 30° superlattice of the (111) plane of an FCC crystal. If this is Au, then the full Wood s notation designation is Au(111) (31/2x31/2) R 30°. Shown are an acute surface unit cell (center) and the preferred obtuse unit cell (lower left) with basis vectors a-m = a/2+6/2 and = b/2 + c/2, with lengths 2 1/2a and 2 1/2a each and with an included angle of 120°. Also shown are two alternate but equivalent settings of the primitive (31/2 x 31/2) R 30° supercell The bottom one is anchored at interstitial sites (shaded circles) with basis vectors... 

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 d notation indicates a diamond glide plane, found in diamond or zinc blende extended crystal structures. Whereas ghde planes are found in many inorganic-based crystals, screw axes are found predominantly in protein structures. 

---