Determination of lattice constant

X-ray line-width measurements, while a Rich Seifert Debyeflex unit was used for the other powder patterns. A North American Philips back-reflection camera was employed for the precision determination of lattice constants. The uncertainty in the lattice constants varies between 0-001 and 0-0025 A. Anatase samples were heated in quartz tubes placed in a vertical tubular furnace fitted with a thermo-regulator. The fraction of anatase in a mixture was determined by the equation,... 

The methods by which the phenomenon of interconfiguration fluctuations may be studied are (i) determination of lattice constant, (ii) magnetic susceptibility measurements, (iii) Mossbauer spectroscopy, (iv) measurement of electrical resistivity, (v) Hall effect, (vi) X-ray absorption spectroscopy and (vii) X-ray photoelectron emission spectroscopy. It is useful to note that a suite of techniques must be used to detect ICF phenomenon in a system. Nuclear magnetic resonance is sparingly used because not all the systems exhibiting ICF contain magnetically active nuclei. 

Thus, although vacancies and interstitials exist in thermal equilibrium in ice, their concentrations are small even at the melting point and for this reason the density of ice crystals, measured by macroscopic means, agrees closely with the value derived from X-ray determinations of lattice constants. 

Boswell FWC (1951) Precise determination of lattice constants by electron diffraction and variations in the lattice constants of very small crystallites. Proc Phys Soc London A 64 465-475... 

At each temperature one can determine the equilibrium lattice constant aQ for the minimum of F. This leads to the thermal expansion of the alloy lattice. At equilibrium the probability f(.p,6=0) of finding an atom away from the reference lattice point is of a Gaussian shape, as shown in Fig. 1. In Fig.2, we present the temperature dependence of lattice constants of pure 2D square and FCC crystals, calculated by the present continuous displacement treatment of CVM. One can see in Fig.2 that the lattice expansion coefficient of 2D lattice is much larger than that of FCC lattice, with the use of the identical Lennard-Lones (LJ) potential. It is understood that the close packing makes thermal expansion smaller. 

In table 2 and 3 we present our results for the elastic constants and bulk moduli of the above metals and compare with experiment and first-principles calculations. The elastic constants are calculated by imposing an external strain on the crystal, relaxing any internal parameters (case of hep crystals) to obtain the energy as a function of the strain[8]. These calculations are also an output of onr TB approach, and especially for the hep materials, they would be very costly to be performed from first-principles. For the cubic materials the elastic constants are consistent with the LAPW values and are to within 1.5% of experiment. This is the accepted standard of comparison between first-principles calculations and experiment. An exception is Sr which has a very soft lattice and the accurate determination of elastic constants is problematic. For the hep materials our results are less accurate and specifically in Zr the is seriously underestimated. ... 

Some other points worth noting in connection with alloy film composition are The loss in weight from separate sources is a guide to mean composition but not an exact measure because the sources become themselves alloyed. It is often important to determine the composition of the actual specimen on which other characterizing measurements have been made. If there is confidence that the films are reasonably homogeneous, lattice constants determined by X-ray diffraction can be used to examine the uniformity of composition (69), but the change of lattice constant with composition may be inconveniently small. 

You now know how to define a supercell for a DFT calculation for a material with the simple cubic crystal structure. We also said at the outset that we assume for the purposes of this chapter that we have a DFT code that can give us the total energy of some collection of atoms. How can we use calculations of this type to determine the lattice constant of our simple cubic metal that would be observed in nature The sensible approach would be to calculate the total energy of our material as a function of the lattice constant, that is, tot(a). A typical result from doing this type of calculation is shown in Fig. 2.1. The details of how these calculations (and the other calculations described in the rest of the chapter) were done are listed in the Appendix at the end of the chapter. 

