Parameter lattice

Lattice parameters of ZnO have been investigated over many decades [22-30]. The lattice parameters of a semiconductor usually depend on the following factors (i) free electron concentration acting via deformation potential of a conduction hand minimum occupied by these electrons, (u) concentration of foreign atoms and defects and their difference of ionic radii with respect to the substituted matrix ion, (iii) external strains (e.g., those induced by substrate), and (iv) temperature. The lattice parameters of any crystalline material are commonly and most accurately measured by high-resolution X-ray diffraction (HRXRD) using the Bond method [31] for a set of symmetrical and asymmetrical reflections. Table 1.2 tabulates measured and calculated lattice parameters, cja ratio, and u parameter reported by several groups for ZnO crystallized in wurtzite, zinc blende, and rocksalt structures for comparison. 

High-pressure phase transition from the wurtzite to the rocksalt structure decreases the lattice constant down to the range of4.271-4.294A. The experimental values obtained by X-ray diffraction are in close agreement. The predicted lattice parameters of4.058-4.316 A, using various calculation techniques such as the HF-PI, GGA, and HF, are about 5% smaller or larger than the experimental values. The dispersion in the calculated values is larger than the measured ones. 

The lattice parameters (or lattice constants) that define the unit cell geometry are the lengths a, b, and c of the unit cell edges and the angles a, f, and y between the b and c axes, c and a axes, and a and b axes, respectively. The first step in 

Carrying out the scalar multiplications on the right of (3.3) and expressing the results in terms of a, b, c, a, P, and y leads, after some heavy arithmetic (see Warren10), to 

For most of the semicrystalline polymers that have been synthesized and studied to any extent to date the unit cell parameters are known at least approximately, and a good tabulation of such data is found in reference books such as Polymer Handbook.11 When the unit cell geometry is entirely unknown, indexing the Bragg reflections is a more involved process. Having a fiber diagram rather than a powder diagram alone is then useful, since the two-dimensional information about the x and y coordinates of 

This lanthanide contraction is associated with the filling of the 4f shell across the lanthanide series. The effect is mainly due to an incomplete shielding of the nuclear charge by the 4f electrons and yields a contraction of the radii of outer electron shells. 

FIGURE 1 Lattice constants of the elemental lanthanides (top), their chalcogenides (middle) (after Jayaraman, 1979), and pnictides (bottom). For the elemental lanthanides, it is the atomic sphere radius, 5, that is shown instead of the lattice parameter, where S is defined as V = 4/3tiS with V the unit cell volume. 

Pressure studies have been able to unravel a lot of the physics of the rare earths. Not only have pressure experiments seen changes of valence from divalent to trivalent, but also changes in the structural properties. In the case of Ce and Ce compounds, the valence changes under pressure from trivalent to tetravalent or from one localized f-state to a delocalized state have been observed. This will be discussed in greater detail in Section 4 of this chapter. 

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An indirect estimate of surface tension may be obtained from the change in lattice parameters of small crystals such as magnesium oxide and sodium chloride owing to surface tensional compression [121] however, these may represent nonequilibrium surface stress rather than surface tension [68]. Surface stresses may produce wrinkling in harder materials [122]. 

The FCC structure is illustrated in figure Al.3.2. Metallic elements such as calcium, nickel, and copper fonu in the FCC structure, as well as some of the inert gases. The conventional unit cell of the FCC structure is cubic with the lengdi of the edge given by the lattice parameter, a. There are four atoms in the conventional cell. In the primitive unit cell, there is only one atom. This atom coincides with the lattice pomts. The lattice vectors for the primitive cell are given by... 

Figure C 1.2.7. Superconducting transition temperature plotted as a function of the a lattice parameter for a variety of A Cgg phases [55].

The picture therefore remains obscure. The degree of localization may well depend on variable factors such as the purity of the surface (ultra high vacuum is now known to be essential), the temperature, and the magnitude of the lattice parameters relative to the (rather large) size of the xenon atom. 

Both sohd-solution hardening and precipitation hardening can be accounted for by internal strains generated by inserting either solute atoms or particles in an elastic matrix (11). The degree of elastic misfit, 5, produced by the difference, Ai , between the lattice parameter, of the pure matrix and a, the lattice parameter of the solute atom is given by... 

Whereas a linear relation between flow stress and lattice-parameter change is obeyed for any single solute element in nickel, the change in yield stress for various solutes in nickel is not a single-valued function of the lattice parameter, but depends directly on the position of the solute in the Periodic Table... 

