Lattice constant calculations

Sintered alloy films of reasonable thickness, e.g., opaque, mirrorlike films, can provide an adequate number of diffraction peaks for the determination of a lattice constant of adequate accuracy for present purposes. Thus, the apparent lattice constants calculated from the centroids of individual diffraction peaks, observed with a counter-diffractometer, may be extrapolated to 0 = 90°, using the Nelson-Riley function to give a value of a0. There has been some discussion about differences in lattice constants for thin films compared with bulk metals values of ao for pure silver films ( 1000 A nominal thickness) were found (74) to be consistently small compared with bulk silver but only by 0.05%. For alloy films a similar deviation would correspond to a variation of 1% in the composition of the alloy. Larger deviations have been reported for very thin films, e.g., —0.2% in copper films of 100 A nominal thickness (75).]... 

It has been observed by some experimenters, but not by the others, that the experimental lattice constant a in crystals of ordinary size was different from that, a + A a, found in extremely small crystals. A recent example72 refers to vacuum-deposited copper grains whose diameter D (they were, of course, not spherical) varied from 24 to 240 angstroms. The lattice constants calculated from the (111) reflexions increased from 3.577 to 3.6143 angstroms when the grain volume decreased, but the particle size had no definite effect on the reflexions from the 220 plane. 

A L6 A and Rietveld fit (solid line) of best fitting model the ticks mark theoretical peak positions for ice Ic (above) and Ih (below), with lattice constants calculated from those of the refined model... 

Lattice constants, calculated and measured densities at 23 °C for several samples of the NaCl-type phase Agi jSn,+,Se2 y. The measured values represent the average of six determinations for each composition. 

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Figure B3.2.11. Total energy versus lattice constant of gallium arsenide from a VMC calculation including 256 valence electrons [118] the curve is a quadratic fit. The error bars reflect the uncertainties of individual values. The experimental lattice constant is 10.68 au, the QMC result is 10.69 (+ 0.1) an (Figure by Professor W Schattke).

In this paper, the electronic structure of disordered Cu-Zn alloys are studied by calculations on models with Cu and Zn atoms distributed randomly on the sites of fee and bcc lattices. Concentrations of 10%, 25%, 50%, 75%, and 90% are used. The lattice spacings are the same for all the bcc models, 5.5 Bohr radii, and for all the fee models, 6.9 Bohr radii. With these lattice constants, the atomic volumes of the atoms are essentially the same in the two different crystal structures. Most of the bcc models contain 432 atoms and the fee models contain 500 atoms. These clusters are periodically reproduced to fill all space. Some of these calculations have been described previously. The test that is used to demonstrate that these clusters are large enough to be self-averaging is to repeat selected calculations with models that have the same concentration but a completely different arrangement of Cu and Zn atoms. We found differences that are quite small, and will be specified below in the discussions of specific properties. 

The functions Pcu(c,q) for the fee alloys can be averaged over the concentration c to obtain a probability Pcu(Q) that a Cu atom in any fee alloy will have a charge between q and q+dq. Recall that all of our calculations for fee were carried out with the same lattice constant. We approximate this function by averaging over the five concentrations that we considered, giving equal weight to all of them. It can be seen from the plot of this function in Fig. 5 that the probability is not uniform in q, but has thirteen prominent peeks. Since there are twelve atoms on the nearest-neighbor (nn) shell in a fee lattice, it is reasonable to write Pcu(q) as the sum of conditional probabilities Pcu(ci.q) where ci is the concentration of Cu atoms on the nn-shell. Five of the possible thirteen conditional probabilities are also plotted in... 

Table 2 shows that in the case of ratile the GGA overestimation of lattice constants is less important in the present calculation than in Ref. 3. Most likely explanation is that the GGA functional is used here only for solid state calculations and not for the pseudopotential generation from the free atom. This procedure has been shown to give more accurate structural results than with the GGA applied both in the potential generation and solid state... 

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 Figs.3 and 4, we show the concentration dependence of the lattice constants of 2D binary alloys. For the calculation of Fig.3, the LJ potential parameters =... 

Elastic constants calculated and the experimental lattice constant. Experimental data from Ref. [36]. 

Obviously, to discuss the possibility of such a phase transition one should at least carry out a fully self-consistent calculation for different lattice constants. 

To be consistent, the minimum energies from the LMTO-program were used even though this underestimated the lattice constant. Murnaghan s equation of state was used to determine bulk moduli and equilibrium volumes. The energy calculations were converged within 1 mRy/atom. 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

Table I. Experimental and calculated lattice constants a (in A), elastic constants, bulk and shear moduli (in units of 10 ) for the M3X (X = Mn, Al, Ga, Ge, Si) intermetallic series. Also listed are values of the anisotropy factor A and Poisson s ratio V. The experimental data for a are from Ref. . The experimental data for B, the elastic constants, A and v are taken from Ref. . The theoretical values for NiaSi are from Ref.. Also listed in the table are values of the polycrystalline elastic quantities-shear moduli G, Yoimg moduli (in units of and the ratio The experimental data for these quantities are from Ref. ...

We have carried out impurity calculations for a zinc atom embedded in a copper matrix. We first perform self consistent band theory calculations on pure Cu and Zn on fee lattices with the lattice constant of pure Cu, 6.76 Bohr radii. This yields Fermi energies, self consistent potentials, scattering matrices, and wave functions for both metals. The Green s function for a system with a Zn atom embedded in a Cu matrix... 

Similar calculations were carried out for the single impurity systems, niobium in Cu, vanadium in Cu, cobalt in Cu, titanium in Cu and nickel in Cu. In each of these systems the scattering parameters for the impurity atom (Nb, V, Co, Ti or Ni) were obtained from a self consistent calculation of pure Nb, pure V, pure Co, pure Ti or pure Ni respectively, each one of the impurities assumed on an fee lattice with the pure Cu lattice constant. The intersection between the calculated variation of Q(A) versus A (for each impurity system) with the one describing the charge Qi versus the shift SVi according to eqn.(l) estimates the charge flow from or towards the impurity cell.The results are presented in Table 2 and are compared with those from Ref.lc. A similar approach was also found succesful for the case of a substitutional Cu impurity in a Ni host as shown in Table 2. 

The valences of the rare-earth metals are calculated from their magnetic properties, as reported by Klemm and Bommer.14 It is from the fine work of these investigators that the lattice constants of the rare-earth metals have in the main been taken. The metals lutecium and ytterbium have only a very small paramagnetism, indicating a completed 4/ subshell and hence the valences 3 and 2, respectively (with not over 3% of trivalent ytterbium present in the metal). The observed paramagnetism of cerium at room temperature corresponds to about 20% Ce4+ and 80% Ce3+, that of praseodymium and that of neodymium to about 10% of the quadripositive ion in each case, and that of samarium to about 20% of the bipositive ion in equilibrium with the tripositive ion. 

In connection with a discussion of alloys of aluminum and zinc (Pauling, 1949) it was pointed out that an element present in very small quantity in solid solution in another element would have a tendency to assume the valence of the second element. The upper straight line in Fig. 2 is drawn between the value of the lattice constant for pure lead and that calculated for thallium with valence 2-14, equal to that of lead in the state of the pure element. It is seen that it passes through the experimental values of aQ for the alloys with 4-9 and 11-2 atomic percent thallium, thus supporting the suggestion that in these dilute alloys thallium has assumed the same valence as its solvent, lead. 

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