The calculations above allowed the positions of atoms to change within a supercell while holding the size and shape of the supercell constant. But in the calculations we introduced in Chapter 2, we varied the size of the supercell to determine the lattice constant of several bulk solids. Hopefully you can see that the numerical optimization methods that allow us to optimize atomic positions can also be extended to optimize the size of a supercell. We will not delve into the details of these calculations—you should read the documentation of the DFT package you are using to find out how to use your package to do these types of calculations accurately. Instead, we will give an example. In Chapter 2 we attempted to find the lattice constant of Cu in the hep crystal structure by doing individual calculations for many different values of the lattice parameters a and c (you should look back at Fig. 2.4). A much easier way to tackle this task is to create an initial supercell of hep Cu with plausible values of a and c and to optimize the supercell volume and shape to minimize... 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

Fig. 4.17 The total energy ol a cubic lattice of rigid anions and cations as a function of r+ with r fixed, for different coordination configurations. When the anions come into mutual contact as a result of decreasing r+ their repulsion determines the lattice constant and the cohesive energy becomes constant when expressed in terms or r. Thus near the values of r+/r at which anion-anion contact takes place, the radius ratio model predicta phase transitions to structures of successively lower coordination numbers. Note that the breaks" in the curves correspond to the values listed in Table 4.6. [From Treatise on Solid State Chemistry Hannay, N. B., Ed. Plenum New York, 1973.]...

Phonon vibration spectrum was determined from force constant k which was determined from dependence of the calculated molecule average energy on volume ( a3), i.e. from compressibility k d2Etot(a,T)lda2. The pressure in the system was determined conventionally as P(a,T) = -dF(a,T) / 8V One can determine the lattice constant a(T) for every value of (P,T) by numerical inversion of the dependence P a,T) => a(P,T) ... 

At every chosen values of P and T we determined the lattice constant a and hence the value of m(P, T). It was found this potential can have several minima at given pressure P and temperature T, therefore phase transitions of adsorbed hydrogen density m(P, T) are possible in this system. 

Indeed, lattice parameters of both the copper and the zinc oxide were found to depend on the catalyst composition. The lattice extension of copper was attributed to alpha brass formation upon partial reduction of zine oxide, and an attempt was made to correlate the lattice constant of copper with the decomposition rate of methanol to methyl formate. Furthermore, the decomposition rate of methanol to carbon monoxide was found to correlate with the changes of lattice constant of zinc oxide. Although such correlations did not establish the cause of the promotion in the absence of surface-area measurements and of correlations of specific activities, the changes of lattice parameters determined by Frolich et al. are real and indicate for the first time that the interaction of catalyst components can result in observable changes of bulk properties of the individual phases. Frolich et al. did not offer an interpretation of the observed changes in lattice parameters of zinc oxide. Yet these changes accompany the formation of an active catalyst, and much of this review will be devoted to the origin, physicochemical nature, and catalytic activity of the active phase in the zinc oxide-copper catalysts. 

Fig. 4.17 The total energy ol a cubic lattice of rigid anions and cations as a function of r+ with r. fixed, for different coordination configurations. When the anions come into mutual contact as a result of decreasing their repulsion determines the lattice constant and the...

Whereas within the family of the cubic Prussian blue analogs a large number of lattice constants have been determined, little attention has been devoted so far to polymeric cyanides not belonging to the cubic system. It must be emphasized, however, that polynuclear cyanides having unit cell symmetries other than cubic are by no means rare exceptions. Hexacyanometalates(III) of Zn2+ and Cd2+ are obtained not only in a cubic modification but also as samples with complicated and not yet resolved X-ray patterns of definitely lower symmetry than cubic (55). The exact conditions for obtaining either modification are not yet known in detail. The hexacyanoferrates(II), -ruthenates(II), and -osmates(II) of Mn2+ and several modifications of the corresponding Co 2+ salts show very complicated X-ray powder patterns which cannot be indexed in the cubic system (55). Preliminary spectroscopic studies show the presence of nearly octahedral M C6-units in these compounds, too. 

The accumulation of lattice constants gave rise to a growing Hbrary of interatomic (and interionic) distances, providing atomic and ionic radii. In 1929 Pauling published five principles (rules) that formed the first rational basis for understanding aystal structures. For example, the ratio of the ionic radii of cations to anions determines coordination number in crystals coordination number 6 for each chlorine and sodium ion in NaCl coordination number 8 for each ion in CsCl. 

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