Eig. 6. Plot of band gap energy vs lattice parameter for (a) common III—V materials employed for LEDs where (—) corresponds to direct and (—) to indirect band gaps. Both Al Gaj As and (Al Gaaj )q lattice matched to GaAs, whereas In Gaj As P can be matched to InP. (b)... 

III—V nitride compounds suitable for fabricating blue/uv emitters. Also shown is the lattice parameter of various materials proposed as substrates. 

Above the solution treatment temperature (ca 1250°C), the alloy is single phase with a bcc crystal stmcture. During cooling to ca 750—850°C, the sohd solution decomposes spinodally into two other bcc phases a and lattice parameter composition. The matrix a-phase is rich in Ni and Al and weakly magnetic as compared with which is rich in Fe and Co. The a -phase tends to be rod-like in the (100) dkection and ca 10 nm in diameter and ca 100 nm long. As the temperature is decreased, segregation of the elements becomes mote pronounced and the difference between the saturation polarizations of the two phases increases. 

AEotrope Stability Crystal stmcture Lattice parameters, Reference... 

In the 2incblende stmcture the bond length is related to the cubic lattice parameter as (3a/4). ... 

Crystals of the dihydrate belong to the monoclinic system and have lattice parameters a = 659 pm, b = 1020 pm, and c = 651 pm. The anhydrous crystal belongs to the cubic system, a = 596 pm. Other physical properties of the anhydrous salt are Hsted iu Table 1. The anhydrous salt is hygroscopic but not dehquescent. 

The most important of these is the diboride, TiB2, which has a hexagonal stmeture and lattice parameters of a = 302.8 pm and c = 322.8 pm. Titanium diboride is a gray crystalline soUd. It is not attacked by cold concentrated hydrochloric or sulfuric acids, but dissolves slowly at boiling temperatures. It dissolves mote readily in nitric acid/hydrogen peroxide or nitric acid/sulfuric acid mixtures. It also decomposes upon fusion with alkaU hydroxides, carbonates, or bisulfates. 

Compoun d Stmeture Lattice parameter, pm Density, kg/m Melting point, °C Electrical resistivity, 25°C, n-mx 1Q- Hardness, Mohs scale Microhardness, GPa... 

Titanium Monoxide. Titanium monoxide [12137-20-17, TiO, has a rock-salt stmcture but can exist with both oxygen and titanium vacancies. For stoichiometric TiO, the lattice parameter is 417 pm, but varies from ca 418 pm at 46 atom % to 4I62 pm at 54 atom % oxygen. Apparendy, stoichiometric TiO has ca 15% of the Ti and O sites vacant. At high temperatures (>900° C), these vacancies are randomly distributed at low temperatures, they become ordered. Titanium monoxide may be made by heating a stoichiometric mixture of titanium metal and titanium dioxide powders at 1600°C... 

Titanium Tetraiodide. Titanium tetraiodide [7720-83 ] forms reddish-brown crystals, cubic at room temperature, having reported lattice parameter of either 1200 (149) or 1221 (150) pm. Til melts at 150°C, boils at 377°C, and has a density of 440(0) kg/m. It forms adducts with a number of donor molecules and undergoes substitution reactions (151). It also hydrolyzes in water and is readily soluble in nonpolar organic solvents. 

Indexings and Lattice Parameter Determination. From a powder pattern of a single component it is possible to determine the indices of many reflections. From this information and the 20-values for the reflections, it is possible to determine the unit cell parameters. As with single crystals this information can then be used to identify the material by searching the NIST Crystal Data File (see "SmaU Molecule Single Stmcture Determination" above). 

The electronic stmcture of cobalt is [Ar] 3i/4A. At room temperature the crystalline stmcture of the a (or s) form, is close-packed hexagonal (cph) and lattice parameters are a = 0.2501 nm and c = 0.4066 nm. Above approximately 417°C, a face-centered cubic (fee) aHotrope, the y (or P) form, having a lattice parameter a = 0.3544 nm, becomes the stable crystalline form. The mechanism of the aHotropic transformation has been well described (5,10—12). Cobalt is magnetic up to 1123°C and at room temperature the magnetic moment is parallel to the ( -direction. Physical properties are Hsted in Table 2. 

Here ao is the lattice parameter of die ctystal. An approximate value for the bond energy, , for this structure where the co-ordination number, Z, equals twelve is given by... 

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