Experimental lattice

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).

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

As examples for our investigation we have chosen Fe/Cu/Fe bcc (001) and Co/Cu/Co fee (001) trilayers. A trilayer consists of two semi infinite crystals of Fe separated by a paramagnetic Cu spacer. The entire trilayer has the same crystal structure which means that all effects from lattice relaxations are excluded. The experimental lattice constant of bcc Fe was chosen for the Fe/Cu/Fe bcc trilayers, whereas for the Co/Cu/Co fee trilayers we used the lattice constant of fee Cu. 

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 systematic analysis of the electrostatic interactions in the crystals of 40 rigid organic molecules was undertaken by Price and coworkers (D. S. Coombes et al. 1996). In this work, distributed (i.e., local) multipoles up to hexadecapoles, obtained from SCF calculations with 6-31G basis sets, scaled by a factor of 0.9 to allow for the omission of electron correlation, are used in the evaluation of the electrostatic interactions. The experimental lattice constants and structures are reproduced successfully, the former to within a few percent of the experimental... 

Experimental lattice parameters for bulk kaolinite, together with those calculated in the static limit, are listed in Table 1. A difference in the length of parameter b of 2.9% is the largest discrepancy, which is reasonable since our calculation relates to an idealised clay structure. 

Table 1 Calculated and experimental lattice parameters for kaolinite...

Figure 13 compares the results of the calculations with Eq. (22), for the AnN and AnAs systems with the experimental lattice parameters and with the corresponding lanthanide systems. This latter comparison evidences, also, in these compounds, that the presence of a departure from a monotonous, almost linear curve, as found for lanthanides, is a clear sign of metallic 5 f bonding. 

Fig. 13. Lattice parameters of the actinide nitrides from LMTO (labelled Pauli pramagnetic), RLMTO (labelled Dirac) and LMTO spin polarized (labelled Pauli spin polarized) calculations. The black filled circles are the experimental lattice parameters...

Given the enthalpy of formation of an ionic solid, an experimental lattice energy can be obtained by thermochemical analysis. For example, the formation of crystalline sodium chloride is broken down as follows ... 

The atomisation enthalpy of elemental sodium Afl%tom, the first ionisation energy of atomic sodium Iu the dissociation enthalpy D of gaseous chlorine, the electron attachment energy Ex of atomic chlorine and the enthalpy of formation A//)1 of crystalline sodium chloride can all be taken from standard tabulations of experimental data. An experimental lattice energy UL is thus given by ... 

As a starting point, we take the experimental enthalpies of formation of MF4. From these, experimental lattice energies can be obtained. The thermochemical breakdown of the disproportionation is as follows ... 

Experimental data are available for all of these quantities except for the lattice energy of A1F. This we estimate as —950 kJ mol-1, which is consistent with the experimental lattice energies of NaF and MgF2 (-923 and -2957 kJ mol-1 respectively). It also leads to about equal ratios... 

Table 5.5 Experimental lattice energies for some oxides, comparing calculated (Kapustinskii) and experimental ratios for different oxidation states of elements (all lattice energies in kJ mol )...

Table V. Band energies of Si calculated at the experimental lattice constant and compared with experimental values in eV. The column headings are defined in the caption of Table I.

Experimental AHf MO crystal (kcal mole-1) Experimental lattice energy (kcal mole-1) —152 - 61 + 91... 

TABLE 5 shows the experimental lattice parameters for the following GaN samples ... 

It must be remembered that the lattice energy given by the Bom-Haber cycle is an experimental lattice energy and is not,dependent upon the nature of the assumptions made about the bonding in the crystal. The classical theoretical calculations are of course dependent upon the assumption of the ionic nature of the bonding in the lattice. Because of this the Bom-Haber cycle has been used mainly for three purposes. 

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